Initial version as published at http://jcgt.org/published/0005/02/01/
authorMartin Lambers <marlam@marlam.de>
Mon, 26 Aug 2019 12:06:01 +0000 (14:06 +0200)
committerMartin Lambers <marlam@marlam.de>
Mon, 26 Aug 2019 12:06:01 +0000 (14:06 +0200)
34 files changed:
CMakeLists.txt [new file with mode: 0644]
LICENSE [new file with mode: 0644]
README [new file with mode: 0644]
disk-square-analyzer.cpp [new file with mode: 0644]
disk-square-common.hpp [new file with mode: 0644]
disk-square-plotter.cpp [new file with mode: 0644]
disk-square-precision-test.cpp [new file with mode: 0644]
libproj/CMakeLists.txt [new file with mode: 0644]
libproj/disk-square.cpp [new file with mode: 0644]
libproj/disk-square.hpp [new file with mode: 0644]
libproj/hemisphere-mix-lambert-stereographic.cpp [new file with mode: 0644]
libproj/hemisphere-mix-lambert-stereographic.hpp [new file with mode: 0644]
libproj/hemisphere-perspective.cpp [new file with mode: 0644]
libproj/hemisphere-perspective.hpp [new file with mode: 0644]
libproj/hemisphere-stereographic.cpp [new file with mode: 0644]
libproj/hemisphere-stereographic.hpp [new file with mode: 0644]
libproj/proj.cpp [new file with mode: 0644]
libproj/proj.hpp [new file with mode: 0644]
libproj/sphere-breusing-harmonic.cpp [new file with mode: 0644]
libproj/sphere-breusing-harmonic.hpp [new file with mode: 0644]
libproj/sphere-equidistant-azimuthal.cpp [new file with mode: 0644]
libproj/sphere-equidistant-azimuthal.hpp [new file with mode: 0644]
libproj/sphere-lambert-azimuthal-equal-area.cpp [new file with mode: 0644]
libproj/sphere-lambert-azimuthal-equal-area.hpp [new file with mode: 0644]
pdfplot.cpp [new file with mode: 0644]
pdfplot.hpp [new file with mode: 0644]
pngvis.cpp [new file with mode: 0644]
pngvis.hpp [new file with mode: 0644]
sphere-map-analyzer.cpp [new file with mode: 0644]
sphere-map-common.hpp [new file with mode: 0644]
sphere-map-plotter.cpp [new file with mode: 0644]
sphere-map-plotter.qrc [new file with mode: 0644]
sphere-map-precision-test.cpp [new file with mode: 0644]
tm-world-borders-0.3.csv [new file with mode: 0644]

diff --git a/CMakeLists.txt b/CMakeLists.txt
new file mode 100644 (file)
index 0000000..5cce59b
--- /dev/null
@@ -0,0 +1,53 @@
+# Copyright (C) 2015
+# Computer Graphics Group, University of Siegen
+# Written by Martin Lambers <martin.lambers@uni-siegen.de>
+#
+# Copying and distribution of this file, with or without modification, are
+# permitted in any medium without royalty provided the copyright notice and this
+# notice are preserved. This file is offered as-is, without any warranty.
+
+cmake_minimum_required(VERSION 2.8)
+set(CMAKE_AUTOMOC ON)
+set(CMAKE_INCLUDE_CURRENT_DIR ON)
+
+if(CMAKE_COMPILER_IS_GNUCXX)
+       set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -std=gnu++1y -Wall -Wextra")
+endif()
+
+# Required libraries
+find_package(Qt5Gui REQUIRED)
+
+# The projection library
+add_subdirectory(libproj)
+
+# The programs
+include_directories(${CMAKE_SOURCE_DIR}/libproj)
+
+add_executable(disk-square-precision-test disk-square-precision-test.cpp disk-square-common.hpp)
+target_link_libraries(disk-square-precision-test libproj)
+install(TARGETS disk-square-precision-test RUNTIME DESTINATION bin)
+
+add_executable(disk-square-analyzer disk-square-analyzer.cpp disk-square-common.hpp pngvis.hpp pngvis.cpp)
+target_link_libraries(disk-square-analyzer libproj Qt5::Gui)
+install(TARGETS disk-square-analyzer RUNTIME DESTINATION bin)
+
+add_executable(disk-square-plotter disk-square-plotter.cpp disk-square-common.hpp pdfplot.hpp pdfplot.cpp)
+target_link_libraries(disk-square-plotter libproj Qt5::Gui)
+install(TARGETS disk-square-plotter RUNTIME DESTINATION bin)
+
+add_executable(sphere-map-precision-test sphere-map-precision-test.cpp sphere-map-common.hpp)
+target_link_libraries(sphere-map-precision-test libproj)
+install(TARGETS sphere-map-precision-test RUNTIME DESTINATION bin)
+
+add_executable(sphere-map-analyzer sphere-map-analyzer.cpp sphere-map-common.hpp pngvis.hpp pngvis.cpp)
+target_link_libraries(sphere-map-analyzer libproj Qt5::Gui)
+install(TARGETS sphere-map-analyzer RUNTIME DESTINATION bin)
+
+qt5_add_resources(SPHERE_MAP_PLOTTER_RESOURCES sphere-map-plotter.qrc)
+add_executable(sphere-map-plotter sphere-map-plotter.cpp sphere-map-common.hpp
+       pdfplot.hpp pdfplot.cpp ${SPHERE_MAP_PLOTTER_RESOURCES})
+target_link_libraries(sphere-map-plotter libproj Qt5::Gui)
+install(TARGETS sphere-map-plotter RUNTIME DESTINATION bin)
+
+# The data
+#configure_file(land-polygons-z5.csv land-polygons-z5.csv COPYONLY)
diff --git a/LICENSE b/LICENSE
new file mode 100644 (file)
index 0000000..1f80ba9
--- /dev/null
+++ b/LICENSE
@@ -0,0 +1,23 @@
+The MIT License (MIT)
+
+Copyright (c) 2015 Martin Lambers, Computer Graphics Group,
+University of Siegen, Germany.
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+
diff --git a/README b/README
new file mode 100644 (file)
index 0000000..80f79ed
--- /dev/null
+++ b/README
@@ -0,0 +1,5 @@
+All mapping methods are implemented in the included libproj library.
+No external libraries are required.
+
+The analysis methods are grouped into disk-square methods and sphere-map
+methods. The pngvis and pdfplot modules require Qt.
diff --git a/disk-square-analyzer.cpp b/disk-square-analyzer.cpp
new file mode 100644 (file)
index 0000000..16e195a
--- /dev/null
@@ -0,0 +1,166 @@
+#include <cmath>
+#include <cstdio>
+#include <vector>
+#include <algorithm>
+
+#include "proj.hpp"
+
+#include "disk-square-common.hpp"
+#include "pngvis.hpp"
+
+
+using std::isfinite;
+using std::min;
+using std::max;
+using Projection::degrees;
+using Projection::radians;
+
+void derivatives(int method, double u, double v,
+        double& dx_dv, double& dx_du, double& dy_dv, double& dy_du, double delta = 1e-5)
+{
+    double mv = v - delta;
+    double pv = v + delta;
+    double mu = u - delta;
+    double pu = u + delta;
+
+    if (mv * mv + mu * mu > 1 || mv * mv + pu * pu > 1
+            || pv * pv + mu * mu > 1 || pv * pv + pu * pu > 1)
+        goto fail;
+
+    double x_mv_mu, x_mv_pu, x_pv_mu, x_pv_pu;
+    double y_mv_mu, y_mv_pu, y_pv_mu, y_pv_pu;
+
+    projections[method].disk_to_square(mu, mv, x_mv_mu, y_mv_mu);
+    projections[method].disk_to_square(pu, mv, x_mv_pu, y_mv_pu);
+    projections[method].disk_to_square(mu, pv, x_pv_mu, y_pv_mu);
+    projections[method].disk_to_square(pu, pv, x_pv_pu, y_pv_pu);
+
+    if (!isfinite(x_mv_mu) || !isfinite(x_mv_pu) || !isfinite(x_pv_mu) || !isfinite(x_pv_pu)
+            || !isfinite(y_mv_mu) || !isfinite(y_mv_pu) || !isfinite(y_pv_mu) || !isfinite(y_pv_pu))
+        goto fail;
+
+    dx_dv = (- x_pv_pu + x_mv_pu + x_mv_mu - x_pv_mu) / (4 * delta);
+    dx_du = (+ x_pv_pu + x_mv_pu - x_mv_mu - x_pv_mu) / (4 * delta);
+    dy_dv = (+ y_pv_pu - y_mv_pu - y_mv_mu + y_pv_mu) / (4 * delta);
+    dy_du = (- y_pv_pu - y_mv_pu + y_mv_mu + y_pv_mu) / (4 * delta);
+    return;
+
+fail:
+    dx_dv = dx_du = dy_dv = dy_du = NAN;
+}
+
+void analysis(int method, double u, double v,
+        double& h, double& k,
+        double& theta_prime,
+        double& a, double& b,
+        double& omega, double& s,
+        double delta = 1e-5)
+{
+    double dx_dv, dx_du, dy_dv, dy_du;
+
+    derivatives(method, u, v, dx_dv, dx_du, dy_dv, dy_du, delta);
+    if (!isfinite(dx_dv) || !isfinite(dx_du) || !isfinite(dy_dv) || !isfinite(dy_du)) {
+        h = k = theta_prime = a = b = omega = s = NAN;
+        return;
+    }
+
+    h = hypot(dx_dv, dy_dv);
+    k = hypot(dx_du, dy_du);
+    double sin_theta_prime = (dy_dv * dx_du - dx_dv * dy_du) / (h * k);
+    s = h * k * sin_theta_prime; // equivalent to s = a * b
+    theta_prime = asin(min(max(sin_theta_prime, -1.0), +1.0));
+    double a_prime = sqrt(max(h * h + k * k + 2 * s, 0.0));
+    double b_prime = sqrt(max(h * h + k * k - 2 * s, 0.0));
+    a = (a_prime + b_prime) / 2;
+    b = (a_prime - b_prime) / 2;
+    omega = 2 * asin(min(max(b_prime / a_prime, -1.0), +1.0));
+}
+
+void analyze(int method, int w, int h, double* dist_area, double* dist_isot)
+{
+    for (int iy = 0; iy < h; iy++) {
+        double y = ((iy + 0.5) / h) * 2 - 1;
+        for (int ix = 0; ix < w; ix++) {
+            double x = ((ix + 0.5) / h) * 2 - 1;
+            double u, v;
+            projections[method].square_to_disk(x, y, u, v);
+            if (std::isfinite(u) && std::isfinite(v)) {
+                double h, k, theta_prime, a, b, omega, s;
+                analysis(method, u, v, h, k, theta_prime, a, b, omega, s);
+                dist_area[iy * w + ix] = s / (4 / M_PI);
+                dist_isot[iy * w + ix] = a / b;
+            } else {
+                dist_area[iy * w + ix] = NAN;
+                dist_isot[iy * w + ix] = NAN;
+            }
+        }
+    }
+}
+
+void scan_array(const double* data, int size,
+        double& min, double& min_1percent,
+        double& max, double& max_1percent,
+        double& mean, double& median)
+{
+    std::vector<double> valid_data;
+    valid_data.reserve(size);
+    min = max = NAN;
+    mean = 0;
+    for (int i = 0; i < size; i++) {
+        if (std::isfinite(data[i])) {
+            if (!std::isfinite(min) || data[i] < min)
+                min = data[i];
+            if (!std::isfinite(max) || data[i] > max)
+                max = data[i];
+            mean += data[i];
+            valid_data.push_back(data[i]);
+        }
+    }
+    if (valid_data.size() == 0) {
+        mean = NAN;
+        min_1percent = NAN;
+        max_1percent = NAN;
+        median = NAN;
+    } else {
+        mean /= valid_data.size();
+        std::sort(valid_data.begin(), valid_data.end());
+        if (valid_data.size() % 2 == 0) {
+            median = (valid_data[valid_data.size() / 2 - 1]
+                    + valid_data[valid_data.size() / 2]) / 2;
+        } else {
+            median = valid_data[valid_data.size() / 2];
+        }
+        min_1percent = valid_data[valid_data.size() / 100];
+        max_1percent = valid_data[valid_data.size() - 1 - valid_data.size() / 100];
+    }
+}
+
+/*
+ * Main function.
+ */
+
+int main(void)
+{
+    const int w = 512;
+    const int h = 512;
+
+    double *dist_area = new double[w * h];
+    double *dist_isot = new double[w * h];
+    double min, max, mean;
+    double min_1percent, max_1percent, median;
+
+    // Loop over the projection methods
+    for (int method = 0; method < n_projections; method++) {
+        fprintf(stderr, "%s\n", projections[method].name);
+        analyze(method, w, h, dist_area, dist_isot);
+
+        scan_array(dist_area, w * h, min, min_1percent, max, max_1percent, mean, median);
+        fprintf(stderr, "  area dist min=%g min_1percent=%g max=%g max_1percent=%g mean=%g median=%g\n", min, min_1percent, max, max_1percent, mean, median);
+        png_vis(std::string(projections[method].name) + "-dist-area", w, h, dist_area, true, 2.5);
+
+        scan_array(dist_isot , w * h, min, min_1percent, max, max_1percent, mean, median);
+        fprintf(stderr, "  isot dist min=%g min_1percent=%g max=%g max_1percent=%g mean=%g median=%g\n", min, min_1percent, max, max_1percent, mean, median);
+        png_vis(std::string(projections[method].name) + "-dist-isot", w, h, dist_isot, true, 2.5);
+    }
+    return 0;
+}
diff --git a/disk-square-common.hpp b/disk-square-common.hpp
new file mode 100644 (file)
index 0000000..4afaf5b
--- /dev/null
@@ -0,0 +1,36 @@
+#ifndef COMMON_2D_HPP
+#define COMMON_2D_HPP
+
+#include "disk-square.hpp"
+
+
+static void disk_to_square_noop(double u, double v, double& x, double& y)
+{
+    x = u;
+    y = v;
+}
+
+static void square_to_disk_noop(double x, double y, double& u, double& v)
+{
+    u = x;
+    v = y;
+}
+
+typedef struct {
+    const char* name;
+    void (*disk_to_square)(double u, double v, double& x, double& y);
+    void (*square_to_disk)(double x, double y, double& u, double& v);
+} projection_t;
+
+projection_t projections[] = {
+    { "original",  disk_to_square_noop,                   square_to_disk_noop                   },
+    { "stretch",   Projection::disk_to_square_stretch,    Projection::square_to_disk_stretch    },
+    { "shirley",   Projection::disk_to_square_shirley,    Projection::square_to_disk_shirley    },
+    { "squircle",  Projection::disk_to_square_squircle,   Projection::square_to_disk_squircle   },
+    { "elliptic",  Projection::disk_to_square_elliptical, Projection::square_to_disk_elliptical },
+    { "conformal", Projection::disk_to_square_conformal,  Projection::square_to_disk_conformal  },
+};
+
+int n_projections = sizeof(projections) / sizeof(projections[0]);
+
+#endif
diff --git a/disk-square-plotter.cpp b/disk-square-plotter.cpp
new file mode 100644 (file)
index 0000000..4335ed4
--- /dev/null
@@ -0,0 +1,331 @@
+#include <vector>
+#include <cmath>
+
+#include <QCoreApplication>
+#include <QPainter>
+
+#include "disk-square-common.hpp"
+#include "pdfplot.hpp"
+
+/*
+ * Get polygon data
+ */
+
+// The teapot outlines as polygons were obtained by rendering a teapot into a
+// black/white bitmap and then running
+// $ potrace -b geojson -a 0 --flat teapot.pbm -o teapot.geojson
+const std::vector<std::vector<std::pair<double, double>>>& get_teapot_polygons()
+{
+    static std::vector<std::vector<std::pair<double, double>>> polygons;
+    if (polygons.size() == 0) {
+        static const float p0[] = {
+            413.5, 478.4, 409.5, 478.0, 400.0, 476.9, 390.5, 475.9, 386.5,
+            474.9, 382.5, 473.8, 377.8, 470.7, 373.1, 467.5, 373.0, 463.1,
+            373.0, 458.7, 375.2, 455.1, 377.4, 451.5, 385.0, 443.0, 392.5,
+            434.5, 394.9, 431.2, 397.2, 427.8, 398.1, 423.9, 399.0, 419.9,
+            397.3, 416.8, 395.5, 413.8, 389.9, 411.0, 384.4, 408.2, 378.4,
+            406.5, 372.5, 404.9, 348.5, 404.0, 324.5, 403.1, 314.0, 402.5,
+            303.5, 401.9, 286.0, 400.4, 268.5, 399.0, 257.0, 397.4, 245.5,
+            395.8, 237.5, 394.0, 229.5, 392.1, 226.0, 390.6, 222.5, 389.2,
+            220.0, 386.8, 217.4, 384.5, 214.1, 379.6, 210.8, 374.8, 204.9,
+            362.5, 199.1, 350.3, 155.3, 349.7, 111.5, 349.1, 95.0, 347.6,
+            78.6, 346.1, 68.0, 344.5, 57.5, 342.9, 47.0, 339.8, 36.5,
+            336.8, 28.5, 332.4, 20.5, 328.1, 15.8, 323.3, 11.1, 318.5, 8.0,
+            311.7, 5.0, 305.0, 3.9, 297.2, 2.8, 289.5, 3.4, 279.6, 4.1,
+            269.7, 6.1, 261.1, 8.1, 252.5, 12.1, 242.7, 16.1, 232.9, 21.7,
+            223.7, 27.2, 214.5, 34.1, 206.0, 40.9, 197.5, 49.7, 188.9,
+            58.5, 180.3, 67.0, 173.4, 75.5, 166.4, 86.7, 158.8, 97.9,
+            151.1, 111.7, 143.2, 125.5, 135.3, 133.7, 131.2, 141.8, 127.1,
+            143.5, 121.9, 145.2, 116.8, 149.3, 109.6, 153.4, 102.5, 159.1,
+            95.5, 164.8, 88.5, 172.6, 80.9, 180.5, 73.3, 195.3, 61.6,
+            210.1, 49.8, 212.4, 44.6, 214.7, 39.5, 218.7, 36.4, 222.6,
+            33.2, 230.0, 29.5, 237.3, 25.8, 247.6, 22.4, 257.9, 19.1,
+            272.7, 16.0, 287.5, 12.9, 300.2, 10.9, 312.9, 9.0, 327.2, 7.5,
+            341.5, 5.9, 365.0, 4.5, 388.5, 3.0, 421.5, 3.0, 454.5, 3.0,
+            478.0, 4.5, 501.5, 6.0, 515.5, 7.4, 529.5, 8.9, 540.0, 10.5,
+            550.5, 12.0, 566.0, 15.1, 581.5, 18.2, 592.0, 21.3, 602.5,
+            24.3, 611.0, 28.2, 619.5, 32.1, 624.6, 36.1, 629.8, 40.1,
+            631.8, 44.8, 633.9, 49.5, 637.2, 52.6, 640.5, 55.7, 651.2,
+            63.8, 661.8, 72.0, 670.0, 79.7, 678.2, 87.4, 683.9, 94.4,
+            689.7, 101.5, 694.0, 109.0, 698.4, 116.5, 701.1, 124.0, 703.7,
+            131.5, 709.3, 133.1, 714.9, 134.8, 732.5, 144.6, 750.2, 154.4,
+            762.2, 167.2, 774.1, 179.9, 781.9, 194.2, 789.7, 208.5, 795.3,
+            225.5, 800.9, 242.5, 803.5, 253.0, 806.1, 263.5, 811.6, 282.3,
+            817.2, 301.0, 824.0, 315.3, 830.9, 329.5, 841.3, 341.5, 851.7,
+            353.5, 866.1, 362.1, 880.5, 370.8, 889.1, 374.8, 897.7, 378.9,
+            898.9, 380.4, 900.2, 381.9, 897.5, 382.6, 894.9, 383.2, 865.0,
+            382.1, 835.2, 381.0, 823.8, 379.9, 812.5, 378.9, 806.1, 375.9,
+            799.7, 373.0, 791.2, 364.7, 782.7, 356.5, 777.5, 348.5, 772.3,
+            340.4, 768.2, 331.0, 764.1, 321.5, 759.0, 307.5, 753.9, 293.5,
+            748.2, 282.1, 742.5, 270.8, 736.0, 263.3, 729.5, 255.8, 721.1,
+            250.3, 712.6, 244.8, 703.2, 241.9, 693.8, 239.0, 693.1, 239.0,
+            692.4, 239.0, 690.0, 245.8, 687.7, 252.5, 682.0, 267.5, 676.2,
+            282.5, 669.2, 298.5, 662.1, 314.5, 652.7, 334.5, 643.3, 354.5,
+            637.0, 367.2, 630.6, 380.0, 625.9, 384.8, 621.1, 389.7, 614.3,
+            391.9, 607.5, 394.0, 596.0, 395.9, 584.5, 397.9, 568.5, 399.4,
+            552.5, 400.9, 536.0, 402.0, 519.5, 403.1, 495.5, 404.0, 471.5,
+            404.8, 465.0, 406.8, 458.4, 408.7, 454.3, 410.8, 450.2, 413.0,
+            448.0, 416.0, 445.8, 419.1, 446.4, 423.3, 447.1, 427.4, 450.9,
+            432.5, 454.7, 437.5, 460.9, 444.3, 467.0, 451.0, 469.5, 455.0,
+            472.0, 459.0, 472.0, 462.9, 472.0, 466.8, 468.5, 469.4, 465.0,
+            472.1, 461.5, 473.6, 457.9, 475.1, 447.7, 476.5, 437.5, 477.9,
+            427.5, 478.3, 417.5, 478.8, 413.5, 478.4 ,
+        };
+        static const float p1[] = {
+            184.0, 318.7, 184.0, 318.4, 178.1, 305.5, 172.3, 292.5, 165.2,
+            275.0, 158.2, 257.5, 153.2, 242.5, 148.2, 227.5, 145.0, 214.5,
+            141.8, 201.5, 139.9, 190.0, 138.0, 178.5, 138.0, 173.2, 138.0,
+            168.0, 137.5, 168.0, 136.9, 168.0, 132.7, 170.1, 128.5, 172.2,
+            119.0, 178.8, 109.5, 185.3, 102.7, 191.3, 95.9, 197.3, 89.4,
+            204.0, 82.9, 210.7, 77.1, 218.6, 71.2, 226.5, 66.1, 235.4,
+            61.0, 244.3, 57.9, 251.8, 54.8, 259.4, 52.3, 268.9, 49.9,
+            278.5, 50.2, 287.1, 50.6, 295.8, 53.3, 299.4, 56.1, 302.9,
+            60.3, 305.8, 64.5, 308.6, 70.5, 310.8, 76.5, 312.9, 84.5,
+            314.4, 92.5, 315.9, 107.0, 317.0, 121.5, 318.1, 130.0, 318.3,
+            138.5, 318.6, 161.2, 318.8, 184.0, 319.0, 184.0, 318.7
+        };
+        static const float* p[] = { p0, p1 };
+        static const size_t n[] = { sizeof(p0) / sizeof(float), sizeof(p1) / sizeof(float) };
+        for (size_t i = 0; i < sizeof(p) / sizeof(float*); i++) {
+            std::vector<std::pair<double, double>> polygon;
+            for (size_t j = 0; j < n[i] / 2; j++) {
+                float x = p[i][2 * j];
+                float y = p[i][2 * j + 1];
+                x = ((x - 450) / 500);
+                y = ((y - 250) / 500);
+                polygon.push_back(std::pair<double, double>(x, y));
+            }
+            polygons.push_back(polygon);
+        }
+    }
+    return polygons;
+}
+
+/*
+ * Main function.
+ */
+
+int main(int argc, char* argv[])
+{
+    QCoreApplication app(argc, argv);
+    QPdfWriter* writer;
+    QPainter* painter;
+    QPen pen;
+    QBrush brush;
+
+    const std::vector<std::vector<std::pair<double, double>>>& polygons = get_teapot_polygons();
+
+    const int r_lines = 12;
+    const int phi_lines = 24;
+    const int x_lines = 18;
+    const int y_lines = 18;
+    const int steps = 1000;
+    const double map_sub_point_dist = 1e-3;
+    std::vector<std::pair<float, float>> point_list;
+    double r, phi, u, v, x, y;
+
+    pen.setCapStyle(Qt::FlatCap);
+    pen.setStyle(Qt::SolidLine);
+    pen.setWidth(4);
+    brush.setColor(QColor(224, 224, 224));
+    brush.setStyle(Qt::NoBrush);
+
+    for (int method = 0; method < n_projections; method++) {
+
+        /* Disk -> Square */
+
+        std::string disk_square_id = std::string("plot-disk-to-square-") + projections[method].name;
+        fprintf(stderr, "%s\n", disk_square_id.c_str());
+        make_pdf_painter(disk_square_id, false, &writer, &painter);
+
+        // Polygons
+        pen.setStyle(Qt::NoPen);
+        painter->setPen(pen);
+        brush.setStyle(Qt::SolidPattern);
+        painter->setBrush(brush);
+        for (size_t i = 0; i < polygons.size(); i++) {
+            const std::vector<std::pair<double, double>>& polygon = polygons[i];
+            point_list.clear();
+            if (i % 2 == 0)
+                brush.setColor(QColor(128, 128, 128));
+            else
+                brush.setColor(QColor(255, 255, 255));
+            painter->setBrush(brush);
+            // first valid point
+            u = polygon[0].first;
+            v = polygon[0].second;
+            projections[method].disk_to_square(u, v, x, y);
+            point_list.push_back(std::pair<float, float>(x, y));
+            double last_u = u;
+            double last_v = v;
+            double last_x = x;
+            double last_y = y;
+            for (size_t j = 1; j <= polygon.size(); j++) {
+                size_t jj = (j < polygon.size() ? j : 0);
+                u = polygon[jj].first;
+                v = polygon[jj].second;
+                projections[method].disk_to_square(u, v, x, y);
+                double dist = hypot(x - last_x, y - last_y);
+                if (dist < map_sub_point_dist)
+                    continue;
+                int substeps = steps / 2 * dist;
+                for (int k = 1; k < substeps; k++) {
+                    double alpha = static_cast<double>(k) / substeps;
+                    double subu = alpha * u + (1.0 - alpha) * last_u;
+                    double subv = alpha * v + (1.0 - alpha) * last_v;
+                    double subx, suby;
+                    projections[method].disk_to_square(subu, subv, subx, suby);
+                    point_list.push_back(std::pair<float, float>(subx, suby));
+                }
+                if (jj != 0)
+                    point_list.push_back(std::pair<float, float>(x, y));
+                last_u = u;
+                last_v = v;
+                last_x = x;
+                last_y = y;
+            }
+            plot_points(painter, false, point_list);
+        }
+
+        pen.setStyle(Qt::SolidLine);
+        painter->setPen(pen);
+        brush.setStyle(Qt::NoBrush);
+        painter->setBrush(brush);
+
+        // Lines of constant radius
+        for (int i = 0; i < r_lines; i++) {
+            point_list.clear();
+            r = (i + 1.0) / r_lines;
+            for (int j = 0; j <= steps; j++) {
+                phi = -M_PI + j * 2 * M_PI / steps;
+                u = r * cos(phi);
+                v = r * sin(phi);
+                projections[method].disk_to_square(u, v, x, y);
+                if (point_list.size() > 0) {
+                    float dist = hypot(point_list.back().first - x, point_list.back().second - y);
+                    if (dist < map_sub_point_dist)
+                        continue;
+                }
+                point_list.push_back(std::pair<float, float>(x, y));
+            }
+            plot_points(painter, true, point_list);
+        }
+
+        // Lines of constant phi
+        for (int i = 0; i < phi_lines; i++) {
+            point_list.clear();
+            phi = -M_PI + i * 2 * M_PI / phi_lines;
+            for (int j = 0; j <= steps; j++) {
+                r = static_cast<double>(j) / steps;
+                u = r * cos(phi);
+                v = r * sin(phi);
+                projections[method].disk_to_square(u, v, x, y);
+                if (point_list.size() > 0) {
+                    float dist = hypot(point_list.back().first - x, point_list.back().second - y);
+                    if (dist < map_sub_point_dist)
+                        continue;
+                }
+                point_list.push_back(std::pair<float, float>(x, y));
+            }
+            plot_points(painter, true, point_list);
+        }
+        finish_pdf_painter(writer, painter);
+
+        /* Square -> Disk */
+
+        std::string square_disk_id = std::string("plot-square-to-disk-") + projections[method].name;
+        fprintf(stderr, "%s\n", square_disk_id.c_str());
+        make_pdf_painter(square_disk_id, false, &writer, &painter);
+
+        // Polygons
+        pen.setStyle(Qt::NoPen);
+        painter->setPen(pen);
+        brush.setStyle(Qt::SolidPattern);
+        painter->setBrush(brush);
+        for (size_t i = 0; i < polygons.size(); i++) {
+            const std::vector<std::pair<double, double>>& polygon = polygons[i];
+            point_list.clear();
+            if (i % 2 == 0)
+                brush.setColor(QColor(128, 128, 128));
+            else
+                brush.setColor(QColor(255, 255, 255));
+            painter->setBrush(brush);
+            // first valid point
+            x = polygon[0].first;
+            y = polygon[0].second;
+            projections[method].square_to_disk(x, y, u, v);
+            point_list.push_back(std::pair<float, float>(u, v));
+            double last_x = x;
+            double last_y = y;
+            double last_u = u;
+            double last_v = v;
+            for (size_t j = 1; j <= polygon.size(); j++) {
+                size_t jj = (j < polygon.size() ? j : 0);
+                x = polygon[jj].first;
+                y = polygon[jj].second;
+                projections[method].square_to_disk(x, y, u, v);
+                double dist = hypot(u - last_u, v - last_v);
+                if (dist < map_sub_point_dist)
+                    continue;
+                int substeps = steps / 2 * dist;
+                for (int k = 1; k < substeps; k++) {
+                    double alpha = static_cast<double>(k) / substeps;
+                    double subx = alpha * x + (1.0 - alpha) * last_x;
+                    double suby = alpha * y + (1.0 - alpha) * last_y;
+                    double subu, subv;
+                    projections[method].square_to_disk(subx, suby, subu, subv);
+                    point_list.push_back(std::pair<float, float>(subu, subv));
+                }
+                if (jj != 0)
+                    point_list.push_back(std::pair<float, float>(u, v));
+                last_x = x;
+                last_y = y;
+                last_u = u;
+                last_v = v;
+            }
+            plot_points(painter, false, point_list);
+        }
+
+        pen.setStyle(Qt::SolidLine);
+        painter->setPen(pen);
+        brush.setStyle(Qt::NoBrush);
+        painter->setBrush(brush);
+
+        // Lines of constant x
+        for (int i = 0; i <= x_lines; i++) {
+            point_list.clear();
+            x = -1 + 2.0 * i / x_lines;
+            for (int j = 0; j <= steps; j++) {
+                y = -1 + 2.0 * j / steps;
+                projections[method].square_to_disk(x, y, u, v);
+                if (point_list.size() > 0) {
+                    float dist = hypot(point_list.back().first - u, point_list.back().second - v);
+                    if (dist < map_sub_point_dist)
+                        continue;
+                }
+                point_list.push_back(std::pair<float, float>(u, v));
+            }
+            plot_points(painter, true, point_list);
+        }
+
+        // Lines of constant y
+        for (int i = 0; i <= y_lines; i++) {
+            point_list.clear();
+            y = -1 + 2.0 * i / y_lines;
+            for (int j = 0; j <= steps; j++) {
+                x = -1 + 2.0 * j / steps;
+                projections[method].square_to_disk(x, y, u, v);
+                if (point_list.size() > 0) {
+                    float dist = hypot(point_list.back().first - u, point_list.back().second - v);
+                    if (dist < map_sub_point_dist)
+                        continue;
+                }
+                point_list.push_back(std::pair<float, float>(u, v));
+            }
+            plot_points(painter, true, point_list);
+        }
+        finish_pdf_painter(writer, painter);
+    }
+
+    return 0;
+}
diff --git a/disk-square-precision-test.cpp b/disk-square-precision-test.cpp
new file mode 100644 (file)
index 0000000..d0b1298
--- /dev/null
@@ -0,0 +1,54 @@
+#include <cmath>
+#include <cstdio>
+
+#include "disk-square-common.hpp"
+
+
+double check_precision(int method)
+{
+    const int xsteps = 2000;
+    const int ysteps = 2000;
+
+    double max_error = -1.0;
+    for (int iy = 0; iy < ysteps; iy++) {
+        double y = (iy / (ysteps - 1.0)) * 2.0 - 1.0;
+        for (int ix = 0; ix < xsteps; ix++) {
+            double x = (ix / (xsteps - 1.0)) * 2.0 - 1.0;
+            double u, v, fx, fy, fu, fv;
+            projections[method].square_to_disk(x, y, u, v);
+            projections[method].disk_to_square(u, v, fx, fy);
+            projections[method].square_to_disk(fx, fy, fu, fv);
+            double error = hypot(u - fu, v - fv);
+            if (!std::isfinite(u) || !std::isfinite(v)
+                    || !std::isfinite(fx) || !std::isfinite(fy)
+                    || !std::isfinite(fu) || !std::isfinite(fv)) {
+                fprintf(stderr, "projection failure "
+                        "ix=%d iy=%d x=%g y=%g u=%g v=%g fx=%g fy=%g fu=%g fv=%g\n",
+                        ix, iy, x, y, u, v, fx, fy, fu, fv);
+            }
+            if (error > max_error) {
+                max_error = error;
+            }
+        }
+    }
+    return max_error;
+}
+
+double in_earth_millimeters(double error)
+{
+    return error * 40075.017 * 1e3 * 1e3;
+}
+
+/*
+ * Main function.
+ */
+
+int main(void)
+{
+    // Loop over the projection methods
+    for (int method = 0; method < n_projections; method++) {
+        double max_error = check_precision(method);
+        fprintf(stderr, "%s: %g (ca. %g mm on Earth equatorial disk)\n", projections[method].name, max_error, in_earth_millimeters(max_error));
+    }
+    return 0;
+}
diff --git a/libproj/CMakeLists.txt b/libproj/CMakeLists.txt
new file mode 100644 (file)
index 0000000..f56de33
--- /dev/null
@@ -0,0 +1,31 @@
+# Copyright (C) 2015
+#
+# Copying and distribution of this file, with or without modification, are
+# permitted in any medium without royalty provided the copyright notice and this
+# notice are preserved. This file is offered as-is, without any warranty.
+
+cmake_minimum_required(VERSION 2.8)
+
+if(CMAKE_COMPILER_IS_GNUCXX)
+       set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -std=gnu++1y -Wall -Wextra")
+endif()
+
+#find_package(GSL REQUIRED)
+#find_package(Boost REQUIRED COMPONENTS math_tr1)
+
+#include_directories(${CMAKE_SOURCE_DIR} ${GSL_INCLUDE_DIRS})
+#include_directories(${CMAKE_SOURCE_DIR} ${Boost_INCLUDE_DIRS})
+add_library(libproj STATIC
+       proj.cpp proj.hpp
+        disk-square.cpp disk-square.hpp
+       sphere-lambert-azimuthal-equal-area.cpp sphere-lambert-azimuthal-equal-area.hpp
+       sphere-breusing-harmonic.cpp sphere-breusing-harmonic.hpp
+       sphere-equidistant-azimuthal.cpp sphere-equidistant-azimuthal.hpp
+       hemisphere-stereographic.cpp hemisphere-stereographic.hpp
+       hemisphere-mix-lambert-stereographic.cpp hemisphere-mix-lambert-stereographic.hpp
+        hemisphere-perspective.cpp hemisphere-perspective.hpp)
+#target_link_libraries(libproj ${GSL_LIBRARIES})
+#target_link_libraries(libproj ${Boost_LIBRARIES})
+#target_link_libraries(libproj -lquadmath)
+set_target_properties(libproj PROPERTIES OUTPUT_NAME proj)
+install(TARGETS libproj RUNTIME DESTINATION bin LIBRARY DESTINATION "lib" ARCHIVE DESTINATION "lib")
diff --git a/libproj/disk-square.cpp b/libproj/disk-square.cpp
new file mode 100644 (file)
index 0000000..a6d71d2
--- /dev/null
@@ -0,0 +1,419 @@
+#include <algorithm>
+#include <cmath>
+
+#include "disk-square.hpp"
+
+// Define exactly one of the following to 1 to choose which numerical methods to use
+// for the conformal mapping.
+// The implementation of Stark's method still has bugs, see comments there
+#define WITH_MYOWN 1
+#define WITH_STARK 0
+
+#if WITH_MYOWN
+# include <cfloat>
+#elif WITH_STARK
+# include <complex>
+  using std::complex;
+  using std::polar;
+#endif
+
+namespace Projection {
+
+using std::sin;
+using std::asin;
+using std::cos;
+using std::acos;
+using std::tan;
+using std::atan;
+using std::atan2;
+using std::fabs;
+using std::hypot;
+using std::sqrt;
+using std::isfinite;
+using std::min;
+using std::max;
+using std::floor;
+using std::ceil;
+using std::hypot;
+using std::log;
+using std::exp;
+
+constexpr double sign(double x) { return (x < 0 ? -1 : x > 0 ? +1 : 0); }
+constexpr double sign_not_zero(double x) { return (x < 0 ? -1 : +1); }
+
+constexpr double degrees(double x) { return 180 / M_PI * x; }
+constexpr double radians(double x) { return M_PI / 180 * x; }
+
+
+/* Radius Stretching */
+
+void disk_to_square_stretch(double u, double v, double& x, double& y)
+{
+    double r = hypot(u, v);
+    if (r <= 0.0) {
+        x = 0.0;
+        y = 0.0;
+    } else {
+        x = sign(u) * r;
+        y = sign(v) * r;
+        if (u * u >= v * v)
+            y *= v / u;
+        else
+            x *= u / v;
+    }
+}
+
+void square_to_disk_stretch(double x, double y, double& u, double& v)
+{
+    double t = hypot(x, y);
+    if (t <= 0.0) {
+        u = 0.0;
+        v = 0.0;
+    } else {
+        double c = (x * x >= y * y ? x : y);
+        u = sign(x) * c * x / t;
+        v = sign(y) * c * y / t;
+    }
+}
+
+/* Shirley's equal-area method, described in Shirley, P., & Chiu, K. (1997).
+ * A low distortion map between disk and square. Journal of graphics tools,
+ * 2(3), 45-52.
+ *
+ * Note that this is equivalent to the independently derived mapping described
+ * in Roşca, D. "New uniform grids on the sphere." Astronomy & Astrophysics
+ * 520 (2010): A63.
+ *
+ * For the square-to-disk functino, we use a trick to simplify the original
+ * equations, first introduced by Dave Cline here:
+ * http://psgraphics.blogspot.de/2011/01/improved-code-for-concentric-map.html
+ */
+
+void disk_to_square_shirley(double u, double v, double& x, double& y)
+{
+    double r = hypot(u, v);
+    double phi = atan2(v, u);
+    if (phi < -M_PI_4)
+        phi += 2 * M_PI;
+    if (phi < M_PI_4) {
+        x = r;
+        y = phi * x / M_PI_4;
+    } else if (phi < M_PI_2 + M_PI_4) {
+        y = r;
+        x = -(phi - M_PI_2) * y / M_PI_4;
+    } else if (phi < M_PI + M_PI_4) {
+        x = -r;
+        y = (phi - M_PI) * x / M_PI_4;
+    } else {
+        y = -r;
+        x = -(phi - (M_PI + M_PI_2)) * y / M_PI_4;
+    }
+}
+
+void square_to_disk_shirley(double x, double y, double& u, double& v)
+{
+    double r, phi;
+    if (x * x > y * y) {
+        r = x;
+        phi = M_PI_4 * y / x;
+    } else {
+        r = y;
+        if (fabs(y) > 0.0)
+            phi = M_PI_2 - M_PI_4 * x / y;
+        else
+            phi = 0.0;
+    }
+    u = r * cos(phi);
+    v = r * sin(phi);
+}
+
+/* Method using Guasti's Squircle: map concentric circles to concentric
+ * squircles that become more square-like with growing size.
+ * Details of the mapping are described in Fong, C. (2014). An Indoor
+ * Alternative to Stereographic Spherical Panoramas. In Proceedings of
+ * Bridges 2014: Mathematics, Music, Art, Architecture, Culture
+ * (pp. 103-110).
+ *
+ * This mapping is neither conformal nor equal-area. */
+
+void disk_to_square_squircle(double u, double v, double& x, double& y)
+{
+    double u2 = u * u;
+    double v2 = v * v;
+    double w = sign(u * v) / M_SQRT2 * sqrt(max(u2 + v2 - sqrt((u2 + v2) * (u2 + v2 - 4 * u2 * v2)), 0.0));
+    if (fabs(w) <= 0) {
+        x = u;
+        y = v;
+    } else {
+        x = w / v;
+        y = w / u;
+    }
+}
+
+void square_to_disk_squircle(double x, double y, double& u, double& v)
+{
+    double x2 = x * x;
+    double y2 = y * y;
+    double t = x2 + y2;
+    if (t <= 0) {
+        u = v = 0;
+    } else {
+        double r = sqrt(max(t - x2 * y2, 0.0));
+        double q = r / sqrt(t);
+        u = x * q;
+        v = y * q;
+    }
+}
+
+/* Method using elliptical arc mapping.
+ * This mapping is neither conformal nor equal-area. */
+
+void disk_to_square_elliptical(double u, double v, double& x, double& y)
+{
+    x =   0.5 * sqrt(max(2 + u * u - v * v + u * 2 * M_SQRT2, 0.0))
+        - 0.5 * sqrt(max(2 + u * u - v * v - u * 2 * M_SQRT2, 0.0));
+    y =   0.5 * sqrt(max(2 - u * u + v * v + v * 2 * M_SQRT2, 0.0))
+        - 0.5 * sqrt(max(2 - u * u + v * v - v * 2 * M_SQRT2, 0.0));
+}
+
+void square_to_disk_elliptical(double x, double y, double& u, double& v)
+{
+    u = x * sqrt(1 - y * y / 2);
+    v = y * sqrt(1 - x * x / 2);
+}
+
+/* Conformal mapping based on Schwarz-Christoffel transformation. */
+
+static const double K = 1.854074677301371918433850347195260046217598823521766905586;
+
+#if WITH_STARK
+
+/* This optimized implementation is based on Stark, M. M. (2009). Fast and
+ * Stable Conformal Mapping Between a Disc and a Square. Journal of Graphics,
+ * GPU, and Game Tools, 14(2), 1-23.
+ *
+ * XXX: this has errors in the disk-to-square function that I was unable to find!
+ * Regions 2 and 3 are affected.
+ */
+
+static complex<double> stark_sum_series(complex<double> z, int n)
+{
+    complex<double> s = z;
+    complex<double> p = z;
+    for (int k = 1; k <= n; k++) {
+        p = p * z * z * z * z * (2 * k - 1.0) / (2.0 * k);
+        s = s + p / (4 * k + 1.0);
+    }
+    return s;
+}
+
+static double stark_F(double t)
+{
+    const int n = 4; // use n=3 for single precision
+    double tt = t * t;
+    double ttt = tt * t;
+    double F0 = t - 0.193 * ttt + 0.026 * ttt * tt;
+    double a = 1;
+    double b = M_SQRT1_2;
+    for (int k = 1; k <= n; k++) {
+        t = (a + b) * t / (a - b * tt);
+        a = (a + b) / 2;
+        b = sqrt(a * b);
+    }
+    double phi_n = atan(t);
+    return phi_n + M_PI * floor(0.5 + ((1 << n) * a * F0 - phi_n) / M_PI) / ((1 << n) * a);
+}
+
+void disk_to_square_conformal(double u, double v, double& x, double& y)
+{
+    double r = hypot(u, v);
+    double phi = atan2(v, u);
+    if (phi <= -M_PI)
+        phi = M_PI;
+
+    const double r0 = 0.6436; // sqrt(sqrt(2.0) - 1.0) + epsilon
+    complex<double> z, w;
+    if (r < r0) {
+        z = polar(r, phi + M_PI_4);
+        w = 2.0 * sqrt(complex<double>(0, -1)) / K * stark_sum_series(z, floor(6.5 / (1 - r) - 2));
+    } else {
+        double Q = floor(phi / M_PI_2) + 0.5;
+        z = polar(r, phi - Q * M_PI_2);
+        complex<double> xi = sqrt((1.0 - z * z) / (1.0 + z * z));
+        if (abs(xi) < r0) {
+            w = K / M_SQRT2 - stark_sum_series(xi, floor(6.5 / (1 - abs(xi)) - 2));
+        } else {
+//fprintf(stderr, "X4: Q=%g r=%g phi=%g |z|=%g re(z)=%g im(z)=%g\n", Q, r, phi * RAD_TO_DEG, cabs(z), creal(z), creal(z));
+            double A = hypot(z.real() + 1, z.imag());
+            double B = hypot(z.real() - 1, z.imag());
+//fprintf(stderr, "    A=%g B=%g\n", A, B);
+            double alpha = (A + B) / 2;
+            double beta = (A - B) / 2;
+//fprintf(stderr, "    alpha=%g beta=%g\n", alpha, beta);
+            double gamma = alpha + sqrt(alpha * alpha - 1);
+            double Tphi = (1 - beta * beta) / (beta * beta);
+            double b = 2 / Tphi + (gamma - 1 / gamma) * (gamma - 1 / gamma) / (4 * (1 - beta * beta)) - 1;
+//fprintf(stderr, "    gamma=%g Tphi=%g b=%g\n", gamma, Tphi, b);
+            double Clambda = (b + sqrt(b * b + 8 / Tphi)) / 4;
+            double Tmu = 2 * (Tphi * Clambda - 1);
+//fprintf(stderr, "    Clambda=%g Tmu=%g\n", Clambda, Tmu);
+            w = complex<double>(K - stark_F(sqrt(1 / Clambda)), -stark_F(sqrt(Tmu)));
+            if (z.imag() > 0)
+                w = conj(w);
+//fprintf(stderr, "    |w|=%g re(w)=%g im(w)=%g\n", cabs(w), creal(w), cimag(w));
+        }
+        //w = polar(abs(w), arg(w) + Q * M_PI_2) * (2 / K);
+        w *= polar(1.0, Q * M_PI_2) * (2 / K);
+//fprintf(stderr, "    |w|=%g re(w)=%g im(w)=%g\n", cabs(w), creal(w), cimag(w));
+    }
+    x = w.real();
+    y = w.imag();
+}
+
+static complex<double> stark_inv_int(complex<double> w)
+{
+    const double C0 = 13.750371636040937;
+    const double c1 =  -1.6360491363469976e-1;
+    const double c5 =  -1.5316508620083077e-3;
+    const double c9 =  +5.9455890307966153e-7;
+    const double c13 = +1.7520282395125552e-8;
+    const double c17 = +2.8997255626121623e-11;
+    const double c21 = -1.4786423004927015e-13;
+    const double c25 = -6.1034023099548599e-16;
+    const double c29 = +5.3527850055041484e-19;
+    const double c33 = +7.9773298274614004e-21;
+    const double c37 = +1.1683926152311516e-23;
+    const complex<double> w2 = w * w;
+    const complex<double> w4 = w2 * w2;
+    const complex<double> w8 = w4 * w4;
+    const complex<double> w12 = w8 * w4;
+    const complex<double> w16 = w8 * w8;
+    const complex<double> w20 = w16 * w4;
+    const complex<double> w24 = w16 * w8;
+    const complex<double> w28 = w16 * w12;
+    const complex<double> w32 = w16 * w16;
+    const complex<double> w36 = w32 * w4;
+    return C0 * w / (K * K * K * K + w4)
+        + w * (c1 + c5 * w4 + c9 * w8 + c13 * w12 + c17 * w16
+                + c21 * w20 + c25 * w24 + c29 * w28 + c33 * w32 + c37 * w36);
+}
+
+void square_to_disk_conformal(double x, double y, double& u, double& v)
+{
+    complex<double> w = complex<double>(x, y);
+    complex<double> z = 1.0 / sqrt(complex<double>(0, 1))
+        * stark_inv_int((K * w) / (2.0 * sqrt(complex<double>(0, -1))));
+    u = z.real();
+    v = z.imag();
+}
+
+#else
+
+static const long long myown_max_iter = 63;
+// compute jacobi elliptic functions sn, cn, dn for fixed modulus k=1/sqrt(2)
+// via Arithmetic-Geometric Mean (AGM), Abramowitz & Stegun 16.4
+static void myown_jacobi_sn_cn_dn(double u, double* sn, double* cn, double* dn)
+{
+    static const long long myown_max_iter = 64;
+    double a[myown_max_iter+1], g[myown_max_iter+1], c[myown_max_iter+1];
+    a[0] = 1;
+    g[0] = M_SQRT1_2;
+    c[0] = M_SQRT1_2;
+    long long i = 0;
+    do {
+        a[i+1] = 0.5 * (a[i] + g[i]);
+        g[i+1] = sqrt(a[i] * g[i]);
+        c[i+1] = 0.5 * (a[i] - g[i]);
+        i++;
+    } while (i < myown_max_iter && fabs(a[i]-g[i]) > DBL_EPSILON);
+    if (i == myown_max_iter)
+        fprintf(stderr, "WARNING: max iterations in myown_jacobi_sn_cn_dn()\n");
+
+    double phi = (1LL << i) * a[i] * u;
+    for (; i > 0; i--)
+        phi = 0.5 * (phi + asin(c[i] * sin(phi) / a[i]));
+
+    *sn = sin(phi);
+    *cn = cos(phi);
+    *dn = sqrt(1 - 0.5 * (*sn * *sn));
+}
+// compute incomplete elliptic integral F for fixed modulus k=1/sqrt(2)
+// via Landen transformation, Abramowitz & Stegun 17.5
+static double myown_ellint_F(double phi)
+{
+    double a = 1;
+    double g = M_SQRT1_2;
+    double last_a, last_g;
+    double tan_2n_phi;
+
+    long long i = 0;
+    do {
+        tan_2n_phi = tan(phi);
+        i++;
+        phi += phi - atan((a - g) * tan_2n_phi / (a + g * tan_2n_phi * tan_2n_phi));
+        last_a = a;
+        last_g = g;
+        a = 0.5 * (last_a + last_g);
+        g = sqrt(last_a * last_g);
+    }
+    while (i < myown_max_iter && fabs(a - g) > DBL_EPSILON);
+
+    if (i == myown_max_iter)
+        fprintf(stderr, "WARNING: max iterations in myown_ellint_F()\n");
+
+    phi /= (1LL << i);
+    return phi / g;
+}
+
+static double F(double phi)
+{
+    return myown_ellint_F(phi);
+}
+
+static void ccn(double re, double im, double* ret_re, double* ret_im)
+{
+    double sn_re, cn_re, dn_re;
+    double sn_im, cn_im, dn_im;
+    myown_jacobi_sn_cn_dn(re, &sn_re, &cn_re, &dn_re);
+    myown_jacobi_sn_cn_dn(im, &sn_im, &cn_im, &dn_im);
+    double t = 1 - dn_re * dn_re * sn_im * sn_im;
+    *ret_re = cn_re * cn_im / t;
+    *ret_im = - sn_re * dn_re * sn_im * dn_im / t;
+}
+
+void disk_to_square_conformal(double u, double v, double& x, double& y)
+{
+    // Rotate (u,v) by pi/4;
+    double ru = (u - v) * M_SQRT1_2;
+    double rv = (u + v) * M_SQRT1_2;
+    // Map (ru,rv) to (rx,ry) on a rotated square.
+    double A = ru * ru + rv * rv;
+    double B = ru * ru - rv * rv;
+    double U = 1 + 2 * B - A * A;
+    double T = sqrt((1 + A * A) * (1 + A * A) - 4 * B  * B);
+    double cos_a = (2 * A - T) / U;
+    double cos_b = U / (2 * A + T);
+    double a = acos(min(max(cos_a, -1.0), 1.0));
+    double b = acos(min(max(cos_b, -1.0), 1.0));
+    double rx = sign(ru) * (1 - F(a) / (2 * K));
+    double ry = sign(rv) * (    F(b) / (2 * K));
+    // Rotate square by -45 deg and make it fill [-1,+1]^2
+    x = rx + ry;
+    y = ry - rx;
+}
+
+void square_to_disk_conformal(double x, double y, double& u, double& v)
+{
+    // Rotate by 45 deg and keep the square inside [-1,+1]^2
+    double z_re = x / 2 - y / 2;
+    double z_im = x / 2 + y / 2;
+    // Map z to unit disk
+    double ru, rv;
+    ccn(K * (1 - z_re), - K * z_im, &ru, &rv);
+    // Rotate (u,v) by pi/4;
+    u = (ru + rv) * M_SQRT1_2;
+    v = (rv - ru) * M_SQRT1_2;
+}
+
+#endif
+}
diff --git a/libproj/disk-square.hpp b/libproj/disk-square.hpp
new file mode 100644 (file)
index 0000000..3f05681
--- /dev/null
@@ -0,0 +1,23 @@
+#ifndef PROJ_DISC_SQUARE_H
+#define PROJ_DISC_SQUARE_H
+
+namespace Projection {
+
+void disk_to_square_stretch(double u, double v, double& x, double& y);
+void square_to_disk_stretch(double x, double y, double& u, double& v);
+
+void disk_to_square_shirley(double u, double v, double& x, double& y);
+void square_to_disk_shirley(double x, double y, double& u, double& v);
+
+void disk_to_square_squircle(double u, double v, double& x, double& y);
+void square_to_disk_squircle(double x, double y, double& u, double& v);
+
+void disk_to_square_elliptical(double u, double v, double& x, double& y);
+void square_to_disk_elliptical(double x, double y, double& u, double& v);
+
+void disk_to_square_conformal(double u, double v, double& x, double& y);
+void square_to_disk_conformal(double x, double y, double& u, double& v);
+
+}
+
+#endif
diff --git a/libproj/hemisphere-mix-lambert-stereographic.cpp b/libproj/hemisphere-mix-lambert-stereographic.cpp
new file mode 100644 (file)
index 0000000..6af8eca
--- /dev/null
@@ -0,0 +1,44 @@
+#include "hemisphere-mix-lambert-stereographic.hpp"
+
+namespace Projection {
+
+MixLambertStereographic::MixLambertStereographic(double beta) : beta(beta)
+{
+}
+
+void MixLambertStereographic::forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y)
+{
+    const MixLambertStereographic* me = reinterpret_cast<const MixLambertStereographic*>(imp);
+
+    if (!ctx->is_sphere)
+        lat = ctx->geodetic_to_geocentric(lat);
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    double t = tan((M_PI_2 - lat) / 2);
+    double r = sqrt(1 + me->beta) * t / sqrt(1 + me->beta * t * t);
+    double phi = lon - M_PI_2;
+
+    x = r * cos(phi);
+    y = r * sin(phi);
+}
+
+void MixLambertStereographic::inverse(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon)
+{
+    const MixLambertStereographic* me = reinterpret_cast<const MixLambertStereographic*>(imp);
+
+    double r = hypot(x, y);
+    double phi = atan2(y, x);
+
+    lat = M_PI_2 - 2 * atan(r / sqrt(1 + me->beta * (1 - r * r)));
+    lon = phi + M_PI_2;
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    if (!ctx->is_sphere)
+        lat = ctx->geocentric_to_geodetic(lat);
+}
+
+}
diff --git a/libproj/hemisphere-mix-lambert-stereographic.hpp b/libproj/hemisphere-mix-lambert-stereographic.hpp
new file mode 100644 (file)
index 0000000..26a8926
--- /dev/null
@@ -0,0 +1,20 @@
+#ifndef PROJ_EQAR_CONF_TRADEOFF_H
+#define PROJ_EQAR_CONF_TRADEOFF_H
+
+#include "proj.hpp"
+
+namespace Projection {
+
+class MixLambertStereographic : public Implementation
+{
+public:
+    double beta;
+
+    MixLambertStereographic(double beta);
+    static void forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y);
+    static void inverse(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon);
+};
+
+}
+
+#endif
diff --git a/libproj/hemisphere-perspective.cpp b/libproj/hemisphere-perspective.cpp
new file mode 100644 (file)
index 0000000..d2c1414
--- /dev/null
@@ -0,0 +1,37 @@
+#include "hemisphere-perspective.hpp"
+
+namespace Projection {
+
+Perspective::Perspective(double P, double lat0, double lon0) :
+    _P(P), _lat0(lat0), _lon0(lon0)
+{
+}
+
+void Perspective::forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y)
+{
+    const Perspective* me = reinterpret_cast<const Perspective*>(imp);
+
+    if (!ctx->is_sphere)
+        lat = ctx->geodetic_to_geocentric(lat);
+
+    double P = me->_P;
+    double lat0 = me->_lat0;
+    double lon0 = me->_lon0;
+
+    double cos_c = sin(lat0) * sin(lat) + cos(lat0) * cos(lat) * cos(lon - lon0);
+    if (cos_c < 1.0 / P) {
+        x = y = NAN;
+        return;
+    }
+    double kp = (P - 1.0) / (P - cos_c);
+    double scale = 1.0 / sqrt((P-1)/(P+1));
+    x = scale * kp * cos(lat) * sin(lon - lon0);
+    y = scale * kp * (cos(lat0) * sin(lat) - sin(lat0) * cos(lat) * cos(lon - lon0));
+}
+
+void Perspective::inverse(const Context* ctx, const Implementation* /* imp */, double /* x */, double /* y */, double& lat, double& lon)
+{
+    lat = lon = NAN;
+}
+
+}
diff --git a/libproj/hemisphere-perspective.hpp b/libproj/hemisphere-perspective.hpp
new file mode 100644 (file)
index 0000000..236e539
--- /dev/null
@@ -0,0 +1,20 @@
+#ifndef PROJ_PERSPECTIVE_H
+#define PROJ_PERSPECTIVE_H
+
+#include "proj.hpp"
+
+namespace Projection {
+
+class Perspective : public Implementation
+{
+private:
+    double _P, _lat0, _lon0;
+public:
+    Perspective(double P, double lat0, double lon0);
+    static void forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y);
+    static void inverse(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon);
+};
+
+}
+
+#endif
diff --git a/libproj/hemisphere-stereographic.cpp b/libproj/hemisphere-stereographic.cpp
new file mode 100644 (file)
index 0000000..8446cb7
--- /dev/null
@@ -0,0 +1,39 @@
+#include "hemisphere-stereographic.hpp"
+
+namespace Projection {
+
+Stereographic::Stereographic()
+{
+}
+
+void Stereographic::forward(const Context* ctx, const Implementation* /* imp */, double lat, double lon, double& x, double& y)
+{
+    if (!ctx->is_sphere)
+        lat = ctx->geodetic_to_geocentric(lat);
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    double r = tan((M_PI_2 - lat) / 2);
+    double phi = lon - M_PI_2;
+
+    x = r * cos(phi);
+    y = r * sin(phi);
+}
+
+void Stereographic::inverse(const Context* ctx, const Implementation* /* imp */, double x, double y, double& lat, double& lon)
+{
+    double r = hypot(x, y);
+    double phi = atan2(y, x);
+
+    lat = M_PI_2 - 2 * atan(r);
+    lon = phi + M_PI_2;
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    if (!ctx->is_sphere)
+        lat = ctx->geocentric_to_geodetic(lat);
+}
+
+}
diff --git a/libproj/hemisphere-stereographic.hpp b/libproj/hemisphere-stereographic.hpp
new file mode 100644 (file)
index 0000000..66382d0
--- /dev/null
@@ -0,0 +1,18 @@
+#ifndef PROJ_STEREOGRAPHIC_H
+#define PROJ_STEREOGRAPHIC_H
+
+#include "proj.hpp"
+
+namespace Projection {
+
+class Stereographic : public Implementation
+{
+public:
+    Stereographic();
+    static void forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y);
+    static void inverse(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon);
+};
+
+}
+
+#endif
diff --git a/libproj/proj.cpp b/libproj/proj.cpp
new file mode 100644 (file)
index 0000000..6a0d9e9
--- /dev/null
@@ -0,0 +1,285 @@
+#include "proj.hpp"
+
+#include "sphere-lambert-azimuthal-equal-area.hpp"
+#include "sphere-breusing-harmonic.hpp"
+#include "sphere-equidistant-azimuthal.hpp"
+#include "hemisphere-stereographic.hpp"
+#include "hemisphere-mix-lambert-stereographic.hpp"
+#include "hemisphere-perspective.hpp"
+
+#include "disk-square.hpp"
+
+namespace Projection {
+
+Context::Context(enum Projection P,
+        const std::string& params,
+        enum Center center,
+        enum Layout layout,
+        enum SquareMethod square_method,
+        double semi_major_axis, double semi_minor_axis) :
+    _valid(false),
+    _imp(nullptr), _forward(nullptr), _inverse(nullptr),
+    _disk_to_square(nullptr), _square_to_disk(nullptr),
+    semi_major_axis(semi_major_axis), semi_minor_axis(semi_minor_axis),
+    eccentricity_squared(1 - (semi_minor_axis * semi_minor_axis) / (semi_major_axis * semi_major_axis)),
+    is_sphere(semi_major_axis >= semi_minor_axis && semi_major_axis <= semi_minor_axis),
+    projection(P), center(center), layout(layout), square_method(square_method)
+{
+    switch (P) {
+    case ProjLambertAzimuthalEqualArea:
+        if (center == CenterNorthPole || center == CenterSouthPole) {
+            _imp = new LambertAzimuthalEqualArea();
+            _forward = LambertAzimuthalEqualArea::forward;
+            _inverse = LambertAzimuthalEqualArea::inverse;
+            _valid = true;
+        }
+        break;
+    case ProjStereographic:
+        if (layout != LayoutSphere && (center == CenterNorthPole || center == CenterSouthPole)) {
+            _imp = new Stereographic();
+            _forward = Stereographic::forward;
+            _inverse = Stereographic::inverse;
+            _valid = true;
+        }
+        break;
+    case ProjBreusingHarmonicMean:
+        if (center == CenterNorthPole || center == CenterSouthPole) {
+            _imp = new BreusingHarmonicMean();
+            _forward = BreusingHarmonicMean::forward;
+            _inverse = BreusingHarmonicMean::inverse;
+            _valid = true;
+        }
+        break;
+    case ProjMixLambertStereographic:
+        if (layout != LayoutSphere && (center == CenterNorthPole || center == CenterSouthPole)) {
+            double beta;
+            if (sscanf(params.c_str(), "beta=%lf", &beta) == 1) {
+                _imp = new MixLambertStereographic(beta);
+                _forward = MixLambertStereographic::forward;
+                _inverse = MixLambertStereographic::inverse;
+                _valid = true;
+            }
+        }
+        break;
+    case ProjEquidistantAzimuthal:
+        if (center == CenterNorthPole || center == CenterSouthPole) {
+            _imp = new EquidistantAzimuthal();
+            _forward = EquidistantAzimuthal::forward;
+            _inverse = EquidistantAzimuthal::inverse;
+            _valid = true;
+        }
+        break;
+    case ProjPerspective:
+        if (layout != LayoutSphere && (center == CenterNorthPole || center == CenterSouthPole)) {
+            _imp = new Perspective(-1e10,
+                    50.90592 / 180 * M_PI, 8.0285  / 180 * M_PI); // my office ;)
+            _forward = Perspective::forward;
+            _inverse = Perspective::inverse;
+            _valid = true;
+        }
+        break;
+    }
+    if (layout == LayoutQuincuncial && square_method == SquareMethodNone) {
+        _valid = false;
+    } else {
+        switch (square_method) {
+        case SquareMethodNone:
+            break;
+        case SquareMethodStretch:
+            _disk_to_square = disk_to_square_stretch;
+            _square_to_disk = square_to_disk_stretch;
+            break;
+        case SquareMethodShirley:
+            _disk_to_square = disk_to_square_shirley;
+            _square_to_disk = square_to_disk_shirley;
+            break;
+        case SquareMethodSquircle:
+            _disk_to_square = disk_to_square_squircle;
+            _square_to_disk = square_to_disk_squircle;
+            break;
+        case SquareMethodElliptical:
+            _disk_to_square = disk_to_square_elliptical;
+            _square_to_disk = square_to_disk_elliptical;
+            break;
+        case SquareMethodConformal:
+            _disk_to_square = disk_to_square_conformal;
+            _square_to_disk = square_to_disk_conformal;
+            break;
+        }
+    }
+}
+
+void Context::forward(double lat, double lon, double& x, double& y) const
+{
+    if (projection != ProjPerspective && layout == LayoutHemisphere && !in_hemisphere(center, lat, lon)) {
+        x = y = NAN;
+    } else if (layout == LayoutQuincuncial) {
+        bool center_hemisphere =
+            (center == CenterNorthPole && lat >= 0)
+            || (center == CenterSouthPole && lat <= 0);
+        _forward(this, _imp, center_hemisphere ? lat : -lat, lon + M_PI_4, x, y);
+        double sx, sy, t;
+        // Transform unit disk to [-1,+1]^2 square
+        _disk_to_square(x, y, sx, sy);
+        // rotate by -45 degrees while keeping square in [-1,+1]^2
+        x =  sx / 2 + sy / 2;
+        y = -sx / 2 + sy / 2;
+        // place the southern hemisphere into the four free corners
+        if (!center_hemisphere) {
+            if (x * y >= 0) {
+                // I or III
+                if (x >= 0) {
+                    t = -y + 1;
+                    y = -x + 1;
+                    x = t;
+                } else {
+                    t = -y - 1;
+                    y = -x - 1;
+                    x = t;
+                }
+            } else {
+                // II or IV
+                if (x < 0) {
+                    t = y - 1;
+                    y = x + 1;
+                    x = t;
+                } else {
+                    t = y + 1;
+                    y = x - 1;
+                    x = t;
+                }
+            }
+        }
+    } else {
+        _forward(this, _imp, lat, lon, x, y);
+        if (square_method != SquareMethodNone)
+            _disk_to_square(x, y, x, y);
+    }
+}
+
+void Context::inverse(double x, double y, double& lat, double& lon) const
+{
+    if (layout == LayoutQuincuncial) {
+        bool center_hemisphere = (fabs(x) + fabs(y) <= 1);
+        // remap the southern hemisphere from the corners to a center square
+        if (!center_hemisphere) {
+            double t;
+            if (x * y >= 0) {
+                // I or III
+                if (x >= 0) {
+                    t = -y + 1;
+                    y = -x + 1;
+                    x = t;
+                } else {
+                    t = -y - 1;
+                    y = -x - 1;
+                    x = t;
+                }
+            } else {
+                // II or IV
+                if (x < 0) {
+                    t = y - 1;
+                    y = x + 1;
+                    x = t;
+                } else {
+                    t = y + 1;
+                    y = x - 1;
+                    x = t;
+                }
+            }
+        }
+        // rotate by 45 degrees while keeping square in [-1,+1]^2
+        double sx = x - y;
+        double sy = x + y;
+        // transform square to disk
+        _square_to_disk(sx, sy, x, y);
+        // inverse projection
+        _inverse(this, _imp, x, y, lat, lon);
+        if (!center_hemisphere)
+            lat = -lat;
+        lon -= M_PI_4;
+        if (lon < -M_PI)
+            lon += 2 * M_PI;
+    } else {
+        if (square_method != SquareMethodNone)
+            _square_to_disk(x, y, x, y);
+        _inverse(this, _imp, x, y, lat, lon);
+    }
+}
+
+void Context::derivatives(double lat, double lon,
+        double& dx_dlat, double& dx_dlon, double& dy_dlat, double& dy_dlon, double delta) const
+{
+    double mlat = lat - delta;
+    double plat = lat + delta;
+    double mlon = lon - delta;
+    double plon = lon + delta;
+
+    if (mlat < -M_PI_2 || plat > M_PI_2)
+        goto fail;
+    if (mlon < -M_PI)
+        mlon += 2 * M_PI;
+    if (plon > M_PI)
+        plon -= 2 * M_PI;
+
+    double x_mlat_mlon, x_mlat_plon, x_plat_mlon, x_plat_plon;
+    double y_mlat_mlon, y_mlat_plon, y_plat_mlon, y_plat_plon;
+
+    forward(mlat, mlon, x_mlat_mlon, y_mlat_mlon);
+    forward(mlat, plon, x_mlat_plon, y_mlat_plon);
+    forward(plat, mlon, x_plat_mlon, y_plat_mlon);
+    forward(plat, plon, x_plat_plon, y_plat_plon);
+
+    if (!isfinite(x_mlat_mlon) || !isfinite(x_mlat_plon) || !isfinite(x_plat_mlon) || !isfinite(x_plat_plon)
+            || !isfinite(y_mlat_mlon) || !isfinite(y_mlat_plon) || !isfinite(y_plat_mlon) || !isfinite(y_plat_plon))
+        goto fail;
+
+    dx_dlat = (- x_plat_plon + x_mlat_plon + x_mlat_mlon - x_plat_mlon) / (4 * delta);
+    dx_dlon = (+ x_plat_plon + x_mlat_plon - x_mlat_mlon - x_plat_mlon) / (4 * delta);
+    dy_dlat = (+ y_plat_plon - y_mlat_plon - y_mlat_mlon + y_plat_mlon) / (4 * delta);
+    dy_dlon = (- y_plat_plon - y_mlat_plon + y_mlat_mlon + y_plat_mlon) / (4 * delta);
+    return;
+
+fail:
+    dx_dlat = dx_dlon = dy_dlat = dy_dlon = NAN;
+}
+
+void Context::analysis(double lat, double lon,
+        double& h, double& k,
+        double& theta_prime,
+        double& a, double& b,
+        double& omega, double& s,
+        double delta) const
+{
+    double dx_dlat, dx_dlon, dy_dlat, dy_dlon;
+
+    derivatives(lat, lon, dx_dlat, dx_dlon, dy_dlat, dy_dlon, delta);
+    if (!isfinite(dx_dlat) || !isfinite(dx_dlon) || !isfinite(dy_dlat) || !isfinite(dy_dlon)) {
+        h = k = theta_prime = a = b = omega = s = NAN;
+        return;
+    }
+
+    double coslat = cos(lat);
+    h = hypot(dx_dlat, dy_dlat);
+    k = hypot(dx_dlon, dy_dlon) / coslat;
+    double sin_theta_prime = (dy_dlat * dx_dlon - dx_dlat * dy_dlon) / (h * k * coslat);
+    s = h * k * sin_theta_prime; // equivalent to s = a * b
+    theta_prime = asin(min(max(sin_theta_prime, -1.0), +1.0));
+    double a_prime = sqrt(max(h * h + k * k + 2 * s, 0.0));
+    double b_prime = sqrt(max(h * h + k * k - 2 * s, 0.0));
+    a = (a_prime + b_prime) / 2;
+    b = (a_prime - b_prime) / 2;
+    omega = 2 * asin(min(max(b_prime / a_prime, -1.0), +1.0));
+}
+
+double Context::geodetic_to_geocentric(double lat) const
+{
+    return is_sphere ? lat : atan(tan(lat) * (1 - eccentricity_squared));
+}
+
+double Context::geocentric_to_geodetic(double lat) const
+{
+    return is_sphere ? lat : atan(tan(lat) / (1 - eccentricity_squared));
+}
+
+}
diff --git a/libproj/proj.hpp b/libproj/proj.hpp
new file mode 100644 (file)
index 0000000..21d5b2c
--- /dev/null
@@ -0,0 +1,130 @@
+#ifndef PROJ_H
+#define PROJ_H
+
+#include <cmath>
+#include <string>
+
+namespace Projection {
+
+enum Projection {
+    ProjLambertAzimuthalEqualArea,
+    ProjStereographic,
+    ProjBreusingHarmonicMean,
+    ProjMixLambertStereographic,
+    ProjEquidistantAzimuthal,
+    ProjPerspective
+};
+
+enum Center {
+    CenterNorthPole,
+    CenterSouthPole
+};
+
+enum Layout {
+    LayoutHemisphere,
+    LayoutSphere,
+    LayoutQuincuncial,
+};
+
+enum SquareMethod {
+    SquareMethodNone,
+    SquareMethodStretch,
+    SquareMethodShirley,
+    SquareMethodSquircle,
+    SquareMethodElliptical,
+    SquareMethodConformal
+};
+
+class Implementation;
+
+class Context
+{
+private:
+    bool _valid;
+    Implementation* _imp;
+    void (*_forward)(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y);
+    void (*_inverse)(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon);
+    void (*_disk_to_square)(double u, double v, double& x, double& y);
+    void (*_square_to_disk)(double x, double y, double& u, double& v);
+
+public:
+
+    /* Information about the Ellipsoid */
+    const double semi_major_axis;
+    const double semi_minor_axis;
+    const double eccentricity_squared;
+    const bool is_sphere;
+
+    /* Generic parameters of the projection */
+    const enum Projection projection;
+    const enum Center center;
+    const enum Layout layout;
+    const enum SquareMethod square_method;
+
+    /* User interface */
+    Context(enum Projection P,
+            const std::string& params,
+            enum Center center,
+            enum Layout layout,
+            enum SquareMethod square_method,
+            double semi_major_axis = 1, double semi_minor_axis = 1);
+
+    bool is_valid() const { return _valid; }
+
+    void forward(double lat, double lon, double& x, double& y) const;
+    void inverse(double x, double y, double& lat, double& lon) const;
+
+    void derivatives(double lat, double lon,
+            double& dx_dlat, double& dx_dlon, double& dy_dlat, double& dy_dlon, double delta = 1e-5) const;
+
+    void analysis(double lat, double lon,
+            double& h, double& k, double& theta_prime,
+            double& a, double& b,
+            double& omega, double& s,
+            double delta = 1e-5) const;
+
+    /* Implementation interface */
+    double geodetic_to_geocentric(double lat) const;
+    double geocentric_to_geodetic(double lat) const;
+};
+
+using std::sin;
+using std::asin;
+using std::cos;
+using std::acos;
+using std::tan;
+using std::atan;
+using std::atan2;
+using std::fabs;
+using std::hypot;
+using std::sqrt;
+using std::isfinite;
+using std::min;
+using std::max;
+using std::floor;
+using std::ceil;
+using std::hypot;
+using std::log;
+using std::exp;
+
+constexpr double sign(double x) { return (x < 0 ? -1 : x > 0 ? +1 : 0); }
+constexpr double sign_not_zero(double x) { return (x < 0 ? -1 : +1); }
+
+constexpr double degrees(double x) { return 180 / M_PI * x; }
+constexpr double radians(double x) { return M_PI / 180 * x; }
+
+constexpr bool in_hemisphere(Center c, double lat, double lon)
+{
+    return (c == CenterNorthPole ? (lat >= 0) : (lat <= 0));
+}
+
+class Implementation
+{
+public:
+    static void forward(const Context* /* ctx */, const Implementation* /* imp */, double /* lat */, double /* lon */, double& /* x */, double& /* y */) {}
+    static void inverse(const Context* /* ctx */, const Implementation* /* imp */, double /* x */, double /* y */, double& /* lat */, double& /* lon */) {}
+};
+
+}
+
+#endif
diff --git a/libproj/sphere-breusing-harmonic.cpp b/libproj/sphere-breusing-harmonic.cpp
new file mode 100644 (file)
index 0000000..09c23a7
--- /dev/null
@@ -0,0 +1,50 @@
+#include "sphere-breusing-harmonic.hpp"
+
+namespace Projection {
+
+BreusingHarmonicMean::BreusingHarmonicMean()
+{
+}
+
+void BreusingHarmonicMean::forward(const Context* ctx, const Implementation* /* imp */, double lat, double lon, double& x, double& y)
+{
+    if (!ctx->is_sphere)
+        lat = ctx->geodetic_to_geocentric(lat);
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    // Lambert az. eq-ar: r=sin((pi/2-lat)/2)
+    // Stereographic: r=tan((pi/2-lat)/2)
+    // Harmonic mean: r = 2*tan((pi/2-lat)/4)
+    // Geometric mean: r = 2*sqrt(sin((pi/2-lat)/2)*tan((pi/2-lat)/2))
+    // Arithm mean: r = (sin((pi/2-lat)/2)+tan((pi/2-lat)/2))/2
+    double r = tan((M_PI_2 - lat) / 4); // harmonic mean
+    double phi = lon - M_PI_2;
+
+    if (ctx->layout != LayoutSphere)
+        r /= (M_SQRT2 - 1);
+
+    x = r * cos(phi);
+    y = r * sin(phi);
+}
+
+void BreusingHarmonicMean::inverse(const Context* ctx, const Implementation* /* imp */, double x, double y, double& lat, double& lon)
+{
+    double r = hypot(x, y);
+    double phi = atan2(y, x);
+
+    if (ctx->layout != LayoutSphere)
+        r *= (M_SQRT2 - 1);
+
+    lat = M_PI_2 - 4 * atan(r);
+    lon = phi + M_PI_2;
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    if (!ctx->is_sphere)
+        lat = ctx->geocentric_to_geodetic(lat);
+}
+
+}
diff --git a/libproj/sphere-breusing-harmonic.hpp b/libproj/sphere-breusing-harmonic.hpp
new file mode 100644 (file)
index 0000000..8fa0341
--- /dev/null
@@ -0,0 +1,18 @@
+#ifndef PROJ_BREUSING_HARMONIC_H
+#define PROJ_BREUSING_HARMONIC_H
+
+#include "proj.hpp"
+
+namespace Projection {
+
+class BreusingHarmonicMean : public Implementation
+{
+public:
+    BreusingHarmonicMean();
+    static void forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y);
+    static void inverse(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon);
+};
+
+}
+
+#endif
diff --git a/libproj/sphere-equidistant-azimuthal.cpp b/libproj/sphere-equidistant-azimuthal.cpp
new file mode 100644 (file)
index 0000000..de044f4
--- /dev/null
@@ -0,0 +1,45 @@
+#include "sphere-equidistant-azimuthal.hpp"
+
+namespace Projection {
+
+EquidistantAzimuthal::EquidistantAzimuthal()
+{
+}
+
+void EquidistantAzimuthal::forward(const Context* ctx, const Implementation* /* imp */, double lat, double lon, double& x, double& y)
+{
+    if (!ctx->is_sphere)
+        lat = ctx->geodetic_to_geocentric(lat);
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    double r = 1 - lat / M_PI_2;
+    double phi = lon - M_PI_2;
+
+    if (ctx->layout == LayoutSphere)
+        r /= 2;
+
+    x = r * cos(phi);
+    y = r * sin(phi);
+}
+
+void EquidistantAzimuthal::inverse(const Context* ctx, const Implementation* /* imp */, double x, double y, double& lat, double& lon)
+{
+    double r = hypot(x, y);
+    double phi = atan2(y, x);
+
+    if (ctx->layout == LayoutSphere)
+        r *= 2;
+
+    lat = (1 - r) * M_PI_2;
+    lon = phi + M_PI_2;
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    if (!ctx->is_sphere)
+        lat = ctx->geocentric_to_geodetic(lat);
+}
+
+}
diff --git a/libproj/sphere-equidistant-azimuthal.hpp b/libproj/sphere-equidistant-azimuthal.hpp
new file mode 100644 (file)
index 0000000..fa882fa
--- /dev/null
@@ -0,0 +1,18 @@
+#ifndef PROJ_EQUIDISTANT_AZ_H
+#define PROJ_EQUIDISTANT_AZ_H
+
+#include "proj.hpp"
+
+namespace Projection {
+
+class EquidistantAzimuthal : public Implementation
+{
+public:
+    EquidistantAzimuthal();
+    static void forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y);
+    static void inverse(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon);
+};
+
+}
+
+#endif
diff --git a/libproj/sphere-lambert-azimuthal-equal-area.cpp b/libproj/sphere-lambert-azimuthal-equal-area.cpp
new file mode 100644 (file)
index 0000000..d52ac99
--- /dev/null
@@ -0,0 +1,45 @@
+#include "sphere-lambert-azimuthal-equal-area.hpp"
+
+namespace Projection {
+
+LambertAzimuthalEqualArea::LambertAzimuthalEqualArea()
+{
+}
+
+void LambertAzimuthalEqualArea::forward(const Context* ctx, const Implementation* /* imp */, double lat, double lon, double& x, double& y)
+{
+    if (!ctx->is_sphere)
+        lat = ctx->geodetic_to_geocentric(lat);
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    double r = sin((M_PI_2 - lat) / 2);
+    double phi = lon - M_PI_2;
+
+    if (ctx->layout != LayoutSphere)
+        r *= M_SQRT2;
+
+    x = r * cos(phi);
+    y = r * sin(phi);
+}
+
+void LambertAzimuthalEqualArea::inverse(const Context* ctx, const Implementation* /* imp */, double x, double y, double& lat, double& lon)
+{
+    double r = hypot(x, y);
+    double phi = atan2(y, x);
+
+    if (ctx->layout != LayoutSphere)
+        r /= M_SQRT2;
+
+    lat = M_PI_2 - 2 * asin(r);
+    lon = phi + M_PI_2;
+
+    if (ctx->center == CenterSouthPole)
+        lat = -lat;
+
+    if (!ctx->is_sphere)
+        lat = ctx->geocentric_to_geodetic(lat);
+}
+
+}
diff --git a/libproj/sphere-lambert-azimuthal-equal-area.hpp b/libproj/sphere-lambert-azimuthal-equal-area.hpp
new file mode 100644 (file)
index 0000000..8c2b813
--- /dev/null
@@ -0,0 +1,18 @@
+#ifndef PROJ_LAMBERT_AZ_AE_H
+#define PROJ_LAMBERT_AZ_AE_H
+
+#include "proj.hpp"
+
+namespace Projection {
+
+class LambertAzimuthalEqualArea : public Implementation
+{
+public:
+    LambertAzimuthalEqualArea();
+    static void forward(const Context* ctx, const Implementation* imp, double lat, double lon, double& x, double& y);
+    static void inverse(const Context* ctx, const Implementation* imp, double x, double y, double& lat, double& lon);
+};
+
+}
+
+#endif
diff --git a/pdfplot.cpp b/pdfplot.cpp
new file mode 100644 (file)
index 0000000..822684f
--- /dev/null
@@ -0,0 +1,46 @@
+#include <cmath>
+
+#include <QPdfWriter>
+#include <QPainter>
+
+#include "pdfplot.hpp"
+
+void make_pdf_painter(const std::string& name, bool clip_circular, QPdfWriter** writer, QPainter** painter)
+{
+    *writer = new QPdfWriter((name + ".pdf").c_str());
+    (*writer)->setPageLayout(QPageLayout(
+                QPageSize(QSizeF(210.0f, 210.0f), QPageSize::Millimeter, "A4 square", QPageSize::ExactMatch),
+                QPageLayout::Portrait, QMarginsF(0.0f, 0.0f, 0.0f, 0.0f), QPageLayout::Millimeter));
+    (*writer)->setTitle(name.c_str());
+    *painter = new QPainter(*writer);
+    (*painter)->setClipRegion(QRegion(0, 0, (*painter)->device()->width(), (*painter)->device()->height(),
+                (clip_circular ? QRegion::Ellipse : QRegion::Rectangle)));
+}
+
+void finish_pdf_painter(QPdfWriter* writer, QPainter* painter)
+{
+    painter->end();
+    delete painter;
+    delete writer;
+}
+
+void plot_points(QPainter* painter, bool as_line, const std::vector<std::pair<float, float>>& point_list)
+{
+    float w = painter->device()->width();
+    float h = painter->device()->height();
+    std::vector<QPointF> qpoints;
+    for (size_t i = 0; i < point_list.size(); i++) {
+        float x = point_list[i].first;
+        float y = point_list[i].second;
+        qpoints.push_back(QPointF((x + 1.0f) / 2.0f * w, (-y + 1.0f) / 2.0f * h));
+    }
+    if (as_line)
+        painter->drawPolyline(qpoints.data(), qpoints.size());
+    else
+        painter->drawPolygon(qpoints.data(), qpoints.size());
+}
+
+bool point_is_valid(double x, double y)
+{
+    return (std::isfinite(x) && std::isfinite(y) && std::fabs(x) < 1.001 && std::fabs(y) < 1.001);
+}
diff --git a/pdfplot.hpp b/pdfplot.hpp
new file mode 100644 (file)
index 0000000..aa111b5
--- /dev/null
@@ -0,0 +1,30 @@
+#ifndef PDFPLOT_HPP
+#define PDFPLOT_HPP
+
+#include <vector>
+
+class QPdfWriter;
+class QPainter;
+
+/*
+ * Construct a QPainter that will paint into a PDF file with the specified name
+ * (the ".pdf" extension will be appended).
+ */
+void make_pdf_painter(const std::string& name, bool clip_circular, QPdfWriter** writer, QPainter** painter);
+
+/*
+ * Finish painting and PDF writing and destroy the associated objects.
+ */
+void finish_pdf_painter(QPdfWriter* writer, QPainter* painter);
+
+/*
+ * Plot a set of points, either as a multiline or as a polygon.
+ */
+void plot_points(QPainter* painter, bool as_line, const std::vector<std::pair<float, float>>& point_list);
+
+/*
+ * Check if a point is a valid map point; allow for some tolerance around the borders.
+ */
+bool point_is_valid(double x, double y);
+
+#endif
diff --git a/pngvis.cpp b/pngvis.cpp
new file mode 100644 (file)
index 0000000..227430c
--- /dev/null
@@ -0,0 +1,350 @@
+#include <cmath>
+
+#include <QColor>
+#include <QImage>
+
+#include "pngvis.hpp"
+
+/* This color map was created with gencolormap, https://github.com/marlam/gencolormap
+ * using the following parameters: type=diverging n=301 hue=0 divergence=240
+ * warmth=0.15 contrast=0.88 saturation=0.6 brightness=0.75 */
+static const int colormap_colors = 301;
+static const unsigned char colormap[] {
+87, 7, 35,
+90, 8, 36,
+94, 9, 38,
+98, 10, 40,
+101, 12, 42,
+105, 13, 44,
+108, 15, 46,
+112, 16, 48,
+115, 18, 50,
+119, 19, 52,
+122, 21, 54,
+126, 22, 56,
+129, 24, 58,
+132, 25, 60,
+136, 27, 62,
+139, 29, 63,
+142, 30, 65,
+145, 32, 67,
+148, 34, 69,
+152, 35, 71,
+155, 37, 73,
+158, 39, 75,
+161, 40, 77,
+164, 42, 78,
+166, 44, 80,
+169, 46, 82,
+172, 48, 84,
+175, 49, 86,
+177, 51, 88,
+180, 53, 90,
+183, 55, 91,
+185, 57, 93,
+188, 59, 95,
+190, 60, 97,
+193, 62, 99,
+195, 64, 100,
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+6, 47, 70,
+5, 44, 67
+};
+
+void png_vis(const std::string& name, int w, int h, const double* data,
+        bool is_based_around_1, double maxval)
+{
+    QImage img(w, h, QImage::Format_RGB32);
+    if (is_based_around_1)
+        maxval -= 1;
+    for (int y = 0; y < h; y++) {
+        QRgb* scanline = reinterpret_cast<QRgb*>(img.scanLine(y));
+        for (int x = 0; x < w; x++) {
+            double v = data[y * w + x];
+            if (std::isfinite(v)) {
+                int direction = +1;
+                if (is_based_around_1) {
+                    if (v < 1) {
+                        v = 1 / v;
+                        direction = -1;
+                    }
+                    v -= 1;
+                }
+                v /= maxval;
+                v = std::min(std::max(v, 0.0), 1.0);
+                if (is_based_around_1)
+                    v = 0.5 + direction * v / 2;
+                //scanline[x] = QColor::fromHslF((0 + v * 240) / 360, 1.0f, 0.5f).rgb(); // jet color: never use!
+                int colormap_index = std::round((1.0 - v) * (colormap_colors - 1) + 0.4999);
+                scanline[x] = QColor(colormap[3 * colormap_index],
+                                     colormap[3 * colormap_index + 1],
+                                     colormap[3 * colormap_index + 2]).rgb();
+            } else {
+                scanline[x] = QColor(255, 255, 255).rgb();
+            }
+        }
+    }
+    img.save((name + ".png").c_str(), "png", 0);
+}
diff --git a/pngvis.hpp b/pngvis.hpp
new file mode 100644 (file)
index 0000000..c6e8bd9
--- /dev/null
@@ -0,0 +1,9 @@
+#ifndef PNGVIS_HPP
+#define PNGVIS_HPP
+
+#include <string>
+
+void png_vis(const std::string& name, int w, int h, const double* data,
+        bool is_based_around_1, double maxval);
+
+#endif
diff --git a/sphere-map-analyzer.cpp b/sphere-map-analyzer.cpp
new file mode 100644 (file)
index 0000000..bfe97af
--- /dev/null
@@ -0,0 +1,119 @@
+#include <cmath>
+#include <cstdio>
+#include <vector>
+#include <algorithm>
+
+#include "proj.hpp"
+
+#include "sphere-map-common.hpp"
+#include "pngvis.hpp"
+
+
+void analyze(const Context& ctx, int w, int h, double* dist_area, double* dist_isot)
+{
+    const double area_unit_sphere = 4 * M_PI;
+    const double area_unit_disk = M_PI;
+    const double area_unit_square = 4;
+    double R;
+    switch (ctx.layout) {
+    case LayoutHemisphere:
+        R = (ctx.square_method == SquareMethodNone ? area_unit_disk : area_unit_square) / (0.5 * area_unit_sphere);
+        break;
+    case LayoutSphere:
+        R = (ctx.square_method == SquareMethodNone ? area_unit_disk : area_unit_square) / area_unit_sphere;
+        break;
+    case LayoutQuincuncial:
+        R = area_unit_square / area_unit_sphere;
+        break;
+    }
+    for (int iy = 0; iy < h; iy++) {
+        double y = ((iy + 0.5) / h) * 2 - 1;
+        for (int ix = 0; ix < w; ix++) {
+            double x = ((ix + 0.5) / h) * 2 - 1;
+            double lat, lon;
+            ctx.inverse(x, y, lat, lon);
+            if (std::isfinite(lat) && std::isfinite(lon)) {
+                double h, k, theta_prime, a, b, omega, s;
+                ctx.analysis(lat, lon, h, k, theta_prime, a, b, omega, s);
+                dist_area[iy * w + ix] = s / R;
+                dist_isot [iy * w + ix] = a / b;
+            } else {
+                dist_area[iy * w + ix] = NAN;
+                dist_isot [iy * w + ix] = NAN;
+            }
+        }
+    }
+}
+
+void scan_array(const double* data, int size,
+        double& min, double& min_1percent,
+        double& max, double& max_1percent,
+        double& mean, double& median)
+{
+    std::vector<double> valid_data;
+    valid_data.reserve(size);
+    min = max = NAN;
+    mean = 0;
+    for (int i = 0; i < size; i++) {
+        if (std::isfinite(data[i])) {
+            if (!std::isfinite(min) || data[i] < min)
+                min = data[i];
+            if (!std::isfinite(max) || data[i] > max)
+                max = data[i];
+            mean += data[i];
+            valid_data.push_back(data[i]);
+        }
+    }
+    if (valid_data.size() == 0) {
+        mean = NAN;
+        min_1percent = NAN;
+        max_1percent = NAN;
+        median = NAN;
+    } else {
+        mean /= valid_data.size();
+        std::sort(valid_data.begin(), valid_data.end());
+        if (valid_data.size() % 2 == 0) {
+            median = (valid_data[valid_data.size() / 2 - 1]
+                    + valid_data[valid_data.size() / 2]) / 2;
+        } else {
+            median = valid_data[valid_data.size() / 2];
+        }
+        min_1percent = valid_data[valid_data.size() / 100];
+        max_1percent = valid_data[valid_data.size() - 1 - valid_data.size() / 100];
+    }
+}
+
+/*
+ * Main function.
+ */
+
+int main(void)
+{
+    const int w = 512;
+    const int h = 512;
+
+    double *dist_area = new double[w * h];
+    double *dist_isot = new double[w * h];
+    double min, max, mean;
+    double min_1percent, max_1percent, median;
+
+    // Loop over the projection methods
+    for (int method = 0; method < n_projections; method++) {
+        Context ctx(projections[method].proj,
+                projections[method].params,
+                projections[method].center,
+                projections[method].layout,
+                projections[method].square_method);
+        fprintf(stderr, "%s\n", projections[method].name);
+        analyze(ctx, w, h, dist_area, dist_isot);
+
+        scan_array(dist_area, w * h, min, min_1percent, max, max_1percent, mean, median);
+        fprintf(stderr, "  area dist min=%g min_1percent=%g max=%g max_1percent=%g mean=%g median=%g\n", min, min_1percent, max, max_1percent, mean, median);
+        png_vis(std::string(projections[method].name) + "-dist-area", w, h, dist_area, true, 2.5);
+
+        scan_array(dist_isot , w * h, min, min_1percent, max, max_1percent, mean, median);
+        fprintf(stderr, "  isot dist min=%g min_1percent=%g max=%g max_1percent=%g mean=%g median=%g\n", min, min_1percent, max, max_1percent, mean, median);
+        png_vis(std::string(projections[method].name) + "-dist-isot", w, h, dist_isot, true, 2.5);
+    }
+    return 0;
+}
diff --git a/sphere-map-common.hpp b/sphere-map-common.hpp
new file mode 100644 (file)
index 0000000..651364a
--- /dev/null
@@ -0,0 +1,113 @@
+#ifndef COMMON_H
+#define COMMON_H
+
+#include "proj.hpp"
+using namespace Projection;
+
+
+typedef struct {
+    const char* name;
+    enum Projection proj;
+    const char* params;
+    Center center;
+    Layout layout;
+    SquareMethod square_method;
+} projection_t;
+
+projection_t projections[] = {
+#if 1
+    // Hemisphere Layouts
+    { "lea-hemisphere-north",    ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    //{ "lea-hemisphere-south",    ProjLambertAzimuthalEqualArea,  "", CenterSouthPole, LayoutHemisphere, SquareMethodNone },
+    { "stg-hemisphere-north",     ProjStereographic,              "", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    //{ "stg-hemisphere-south",     ProjStereographic,              "", CenterSouthPole, LayoutHemisphere, SquareMethodNone },
+    { "eqd-hemisphere-north",     ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    //{ "eqd-hemisphere-south",     ProjEquidistantAzimuthal,       "", CenterSouthPole, LayoutHemisphere, SquareMethodNone },
+    { "bhm-hemisphere-north",     ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "bhm-hemisphere-south",     ProjBreusingHarmonicMean,       "", CenterSouthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    //{ "mix-hemisphere-south",     ProjMixLambertStereographic,    "beta=0.4", CenterSouthPole, LayoutHemisphere, SquareMethodNone },
+#endif
+#if 1
+    // Squared Hemisphere Layouts
+    //{ "lea-hemisphere-shirley-north", ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutHemisphere, SquareMethodShirley },
+    //{ "lea-hemisphere-shirley-south", ProjLambertAzimuthalEqualArea,  "", CenterSouthPole, LayoutHemisphere, SquareMethodShirley },
+    //{ "stg-hemisphere-conf-north",     ProjStereographic,              "", CenterNorthPole, LayoutHemisphere, SquareMethodConformal },
+    //{ "stg-hemisphere-conf-south",     ProjStereographic,              "", CenterSouthPole, LayoutHemisphere, SquareMethodConformal },
+    //{ "eqd-hemisphere-ell-north",      ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutHemisphere, SquareMethodElliptical },
+    //{ "eqd-hemisphere-ell-south",      ProjEquidistantAzimuthal,       "", CenterSouthPole, LayoutHemisphere, SquareMethodElliptical },
+    { "bhm-hemisphere-ell-north",      ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutHemisphere, SquareMethodElliptical },
+    { "bhm-hemisphere-ell-south",      ProjBreusingHarmonicMean,       "", CenterSouthPole, LayoutHemisphere, SquareMethodElliptical },
+    //{ "mix-hemisphere-sq-north",       ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutHemisphere, SquareMethodSquircle },
+    //{ "mix-hemisphere-sq-south",       ProjMixLambertStereographic,    "beta=0.4", CenterSouthPole, LayoutHemisphere, SquareMethodSquircle },
+#endif
+#if 1
+    // Sphere Layouts
+    { "lea-sphere-north",    ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutSphere, SquareMethodNone },
+    //{ "lea-sphere-south",    ProjLambertAzimuthalEqualArea,  "", CenterSouthPole, LayoutSphere, SquareMethodNone },
+    { "eqd-sphere-north",     ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutSphere, SquareMethodNone },
+    //{ "eqd-sphere-south",     ProjEquidistantAzimuthal,       "", CenterSouthPole, LayoutSphere, SquareMethodNone },
+    { "bhm-sphere-north",     ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutSphere, SquareMethodNone },
+    //{ "bhm-sphere-south",     ProjBreusingHarmonicMean,       "", CenterSouthPole, LayoutSphere, SquareMethodNone },
+#endif
+#if 1
+    // Squared Sphere Layouts
+    //{ "lea-sphere-shirley-north", ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutSphere, SquareMethodShirley  },
+    //{ "lea-sphere-shirley-south", ProjLambertAzimuthalEqualArea,  "", CenterSouthPole, LayoutSphere, SquareMethodShirley  },
+    { "eqd-sphere-sq-north",       ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutSphere, SquareMethodSquircle },
+    //{ "eqd-sphere-sq-south",       ProjEquidistantAzimuthal,       "", CenterSouthPole, LayoutSphere, SquareMethodSquircle },
+    //{ "bhm-sphere-sq-north",       ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutSphere, SquareMethodSquircle },
+    //{ "bhm-sphere-sq-south",       ProjBreusingHarmonicMean,       "", CenterSouthPole, LayoutSphere, SquareMethodSquircle },
+#endif
+#if 1
+    // Quincuncial Layouts
+    //{ "lea-quinc-stretch",     ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutQuincuncial, SquareMethodStretch    },
+    //{ "lea-quinc-adjstretch",  ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutQuincuncial, SquareMethodAdjStretch },
+    { "lea-quinc-shirley",     ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutQuincuncial, SquareMethodShirley    },
+    //{ "lea-quinc-squircle",    ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutQuincuncial, SquareMethodSquircle   },
+    //{ "lea-quinc-elliptic",    ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutQuincuncial, SquareMethodElliptical },
+    //{ "lea-quinc-conf",        ProjLambertAzimuthalEqualArea,  "", CenterNorthPole, LayoutQuincuncial, SquareMethodConformal  },
+    //{ "stg-quinc-stretch",      ProjStereographic,              "", CenterNorthPole, LayoutQuincuncial, SquareMethodStretch    },
+    //{ "stg-quinc-adjstretch",   ProjStereographic,              "", CenterNorthPole, LayoutQuincuncial, SquareMethodAdjStretch },
+    //{ "stg-quinc-shirley",      ProjStereographic,              "", CenterNorthPole, LayoutQuincuncial, SquareMethodShirley    },
+    //{ "stg-quinc-squircle",     ProjStereographic,              "", CenterNorthPole, LayoutQuincuncial, SquareMethodSquircle   },
+    //{ "stg-quinc-elliptic",     ProjStereographic,              "", CenterNorthPole, LayoutQuincuncial, SquareMethodElliptical },
+    { "stg-quinc-conf",         ProjStereographic,              "", CenterNorthPole, LayoutQuincuncial, SquareMethodConformal  },
+    //{ "mix-quinc-stretch",      ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutQuincuncial, SquareMethodStretch    },
+    //{ "mix-quinc-adjstretch",   ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutQuincuncial, SquareMethodAdjStretch },
+    //{ "mix-quinc-shirley",      ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutQuincuncial, SquareMethodShirley    },
+    { "mix-quinc-squircle",     ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutQuincuncial, SquareMethodSquircle   },
+    //{ "mix-quinc-elliptic",     ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutQuincuncial, SquareMethodElliptical },
+    //{ "mix-quinc-conf",         ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutQuincuncial, SquareMethodConformal  },
+    //{ "bhm-quinc-stretch",      ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodStretch    },
+    //{ "bhm-quinc-adjstretch",   ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodAdjStretch },
+    //{ "bhm-quinc-shirley",      ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodShirley    },
+    //{ "bhm-quinc-squircle",     ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodSquircle   },
+    { "bhm-quinc-elliptic",     ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodElliptical },
+    //{ "bhm-quinc-conf",         ProjBreusingHarmonicMean,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodConformal  },
+    //{ "eqd-quinc-stretch",      ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodStretch    },
+    //{ "eqd-quinc-adjstretch",   ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodAdjStretch },
+    //{ "eqd-quinc-shirley",      ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodShirley    },
+    //{ "eqd-quinc-squircle",     ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodSquircle   },
+    { "eqd-quinc-elliptic",     ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodElliptical },
+    //{ "eqd-quinc-conf",         ProjEquidistantAzimuthal,       "", CenterNorthPole, LayoutQuincuncial, SquareMethodConformal  },
+#endif
+#if 1
+    { "mix-00-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.0", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-01-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.1", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-02-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.2", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-03-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.3", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-04-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.4", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-05-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.5", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-06-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.6", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-07-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.7", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-08-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.8", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-09-hemisphere-north",     ProjMixLambertStereographic,    "beta=0.9", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+    { "mix-10-hemisphere-north",     ProjMixLambertStereographic,    "beta=1.0", CenterNorthPole, LayoutHemisphere, SquareMethodNone },
+#endif
+    { "perspective", ProjPerspective, "", CenterNorthPole, LayoutHemisphere, SquareMethodNone }
+};
+
+int n_projections = sizeof(projections) / sizeof(projections[0]);
+
+#endif
diff --git a/sphere-map-plotter.cpp b/sphere-map-plotter.cpp
new file mode 100644 (file)
index 0000000..d2a4c6f
--- /dev/null
@@ -0,0 +1,309 @@
+#include <vector>
+#include <cmath>
+
+#include <QCoreApplication>
+#include <QPainter>
+#include <QFile>
+#include <QTextStream>
+
+#include "proj.hpp"
+
+#include "sphere-map-common.hpp"
+#include "pdfplot.hpp"
+
+
+/*
+ * Get land mass polygon data from buildin CSV file.
+ */
+
+const std::vector<std::vector<std::pair<double, double>>>& get_landmass_polygons()
+{
+    static std::vector<std::vector<std::pair<double, double>>> polygons;
+    if (polygons.size() == 0) {
+        /* The land mass polygon data (actually country border polygons) was
+         * downloaded from http://thematicmapping.org/downloads/world_borders.php,
+         * unzipped in a directory named 'tm', and converted to CSV with
+         * ogr2ogr -f CSV tm-world-borders-0.3.csv tm -lco GEOMETRY=AS_WKT
+         */
+        QFile f(":tm-world-borders-0.3.csv");
+        if (!f.open(QIODevice::ReadOnly)) {
+            fprintf(stderr, "cannot load land polygon data\n");
+            std::exit(1);
+        }
+        QTextStream in(&f);
+        in.readLine(); // ignore first line 'WKT,...'
+        while (!in.atEnd()) {
+            std::vector<QStringList> line_polygons_as_pair_lists;
+            QString line = in.readLine();
+            if (line.startsWith("\"POLYGON")) {
+                line.remove(0, 11); // ignore '"POLYGON (('
+                line.remove(line.indexOf(')'), (1 << 30)); // ignore ')",...' THIS ALSO THROWS AWAY INTERIOR POLYGONS!
+                QStringList pair_list = line.split(',');
+                line_polygons_as_pair_lists.push_back(pair_list);
+            } else if (line.startsWith("\"MULTIPOLYGON")) {
+                line.remove(0, 17); // ignore '"MULTIPOLYGON ((('
+                line.remove(line.indexOf(")))"), 256); // ignore ')))",<gedoense>'
+                QStringList polygon_list = line.split(")),((");
+                for (int i = 0; i < polygon_list.size(); i++) {
+                    polygon_list[i].remove(polygon_list[i].indexOf(')'), (1 << 30)); // THIS THROWS AWAY INTERIOR POLYGONS!
+                    QStringList pair_list = polygon_list[i].split(',');
+                    line_polygons_as_pair_lists.push_back(pair_list);
+                }
+            }
+            for (size_t i = 0; i < line_polygons_as_pair_lists.size(); i++) {
+                std::vector<std::pair<double, double>> polygon;
+                polygon.reserve(line_polygons_as_pair_lists[i].size());
+                for (int j = 0; j < line_polygons_as_pair_lists[i].size(); j++) {
+                    QStringList pair = line_polygons_as_pair_lists[i][j].split(' ');
+                    double lon = radians(pair[0].toDouble());
+                    double lat = radians(pair[1].toDouble());
+                    polygon.push_back(std::pair<double, double>(lat, lon));
+                }
+                polygons.push_back(polygon);
+            }
+        }
+    }
+    return polygons;
+}
+
+/*
+ * Main function.
+ */
+
+int main(int argc, char* argv[])
+{
+    QCoreApplication app(argc, argv);
+    QPdfWriter* writer;
+    QPainter* painter;
+    QPen pen;
+    QBrush brush;
+
+    const std::vector<std::vector<std::pair<double, double>>>& landmass_polygons = get_landmass_polygons();
+
+    const int lat_lines = 18;
+    const int lon_lines = 36;
+    const double map_sub_point_dist = 1e-3;
+    const int line_steps = 5000;
+
+    std::vector<std::pair<float, float>> point_list;
+    double lat, lon;
+    double x, y;
+
+    pen.setCapStyle(Qt::FlatCap);
+
+    // Loop over the map methods
+    for (int method = 0; method < n_projections; method++) {
+        // Set up the PDF painter and the projection context
+        std::string id = std::string("plot-") + projections[method].name;
+        fprintf(stderr, "%s\n", id.c_str());
+        Context ctx(projections[method].proj,
+                projections[method].params,
+                projections[method].center,
+                projections[method].layout,
+                projections[method].square_method);
+        make_pdf_painter(id,
+                (ctx.layout == LayoutHemisphere || ctx.layout == LayoutSphere) && ctx.square_method == SquareMethodNone,
+                &writer, &painter);
+        // Plot the landmass map
+        pen.setColor(QColor(128, 128, 128));
+        pen.setStyle(Qt::SolidLine);
+        pen.setWidth(1);
+        painter->setPen(pen);
+        brush.setColor(QColor(128, 128, 128));
+        brush.setStyle(Qt::SolidPattern);
+        painter->setBrush(brush);
+        for (size_t i = 0; i < landmass_polygons.size(); i++) {
+            const std::vector<std::pair<double, double>>& polygon = landmass_polygons[i];
+            // Check if this needs special handling of the south pole in the
+            // quincuncial layout.
+            // This is the case if the polygon intersects more than one of the four
+            // parts that represent the southern hemisphere.
+            bool special_handling = false;
+            if (ctx.layout == LayoutQuincuncial) {
+                unsigned int corner_seen = 0;
+                for (size_t j = 0; j < polygon.size(); j++) {
+                    lat = polygon[j].first;
+                    lon = polygon[j].second;
+                    ctx.forward(lat, lon, x, y);
+                    if (std::fabs(x) + fabs(y) <= 1.0)
+                        break;
+                    unsigned int corner;
+                    if (x > 0)
+                        corner = (y > 0 ? (1 << 0) : (1 << 3));
+                    else
+                        corner = (y > 0 ? (1 << 1) : (1 << 2));
+                    if (corner_seen & ~corner) {
+                        special_handling = true;
+                        break;
+                    }
+                    corner_seen = corner;
+                }
+            }
+            point_list.clear();
+            // find first valid point
+            size_t j;
+            for (j = 0; j < polygon.size(); j++) {
+                lat = polygon[j].first;
+                lon = polygon[j].second;
+                ctx.forward(lat, lon, x, y);
+                if (point_is_valid(x, y)) {
+                    point_list.push_back(std::pair<float, float>(x, y));
+                    break;
+                }
+            }
+            size_t first_valid = j;
+            // find next valid point
+            double last_lat = lat;
+            double last_lon = lon;
+            double last_x = x;
+            double last_y = y;
+            for (; j <= polygon.size(); j++) {
+                size_t jj = (j < polygon.size() ? j : first_valid);
+                lat = polygon[jj].first;
+                lon = polygon[jj].second;
+                ctx.forward(lat, lon, x, y);
+                if (point_is_valid(x, y)) {
+                    // plot the line from last point to current point
+                    double dist = hypot(x - last_x, y - last_y);
+                    if (dist < map_sub_point_dist)
+                        continue;
+                    int substeps = dist / map_sub_point_dist;
+                    for (int k = 1; k < substeps; k++) {
+                        double alpha = static_cast<double>(k) / substeps;
+                        double sublat = alpha * lat + (1.0 - alpha) * last_lat;
+                        double sublon = alpha * lon + (1.0 - alpha) * last_lon;
+                        double subx, suby;
+                        ctx.forward(sublat, sublon, subx, suby);
+                        if (point_is_valid(subx, suby))
+                            point_list.push_back(std::pair<float, float>(subx, suby));
+                    }
+                    if (jj != first_valid)
+                        point_list.push_back(std::pair<float, float>(x, y));
+                    last_lat = lat;
+                    last_lon = lon;
+                    last_x = x;
+                    last_y = y;
+                }
+            }
+            if (special_handling) {
+                for (unsigned int corner = 0; corner < 4; corner++) {
+                    std::vector<std::pair<float, float>> special_point_list;
+                    for (size_t j = 0; j < point_list.size(); j++) {
+                        x = point_list[j].first;
+                        y = point_list[j].second;
+                        double t;
+                        // first, map corners back to center
+                        // see inverse quincuncial mapping
+                        if (x * y >= 0) {
+                            if (x >= 0 && y >= 0) {
+                                t = -y + 1;
+                                y = -x + 1;
+                                x = t;
+                            } else {
+                                t = -y - 1;
+                                y = -x - 1;
+                                x = t;
+                            }
+                        } else {
+                            if (x < 0) {
+                                t = y - 1;
+                                y = x + 1;
+                                x = t;
+                            } else {
+                                t = y + 1;
+                                y = x - 1;
+                                x = t;
+                            }
+                        }
+                        // then map the whole polygon into the single current corner
+                        // (see forward quincuncial mapping)
+                        // let the plotting library do the appropriate clipping
+                        if (corner == 0) {
+                            t = -y + 1;
+                            y = -x + 1;
+                            x = t;
+                        } else if (corner == 1) {
+                            t = y - 1;
+                            y = x + 1;
+                            x = t;
+                        } else if (corner == 2) {
+                            t = -y - 1;
+                            y = -x - 1;
+                            x = t;
+                        } else {
+                            t = y + 1;
+                            y = x - 1;
+                            x = t;
+                        }
+                        special_point_list.push_back(std::pair<float, float>(x, y));
+                    }
+                    plot_points(painter, false, special_point_list);
+                }
+            } else {
+                plot_points(painter, false, point_list);
+            }
+        }
+        // Plot the lat/lon grid
+        pen.setColor(QColor(0, 0, 0));
+        pen.setStyle(Qt::SolidLine);
+        painter->setPen(pen);
+        brush.setStyle(Qt::NoBrush);
+        painter->setBrush(brush);
+        for (int ilat = 0; ilat <= lat_lines; ilat++) {
+            lat = M_PI_2 - ilat * M_PI / lat_lines;
+            point_list.clear();
+            for (int i = 0; i <= line_steps; i++) {
+                lon = i * 2.0 * M_PI / line_steps;
+                lon -= M_PI;
+                ctx.forward(lat, lon, x, y);
+                if (point_is_valid(x, y)) {
+                    if (point_list.size() > 0) {
+                        float dist = hypot(point_list.back().first - x, point_list.back().second - y);
+                        if (dist < map_sub_point_dist)
+                            continue;
+                    }
+                    point_list.push_back(std::pair<float, float>(x, y));
+                }
+            }
+            pen.setWidth((ilat % 3 == 0) ? 8 : 2);
+            painter->setPen(pen);
+            plot_points(painter, true, point_list);
+        }
+        for (int ilon = 0; ilon <= lon_lines; ilon++) {
+            lon = -M_PI + ilon * 2.0 * M_PI / lon_lines;
+            point_list.clear();
+            for (int i = 0; i <= line_steps; i++) {
+                if (i == 0 || i == line_steps)
+                    continue;
+#if 0
+                if (ctx.layout == LayoutSphere) {
+                    // latitude range starts 5 deg from center pole and ends at outer pole
+                    if (ctx.center == CenterNorthPole)
+                        lat = -M_PI_2 + i * (M_PI - M_PI / 36) / line_steps;
+                    else
+                        lat = -M_PI_2 + M_PI / 36 + i * (M_PI - M_PI / 36) / line_steps;
+                } else {
+                    // latitude range starts/stops 5 deg from each pole
+                    lat = -M_PI_2 + M_PI / 36 + i * (M_PI - M_PI / 18) / line_steps;
+                }
+#endif
+                lat = -M_PI_2 + i * M_PI / line_steps;
+                ctx.forward(lat, lon, x, y);
+                if (point_is_valid(x, y)) {
+                    if (point_list.size() > 0) {
+                        float dist = hypot(point_list.back().first - x, point_list.back().second - y);
+                        if (dist < map_sub_point_dist)
+                            continue;
+                    }
+                    point_list.push_back(std::pair<float, float>(x, y));
+                }
+            }
+            pen.setWidth((ilon % 3 == 0) ? 8 : 2);
+            painter->setPen(pen);
+            plot_points(painter, true, point_list);
+        }
+        finish_pdf_painter(writer, painter);
+    }
+
+    return 0;
+}
diff --git a/sphere-map-plotter.qrc b/sphere-map-plotter.qrc
new file mode 100644 (file)
index 0000000..7e90d4e
--- /dev/null
@@ -0,0 +1,5 @@
+<!DOCTYPE RCC><RCC version="1.0">
+<qresource>
+  <file>tm-world-borders-0.3.csv</file>
+</qresource>
+</RCC>
diff --git a/sphere-map-precision-test.cpp b/sphere-map-precision-test.cpp
new file mode 100644 (file)
index 0000000..f799ee4
--- /dev/null
@@ -0,0 +1,113 @@
+#include <cmath>
+#include <cstdio>
+
+#include "proj.hpp"
+
+#include "sphere-map-common.hpp"
+
+
+void latlon_to_sphere(double lat, double lon, double& u, double& v, double& w)
+{
+    u = std::cos(lat) * std::cos(lon);
+    v = std::cos(lat) * std::sin(lon);
+    w = std::sin(lat);
+}
+
+double dist_rad_sphere(double lat0, double lon0, double lat1, double lon1)
+{
+    /* angle between vectors: not precise enough!
+    double u0, v0, w0, u1, v1, w1;
+    latlon_to_sphere(lat0, lon0, u0, v0, w0);
+    latlon_to_sphere(lat1, lon1, u1, v1, w1);
+    double angle = std::acos(u0 * u1 + v0 * v1 + w0 * w1);
+    return angle;
+    */
+    double slat2 = std::sin(std::fabs(lat0 - lat1) / 2.0);
+    slat2 *= slat2;
+    double slon2 = std::sin(std::fabs(lon0 - lon1) / 2.0);
+    slon2 *= slon2;
+    double t = slat2 + std::cos(lat0) * std::cos(lat1) * slon2;
+    return 2.0 * std::asin(std::sqrt(t));
+}
+
+double check_precision(Context& ctx)
+{
+    const int xsteps = 2000;
+    const int ysteps = 2000;
+
+    double max_error = -1.0;
+    int max_considered_points = 0;
+    int considered_points = 0;
+    for (int iy = 0; iy < ysteps; iy++) {
+        double y = (iy / (ysteps - 1.0)) * 2.0 - 1.0;
+        for (int ix = 0; ix < xsteps; ix++) {
+            double x = (ix / (xsteps - 1.0)) * 2.0 - 1.0;
+            double lat, lon;
+            if ((ctx.layout == LayoutHemisphere || ctx.layout == LayoutSphere)
+                    && x * x + y * y > 1) {
+                continue;
+            }
+            max_considered_points++;
+            ctx.inverse(x, y, lat, lon);
+            if (!std::isfinite(lat) || !std::isfinite(lon)) {
+                //fprintf(stderr, "!cannot inverse map %g,%g! ", x, y);
+                return NAN;
+            }
+            double fx, fy, flat, flon;
+            ctx.forward(lat, lon, fx, fy);
+            if (!std::isfinite(fx) || !std::isfinite(fy)) {
+                // we cannot compute an error for this point
+                //fprintf(stderr, "!cannot map %.19g,%.19g (comes from %g,%g)! ", degrees(lat), degrees(lon), x, y);
+                continue;
+            }
+            ctx.inverse(fx, fy, flat, flon);
+            if (!std::isfinite(flat) || !std::isfinite(flon)) {
+                //fprintf(stderr, "!cannot inverse map %g,%g! ", fx, fy);
+                return NAN;
+            }
+            considered_points++;
+            double error = dist_rad_sphere(lat, lon, flat, flon);
+            /*
+            if (error > 1e-12) {
+                fprintf(stderr, "suspiciously large error at "
+                        "x=%.17g y=%.17g lat=%.17g lon=%.17g "
+                        "fx=%.17g fy=%.17g flat=%.17g flon=%.17g\n",
+                        x, y, degrees(lat), degrees(lon),
+                        fx, fy, degrees(flat), degrees(flon));
+            }
+            */
+            if (error > max_error) {
+                max_error = error;
+            }
+        }
+    }
+    if (considered_points < max_considered_points - max_considered_points / 100) {
+        fprintf(stderr, "!only %d out of %d points could be considered! ", considered_points, max_considered_points);
+        return NAN;
+    }
+    return max_error;
+}
+
+double in_earth_millimeters(double rad_error)
+{
+    return rad_error * 40075.017 * 1e3 * 1e3;
+}
+
+/*
+ * Main function.
+ */
+
+int main(void)
+{
+    // Loop over the projection methods
+    for (int method = 0; method < n_projections; method++) {
+        Context ctx(projections[method].proj,
+                projections[method].params,
+                projections[method].center,
+                projections[method].layout,
+                projections[method].square_method);
+        double max_error = check_precision(ctx);
+        fprintf(stderr, "%s: %g (ca. %g mm on Earth)\n", projections[method].name, max_error, in_earth_millimeters(max_error));
+    }
+    return 0;
+}
diff --git a/tm-world-borders-0.3.csv b/tm-world-borders-0.3.csv
new file mode 100644 (file)
index 0000000..031693b
--- /dev/null
@@ -0,0 +1,247 @@
+WKT,FIPS,ISO2,ISO3,UN,NAME,AREA,POP2005,REGION,SUBREGION,LON,LAT
+"MULTIPOLYGON (((-61.686668 17.024441000000138,-61.73806 16.989719,-61.82917 16.996944,-61.876114 17.016941,-61.880562 17.019722,-61.883614 17.023609,-61.885834 17.028053,-61.887222 17.033054000000106,-61.891113 17.094166,-61.887222 17.105274,-61.884171 17.109722,-61.832779 17.163887,-61.826393 17.167221,-61.794449 17.16333,-61.784172 17.158333,-61.744171 17.137218,-61.674171 17.093609,-61.67028 17.090275,-61.668892 17.084999,-61.666389 17.04583,-61.667503 17.040554,-61.682503 17.027496,-61.686668 17.024441000000138)),((-61.729172 17.608608,-61.731117 17.547222,-61.73278 17.541111,-61.738892 17.5405540000001,-61.751945 17.549442,-61.815559 17.583885,-61.834724 17.588608,-61.839447 17.586666,-61.842781 17.582775,-61.847504 17.580830000000105,-61.853058 17.583054,-61.856674 17.592499,-61.873894 17.688889,-61.875282 17.698608,-61.873062 17.703888,-61.850281 17.722775000000127,-61.845558 17.724998,-61.839172 17.724720000000104,-61.787224 17.700554,-61.783615 17.69722,-61.74334 17.653053,-61.74028 17.649166,-61.73806 17.644722000000115,-61.731674 17.624996,-61.729172 17.608608)))",AC,AG,ATG,28,Antigua and Barbuda,44,83039,19,29,-61.783,17.078
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