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Complex logarithm projection
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‎README.md‎

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The Cox conformal projection.
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<a href="#geoComplexLog" name="geoComplexLog">#</a> d3.<b>geoComplexLog</b>([<i>planarProjectionRaw</i>[<i>, cutoffLatitude</i>]]) · [Source](src/complexLog.js), [Example](https://cgmi.github.io/complex-log-projection/)
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<br><a href="#geoComplexLogRaw" name="geoComplexLogRaw">#</a> d3.<b>geoComplexLogRaw</b>([<i>planarProjectionRaw</i>])
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[<img src="img/complexLog.png" width="480" height="250">](https://cgmi.github.io/complex-log-projection/)
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Complex logarithmic view. This projection is based on the papers by Joachim Böttger et al.:
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* [Detail‐In‐Context Visualization for Satellite Imagery (2008)](https://doi.org/10.1111/j.1467-8659.2008.01156.x)
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* [Complex Logarithmic Views for Small Details in Large Contexts (2006)](https://doi.org/10.1109/TVCG.2006.126)
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The specified raw projection <i>planarProjectionRaw</i> is used to project onto the complex plane on which the complex logarithm is applied.
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Recommended are [azimuthal equal-area](https://github.com/d3/d3-geo#geoAzimuthalEqualAreaRaw) (default) or [azimuthal equidistant](https://github.com/d3/d3-geo#geoAzimuthalEquidistantRaw).
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<i>cutoffLatitude</i> is the latitude relative to the projection center at which to cutoff/clip the projection, lower values result in more detail around the projection center. Value must be < 0 because complex log projects the origin to infinity.
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<a href="#complexLog_planarProjectionRaw" name="complexLog_planarProjectionRaw">#</a> <i>complexLog</i>.<b>planarProjectionRaw</b>([<i>projectionRaw</i>])
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If <i>projectionRaw</i> is specified, sets the planar raw projection. See above. If <i>projectionRaw</i> is not specified, returns the current planar raw projection.
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<a href="#complexLog_cutoffLatitude" name="complexLog_cutoffLatitude">#</a> <i>complexLog</i>.<b>cutoffLatitude</b>([<i>latitude</i>])
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If <i>latitude</i> is specified, sets the cutoff latitude. See above. If <i>latitude</i> is not specified, returns the current cutoff latitude.

‎img/complexLog.png‎

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‎src/complexLog.js‎

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/*
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* Complex logarithm projection
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*
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* Based on the following papers by Joachim Böttger et al.:
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* - Detail‐In‐Context Visualization for Satellite Imagery (2008) (https://doi.org/10.1111/j.1467-8659.2008.01156.x)
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* - Complex Logarithmic Views for Small Details in Large Contexts (2006) (https://doi.org/10.1109/TVCG.2006.126)
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*
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* Implemented for d3 by Matthias Albrecht and Jochen Görtler (2019)
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*
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*/
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import { geoProjectionMutator as projectionMutator, geoAzimuthalEqualAreaRaw as azimuthalEqualAreaRaw } from "d3-geo";
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import { abs, sin, cos, pi, exp, atan2 } from "./math.js";
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import { complexMul, complexLogHypot } from "./complex.js";
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import { default as clipPolygon } from "./clip/polygon.js";
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// Default planar projection and cutoff latitude, see below for an explanation of these settings.
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var DEFAULT_PLANAR_PROJECTION_RAW = azimuthalEqualAreaRaw;
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var DEFAULT_CUTOFF_LATITUDE = -0.05;
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// Offset used to prevent logarithm of 0.
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var CARTESIAN_OFFSET = 1e-10;
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// Projection parameters for the default 960x500 projection area.
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var DEFAULT_PROJECTION_PARAMS = {
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angle: 90,
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center: [0, 5.022570623227068],
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scale: 79.92959180396787,
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translate: [479.9999905630355, 250.35977064160338]
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}
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// Vertices of the clipping polygon in spherical coordinates.
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// It contains the whole world except a small strip along longitude 0/180 crossing the south pole.
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var CLIP_POLY_SPHERICAL = [
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[-180, -1e-4],
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[180, -1e-4],
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[1e-4, DEFAULT_CUTOFF_LATITUDE],
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[-1e-4, DEFAULT_CUTOFF_LATITUDE]
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]
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// Clipping polygon precision.
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var N_SIDE = 5;
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var N_BOTTOM = 50;
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export function complexLogRaw(planarProjectionRaw = DEFAULT_PLANAR_PROJECTION_RAW) {
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function forward(lambda, phi) {
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// Project on plane.
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// Interpret projected point on complex plane.
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var aziComp = planarProjectionRaw(lambda, phi);
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// Rotate by -90 degrees in complex plane so the following complex log projection will be horizontally centered
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aziComp = complexMul(aziComp, [cos(-pi / 2), sin(-pi / 2)]);
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// Small offset to prevent logarithm of 0.
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if (aziComp[0] == 0 && aziComp[1] == 0) {
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aziComp[0] += CARTESIAN_OFFSET;
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aziComp[1] += CARTESIAN_OFFSET;
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}
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// Apply complex logarithm.
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var logComp = [complexLogHypot(aziComp[0], aziComp[1]), atan2(aziComp[1], aziComp[0])];
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return logComp;
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}
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function invert(x, y) {
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// Inverse complex logarithm (complex exponential function).
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var invLogComp = [exp(x) * cos(y), exp(x) * sin(y)];
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// Undo rotation.
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invLogComp = complexMul(invLogComp, [cos(pi / 2), sin(pi / 2)]);
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// Invert azimuthal equal area.
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return planarProjectionRaw.invert(invLogComp[0], invLogComp[1]);
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}
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forward.invert = invert;
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return forward;
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}
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export default function(planarProjectionRaw = DEFAULT_PLANAR_PROJECTION_RAW, cutoffLatitude = DEFAULT_CUTOFF_LATITUDE) {
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var mutator = projectionMutator(complexLogRaw);
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var projection = mutator(planarProjectionRaw);
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// Projection used to project onto the complex plane.
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projection.planarProjectionRaw = function(_) {
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return arguments.length ? clipped(mutator(planarProjectionRaw = _)) : planarProjectionRaw;
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}
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// Latitude relative to the projection center at which to cutoff/clip the projection, lower values result in more detail around the projection center.
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// Value must be < 0 because complex log projects the origin to infinity.
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projection.cutoffLatitude = function(_) {
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return arguments.length ? (cutoffLatitude = _, clipped(mutator(planarProjectionRaw))) : cutoffLatitude;
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}
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function clipped(projection) {
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var angle = projection.angle();
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var scale = projection.scale();
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var center = projection.center();
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var translate = projection.translate();
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var rotate = projection.rotate();
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projection
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.angle(DEFAULT_PROJECTION_PARAMS.angle)
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.scale(1)
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.center([0, 0])
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.rotate([0, 0])
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.translate([0, 0])
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.preclip();
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// These are corner vertices of a rectangle in the projected complex log view.
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var topLeft = projection(CLIP_POLY_SPHERICAL[0]);
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var topRight = projection(CLIP_POLY_SPHERICAL[1]);
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var bottomRight = projection([CLIP_POLY_SPHERICAL[2][0], cutoffLatitude]);
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var bottomLeft = projection([CLIP_POLY_SPHERICAL[3][0], cutoffLatitude]);
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var width = abs(topRight[0] - topLeft[0]);
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var height = abs(bottomRight[1] - topRight[1]);
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// Prevent overlapping polygons that result from paths that go from one side to the other,
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// so cut along 180°/-180° degree line (left and right in complex log projected view).
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// This means cutting against a rectangular shaped polygon in the projected view.
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// The following generator produces a polygon that is shaped like this:
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//
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// Winding order: ==>
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//
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// ******************|
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// | |
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// | |
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// | |
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// | |
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// | |
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// |------------------
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//
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// N_SIDE determines how many vertices to insert along the sides (marked as | above).
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// N_BOTTOM determines how many vertices to insert along the bottom (marked as - above).
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//
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// The resulting polygon vertices are back-projected to spherical coordinates.
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var polygon = {
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type: "Polygon",
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coordinates: [
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[
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topLeft,
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...Array.from({length: N_SIDE}, (_, t) => [bottomRight[0], bottomRight[1] - height * (N_SIDE- t) / N_SIDE]),
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...Array.from({length: N_BOTTOM}, (_, t) => [bottomRight[0] - width * t / N_BOTTOM, bottomRight[1]]),
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...Array.from({length: N_SIDE}, (_, t) => [bottomLeft[0], bottomLeft[1] - height * t / N_SIDE]),
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topLeft
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].map(point => projection.invert(point))
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]
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};
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return projection
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.angle(angle)
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.scale(scale)
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.center(center)
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.translate(translate)
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.rotate(rotate)
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.preclip(clipPolygon(polygon));
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}
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// The following values are for the default 960x500 projection area
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return clipped(projection)
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.angle(DEFAULT_PROJECTION_PARAMS.angle)
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.center(DEFAULT_PROJECTION_PARAMS.center)
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.scale(DEFAULT_PROJECTION_PARAMS.scale)
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.translate(DEFAULT_PROJECTION_PARAMS.translate);
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}

‎src/index.js‎

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export {default as geoImago, imagoBlock as geoImagoBlock, imagoRaw as geoImagoRaw} from "./imago";
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export {default as geoCubic} from "./cubic";
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export {default as geoCahillKeyes, cahillKeyesRaw as geoCahillKeyesRaw} from "./cahillKeyes";
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export {default as geoComplexLog, complexLogRaw as geoComplexLogRaw} from "./complexLog";

‎test/compare-images‎

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airocean \
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cox \
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cahillKeyes \
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complexLog \
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cubic \
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dodecahedral \
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icosahedral \

‎test/invert-test.js‎

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test.end();
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});
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tape("inverse complex log", function(test) {
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[ d3.geoComplexLog() ]
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.forEach(function(projection) {
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[ [0, 0], [-23, 12], [10,10], [100,-45] ]
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.forEach(function(location) {
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projection.angle(Math.random()*360);
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test.projectionEqual(projection, location, projection(location), 1e-5);
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});
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});
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test.end();
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});
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‎test/render-world‎

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case "modifiedStereographicGs50": outline = graticule.extent([[-180, 15], [-50, 75]]).outline(); break;
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case "modifiedStereographicMiller": outline = graticule.extent([[-40, -40], [80, 80]]).outline(); break;
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case "tetrahedralLeeSouth": projectionSymbol = "geoTetrahedralLee"; rotate = [-30,0]; angle = -30; translate = [599.204, 98.0632]; break;
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// Outline cannot be rendered properly using complex log
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case "complexLog": outline = {type: "Point", coordinates: []}; break;
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}
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var projection = d3[projectionSymbol]().precision(0.1),

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