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Tetrahedral Gnomonic projection Antarctic
license: mit

The Tetrahedral Gnomonic projection, published by A. J. Potter in The Geographical Teacher, Vol. 13, No. 1 (SPRING, 1925), pp. 52-56.

  • Using d3.geoPolyhedral on the tetrahedron

  • Gnomonic projection on each face

  • This is the Antarctic-centered aspect. See Potter’s North Pole aspect


Research by Philippe Rivière for d3-geo-projection issue 107.


[

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<!DOCTYPE html>
<canvas width="960" height="600"></canvas>
<script src="https://d3js.org/d3.v4.js"></script>
<script src="https://d3js.org/d3-geo-projection.v2.min.js"></script>
<script src="https://d3js.org/topojson.v2.min.js"></script>
<script src="versor.js"></script>
<style>
path {fill: none; stroke: #444; }
</style>
<script>
var canvas = d3.select("canvas"),
width = canvas.property("width"),
height = canvas.property("height"),
context = canvas.node().getContext("2d");
// retina display
var devicePixelRatio = window.devicePixelRatio || 1;
canvas.style('width', canvas.attr('width')+'px');
canvas.style('height', canvas.attr('height')+'px');
canvas.attr('width', canvas.attr('width') * devicePixelRatio);
canvas.attr('height', canvas.attr('height') * devicePixelRatio);
context.scale(devicePixelRatio,devicePixelRatio);
var pi = Math.PI, degrees = 180 / pi, asin1_3 = Math.asin(1 / 3);
var centers = [
[0, 90],
[-180, -asin1_3 * degrees],
[-60, -asin1_3 * degrees],
[60, -asin1_3 * degrees]
];
d3.geoTetrahedralGnomonic = function(faceProjection) {
var tetrahedron = [[1, 2, 3], [0, 2, 1], [0, 3, 2], [0, 1, 3]].map(function(
face
) {
return face.map(function(i) {
return centers[i];
});
});
faceProjection =
faceProjection ||
function(face) {
var c = d3.geoCentroid({ type: "MultiPoint", coordinates: face });
return d3
.geoGnomonic()
.scale(1)
.translate([0, 0])
.rotate([-c[0] * (Math.abs(c[1]) != 90), -c[1]]);
};
var faces = tetrahedron.map(function(face) {
return { face: face, project: faceProjection(face) };
});
[-1, 0, 0, 0].forEach(function(d, i) {
var node = faces[d];
node && (node.children || (node.children = [])).push(faces[i]);
});
return d3
.geoPolyhedral(
faces[0],
function(lambda, phi) {
lambda *= degrees;
phi *= degrees;
for (var i = 0; i < faces.length; i++) {
if (
d3.geoContains(
{
type: "Polygon",
coordinates: [[...tetrahedron[i], tetrahedron[i][0]]]
},
[lambda, phi]
)
) {
return faces[i];
}
}
},
pi / 3
)
.clipAngle(180)
.rotate([-30, 0])
.fitExtent([[0, 0], [width, height]], { type: "Sphere" });
};
projection = d3.geoTetrahedralGnomonic();
var init_scale = projection.scale(),
path = d3.geoPath().projection(projection).context(context);
d3.json("https://unpkg.com/world-atlas@1/world/110m.json", function(
error,
world
) {
if (error) throw error;
var land = topojson.feature(world, world.objects.countries);
render = function() {
context.fillStyle = "#fff";
context.fillRect(0, 0, width, height);
context.beginPath();
path({type:"Sphere"});
context.strokeStyle = "black";
context.stroke(), context.clip(), context.closePath();
context.beginPath();
path(land);
context.fillStyle = "#000";
context.fill(), context.closePath();
context.beginPath();
path(d3.geoGraticule()());
context.strokeStyle = "#777";
context.stroke(), context.closePath();
};
render();
});
canvas.call(d3.drag().on("start", dragstarted).on("drag", dragged));
var render = function() {},
v0, // Mouse position in Cartesian coordinates at start of drag gesture.
r0, // Projection rotation as Euler angles at start.
q0; // Projection rotation as versor at start.
function dragstarted() {
v0 = versor.cartesian(projection.invert(d3.mouse(this)));
r0 = projection.rotate();
console.log("r0", r0);
q0 = versor(r0);
}
function dragged() {
var v1 = versor.cartesian(projection.rotate(r0).invert(d3.mouse(this))),
q1 = versor.multiply(q0, versor.delta(v0, v1)),
r1 = versor.rotation(q1);
projection.rotate(r1);
render();
}
</script>
// Version 0.0.0. Copyright 2017 Mike Bostock.
(function(global, factory) {
typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory() :
typeof define === 'function' && define.amd ? define(factory) :
(global.versor = factory());
}(this, (function() {'use strict';
var acos = Math.acos,
asin = Math.asin,
atan2 = Math.atan2,
cos = Math.cos,
max = Math.max,
min = Math.min,
PI = Math.PI,
sin = Math.sin,
sqrt = Math.sqrt,
radians = PI / 180,
degrees = 180 / PI;
// Returns the unit quaternion for the given Euler rotation angles [λ, φ, γ].
function versor(e) {
var l = e[0] / 2 * radians, sl = sin(l), cl = cos(l), // λ / 2
p = e[1] / 2 * radians, sp = sin(p), cp = cos(p), // φ / 2
g = e[2] / 2 * radians, sg = sin(g), cg = cos(g); // γ / 2
return [
cl * cp * cg + sl * sp * sg,
sl * cp * cg - cl * sp * sg,
cl * sp * cg + sl * cp * sg,
cl * cp * sg - sl * sp * cg
];
}
// Returns Cartesian coordinates [x, y, z] given spherical coordinates [λ, φ].
versor.cartesian = function(e) {
var l = e[0] * radians, p = e[1] * radians, cp = cos(p);
return [cp * cos(l), cp * sin(l), sin(p)];
};
// Returns the Euler rotation angles [λ, φ, γ] for the given quaternion.
versor.rotation = function(q) {
return [
atan2(2 * (q[0] * q[1] + q[2] * q[3]), 1 - 2 * (q[1] * q[1] + q[2] * q[2])) * degrees,
asin(max(-1, min(1, 2 * (q[0] * q[2] - q[3] * q[1])))) * degrees,
atan2(2 * (q[0] * q[3] + q[1] * q[2]), 1 - 2 * (q[2] * q[2] + q[3] * q[3])) * degrees
];
};
// Returns the quaternion to rotate between two cartesian points on the sphere.
versor.delta = function(v0, v1) {
var w = cross(v0, v1), l = sqrt(dot(w, w));
if (!l) return [1, 0, 0, 0];
var t = acos(max(-1, min(1, dot(v0, v1)))) / 2, s = sin(t); // t = θ / 2
return [cos(t), w[2] / l * s, -w[1] / l * s, w[0] / l * s];
};
// Returns the quaternion that represents q0 * q1.
versor.multiply = function(q0, q1) {
return [
q0[0] * q1[0] - q0[1] * q1[1] - q0[2] * q1[2] - q0[3] * q1[3],
q0[1] * q1[0] + q0[0] * q1[1] + q0[2] * q1[3] - q0[3] * q1[2],
q0[0] * q1[2] - q0[1] * q1[3] + q0[2] * q1[0] + q0[3] * q1[1],
q0[0] * q1[3] + q0[1] * q1[2] - q0[2] * q1[1] + q0[3] * q1[0]
];
};
function cross(v0, v1) {
return [
v0[1] * v1[2] - v0[2] * v1[1],
v0[2] * v1[0] - v0[0] * v1[2],
v0[0] * v1[1] - v0[1] * v1[0]
];
}
function dot(v0, v1) {
return v0[0] * v1[0] + v0[1] * v1[1] + v0[2] * v1[2];
}
return versor;
})));
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