4D to 3D transformation

TODO.md

@@ -12,4 +12,13 @@ To-do:
 
 * algorithm for 4d -> 3d transform
 
-* 4D shape -> 3D projection -> wireframe -> three.js
+--DONE
+
+* 4D shape -> 3D projection -> wireframe -> three.js
+
+--DONE
+
+Next:
+
+how to do this efficiently so that we can rotate the shape in 4d and have
+this animate in 3d?

main.js

@@ -3,6 +3,72 @@ import * as THREE from 'three';
 const NODE_SIZE = 0.07;
 const LINK_SIZE = 0.02;
 
+const HYPERPLANE = 4;
+
+const r5 = Math.sqrt(5);
+
+const CELL5 = {
+	nodes: [
+      {id:1, x: 1, y: 1, z: 1, w: -1 / r5 },
+      {id:2, x: 1, y: -1, z: -1, w: -1 / r5 },
+      {id:3, x: -1, y: 1, z: -1, w: -1 / r5 },
+      {id:4, x: -1, y: -1, z: 1, w: -1 / r5 },
+      {id:5, x: 0, y: 0, z: 0, w: 4 / r5 },
+	],
+	links: [
+		{ id:1, source:1, target: 2},
+		{ id:2, source:1, target: 3},
+		{ id:3, source:1, target: 4},
+		{ id:4, source:1, target: 5},
+		{ id:5, source:2, target: 3},
+		{ id:6, source:2, target: 4},
+		{ id:7, source:2, target: 5},
+		{ id:8, source:3, target: 4},
+		{ id:9, source:3, target: 5},
+		{ id:10, source:4, target: 5},
+		]
+
+	};
+
+
+const CELL16 = {
+	nodes: [
+		{ id: 1, x: 0, y: -1, z: 0, w: 0 },
+		{ id: 2, x: 0, y: 0, z: -1, w: 0 },
+		{ id: 3, x: -1, y: 0, z: 0, w: 0 },
+		{ id: 4, x: 0, y: 0, z: 1, w: 0 },
+		{ id: 5, x: 1, y: 0, z: 0, w: 0 },
+		{ id: 6, x: 0, y: 1, z: 0, w: 0 },
+		{ id: 7, x: 0, y: 0, z: 0, w: -1 },
+		{ id: 8, x: 0, y: 0, z: 0, w: 1 },
+	],
+	links: [
+		{ id: 1, source: 1, target: 2 },
+		{ id: 2, source: 1, target: 3 },
+		{ id: 3, source: 1, target: 4 },
+		{ id: 4, source: 1, target: 5 },
+		{ id: 5, source: 2, target: 3 },
+		{ id: 6, source: 3, target: 4 },
+		{ id: 7, source: 4, target: 5 },
+		{ id: 8, source: 5, target: 2 },
+		{ id: 9, source: 2, target: 6 },
+		{ id: 10, source: 3, target: 6 },
+		{ id: 11, source: 4, target: 6 },
+		{ id: 12, source: 5, target: 6 },
+		{ id: 13, source: 1, target: 7 },
+		{ id: 14, source: 1, target: 8 },
+		{ id: 15, source: 2, target: 7 },
+		{ id: 16, source: 2, target: 8 },
+		{ id: 17, source: 3, target: 7 },
+		{ id: 18, source: 3, target: 8 },
+		{ id: 19, source: 4, target: 7 },
+		{ id: 20, source: 4, target: 8 },
+		{ id: 21, source: 5, target: 7 },
+		{ id: 22, source: 5, target: 8 },
+		{ id: 23, source: 6, target: 7 },
+		{ id: 25, source: 6, target: 8 },
+	]
+};
 
 function makeLink(link_m, n1, n2) {
 	const length = n1.distanceTo(n2);
@@ -21,10 +87,10 @@ function makeLink(link_m, n1, n2) {
 
 
 
-function makeWireFrame(node_m, link_m, graph) {
+function makeWireFrame(node_m, link_m, graph3) {
 	const nodeids = {}
 	const group = new THREE.Group();
-	for ( const n of graph.nodes ) {
+	for ( const n of graph3.nodes ) {
 		const v3 = new THREE.Vector3(n.x, n.y, n.z);
 		nodeids[n.id] = v3.clone();
 		const geometry = new THREE.SphereGeometry(NODE_SIZE);
@@ -32,7 +98,7 @@ function makeWireFrame(node_m, link_m, graph) {
 		sphere.position.copy(v3);
 		group.add(sphere);
 	}
-	for ( const l of graph.links ) {
+	for ( const l of graph3.links ) {
 		const link = makeLink(link_m, nodeids[l.source], nodeids[l.target]);
 		group.add(link);
 	}
@@ -40,6 +106,44 @@ function makeWireFrame(node_m, link_m, graph) {
 }
 
 
+
+
+function fourDtoV3(x, y, z, w) {
+	const k = HYPERPLANE / (HYPERPLANE + w);
+	return new THREE.Vector3(x * k, y * k, z * k);
+}
+
+
+function nodes4tonodes3(graph4) {
+	return graph4.map((n) => {
+		return { id: n.id, v3: fourDtoV3(n.x, n.y, n.z, n.w) }
+	})
+}
+
+
+// TODO - turn W into a parameter for the node size
+
+function makeShape4D(node_m, link_m, graph4) {
+	const nodes3 = nodes4tonodes3(graph4.nodes);
+	const nodeids = {}
+	const group = new THREE.Group();
+	for ( const n of nodes3 ) {
+		nodeids[n.id] = n.v3;
+		const geometry = new THREE.SphereGeometry(NODE_SIZE);
+		const sphere = new THREE.Mesh(geometry, node_m);
+		sphere.position.copy(n.v3);
+		group.add(sphere);
+	}
+	for ( const l of graph4.links ) {
+		const link = makeLink(link_m, nodeids[l.source], nodeids[l.target]);
+		group.add(link);
+	}
+	return group;
+
+}
+
+
+
 const scene = new THREE.Scene();
 const camera = new THREE.PerspectiveCamera( 75, window.innerWidth / window.innerHeight, 0.1, 1000 );
 
@@ -68,31 +172,12 @@ const link_m = new THREE.MeshStandardMaterial(
 
 link_m.metalness = 0.4;
 link_m.roughness = 0.0;
+link_m.transparent = true;
+link_m.opacity = 0.3;
 
-const shape = makeWireFrame(node_m, link_m, {
+
-	nodes: [
+
-		{ id: 1, x: 0, y: -1, z: 0 },
+const shape = makeShape4D(node_m, link_m, CELL5);
-		{ id: 2, x: 0, y: 0, z: -1 },
-		{ id: 3, x: -1, y: 0, z: 0 },
-		{ id: 4, x: 0, y: 0, z: 1 },
-		{ id: 5, x: 1, y: 0, z: 0 },
-		{ id: 6, x: 0, y: 1, z: 0 },
-	],
-	links: [
-		{ id: 1, source: 1, target: 2 },
-		{ id: 2, source: 1, target: 3 },
-		{ id: 3, source: 1, target: 4 },
-		{ id: 4, source: 1, target: 5 },
-		{ id: 5, source: 2, target: 3 },
-		{ id: 6, source: 3, target: 4 },
-		{ id: 7, source: 4, target: 5 },
-		{ id: 8, source: 5, target: 2 },
-		{ id: 9, source: 2, target: 6 },
-		{ id: 10, source: 3, target: 6 },
-		{ id: 11, source: 4, target: 6 },
-		{ id: 12, source: 5, target: 6 },
-	]
-});
 
 
 
@@ -105,6 +190,8 @@ camera.position.z = 3;
 
 let tick = 0;
 
+// note: for 4d stuff we'll need to redraw the whole thing I think
+
 function animate() {
 	requestAnimationFrame( animate );