### Cubic space division #2

Based on Cubic space division by Escher: wikiart.org/en/m-c-escher/cubic-space-division.

Cubic space division 1: turtletoy.net/turtle/4487884a96

#polygons #3D #Escher

Created by reinder on 2018/12/2
2056
2

```// Cubic space division #2. Created by Reinder Nijhoff 2018
// @reindernijhoff
//
// https://turtletoy.net/turtle/25b7bc4d43
//

const drawImperfection = 0; // try 1`

Canvas.setpenopacity(.75);

const turtle = new Turtle();
let viewProjectionMatrix;
const polygonList = [];

const lw = .25;
const ll =  5;
const dim = 14;

function walk(i) {
if (i == 0) {
setupCamera();
}

for (let j=0; j<dim; j++) {
const x = 18.5 -(i%dim) * (2*ll+1);
const z = 50 -((i/dim)|0) * (2*ll+1);
const y = 32 -(j) * (2*ll+1);

drawCube(turtle,   x,y,z,    1,1,1);
drawCube(turtle, x-ll,y,z, ll,lw,lw);
drawCube(turtle, x,y,z-ll, lw,lw,ll);
drawCube(turtle, x,y-ll,z, lw,ll,lw);
}
return i < dim*dim;
}

function drawCube(turtle, x, y, z, w, h, d) {
const vl = [
[x-w,y-h,z+d,1],
[x+w,y-h,z+d,1],
[x+w,y+h,z+d,1],
[x-w,y+h,z+d,1],
[x-w,y+h,z-d,1],
[x+w,y+h,z-d,1],
[x+w,y-h,z-d,1],
];
const il = [0,1,2,3, 1,2,5,6, 5,2,3,4];

for (let i=0; i<3; i++) {
let vis = true;
let dist = 0;
const p = new Polygon();
for (let j=0; j<4; j++) {
const v = transform4(vl[il[j+i*4]], viewProjectionMatrix);
p.cp.push([v[0]/v[3]*50, -v[1]/v[3]*50]);
if (v[3] < 0) vis = false;
dist = Math.max(dist, v[3]);
}
if (vis) {
vis = false;
for (let j=0; j<p.cp.length; j++) {
if (Math.abs(p.cp[j][0]) < 100 && Math.abs(p.cp[j][1]) < 100) {
vis = true;
}
}
if (vis) {
drawPolygon(turtle, p);
}
}
}
}

function drawPolygon(turtle, p) {
for (let j=0; j<polygonList.length; j++) {
if(!p.boolean(polygonList[j])) {
return;
}
}
p.draw(turtle, drawImperfection);
polygonList.push(p);
}

function setupCamera() {
viewMatrix = lookAt4m([15,30,50], [-10,0,0], [0,1,0]);
projectionMatrix = perspective4m(0.3, 1, .1);
viewProjectionMatrix = multiply4m(projectionMatrix, viewMatrix);
}

// polygon functions
function LineSegment(p1, p2) {
this.p1 = p1;
this.p2 = p2;
}
function Polygon() {
this.cp = []; // clip path: array of [x,y] pairs
this.dp = []; // 2d line to draw: array of linesegments
}
for (let i=s, l=this.cp.length; i<l; i++) {
this.dp.push(new LineSegment(this.cp[i], this.cp[(i+1)%l]));
}
}
Polygon.prototype.createPoly = function(x,y,c,r,a) {
this.cp = [];
for (let i=0; i<c; i++) {
this.cp.push( [x + Math.sin(i*Math.PI*2/c+a) * r, y + Math.cos(i*Math.PI*2/c+a) * r] );
}
}
Polygon.prototype.draw = function(t, inp=0) {
if (this.dp.length ==0) {
return;
}
for (let i=0, l=this.dp.length; i<l; i++) {
const d = this.dp[i];
if (!equal2(d.p1, t.pos())) {
t.penup();
t.goto([d.p1[0]+inp*(Math.random()-.5), d.p1[1]+inp*(Math.random()-.5)]);
t.pendown();
}
t.goto([d.p2[0]+inp*(Math.random()-.5), d.p2[1]+inp*(Math.random()-.5)]);
}
}
Polygon.prototype.inside = function(p) {
// find number of intersections from p to far away - if even you're outside
const p1 = [0, -1000];
let int = 0;
for (let i=0, l=this.cp.length; i<l; i++) {
if (segment_intersect2(p, p1, this.cp[i], this.cp[(i+1)%l])) {
int ++;
}
}
return int & 1;
}
Polygon.prototype.boolean = function(p, diff = true) {
// very naive polygon diff algorithm - made this up myself
const ndp = [];
for (let i=0, l=this.dp.length; i<l; i++) {
const ls = this.dp[i];

// find all intersections with clip path
const int = [];
for (let j=0, cl=p.cp.length; j<cl; j++) {
const pint = segment_intersect2(ls.p1,ls.p2,p.cp[j],p.cp[(j+1)%cl]);
if (pint) {
int.push(pint);
}
}
if (int.length == 0) { // 0 intersections, inside or outside?
if (diff != p.inside(ls.p1)) {
ndp.push(ls);
}
} else {
int.push(ls.p1); int.push(ls.p2);
// order intersection points on line ls.p1 to ls.p2
const cmp = sub2(ls.p2,ls.p1);
int.sort((a,b) => dot2(sub2(a,ls.p1),cmp)-dot2(sub2(b,ls.p1),cmp));

for (let j=0; j<int.length-1; j++) {
if (!equal2(int[j], int[j+1]) &&
ndp.push(new LineSegment(int[j], int[j+1]));
}
}
}
}
this.dp = ndp;
return this.dp.length > 0;
}

// vec2 functions
const equal2=(a,b)=>.001>dist_sqr2(a,b);
const scale2=(a,b)=>[a[0]*b,a[1]*b];
const sub2=(a,b)=>[a[0]-b[0],a[1]-b[1]];
const dot2=(a,b)=>a[0]*b[0]+a[1]*b[1];
const dist_sqr2=(a,b)=>(a[0]-b[0])*(a[0]-b[0])+(a[1]-b[1])*(a[1]-b[1]);
const segment_intersect2=(a,b,d,c)=>{
const e=(c[1]-d[1])*(b[0]-a[0])-(c[0]-d[0])*(b[1]-a[1]);
if(0==e)return false;
c=((c[0]-d[0])*(a[1]-d[1])-(c[1]-d[1])*(a[0]-d[0]))/e;
d=((b[0]-a[0])*(a[1]-d[1])-(b[1]-a[1])*(a[0]-d[0]))/e;
return 0<=c&&1>=c&&0<=d&&1>=d?[a[0]+c*(b[0]-a[0]),a[1]+c*(b[1]-a[1])]:false;
}
// vec3 functions
const scale3=(a,b)=>[a[0]*b,a[1]*b,a[2]*b];
const len3=(a)=>Math.sqrt(dot3(a,a));
const normalize3=(a)=>scale3(a,1/len3(a));
const sub3=(a,b)=>[a[0]-b[0],a[1]-b[1],a[2]-b[2]];
const dot3=(a,b)=>a[0]*b[0]+a[1]*b[1]+a[2]*b[2];
const cross3=(a,b)=>[a[1]*b[2]-a[2]*b[1],a[2]*b[0]-a[0]*b[2],a[0]*b[1]-a[1]*b[0]]
// vec4 functions
const transform4=(a,b)=>{
const d=new Float32Array(4);
for(let c=0;4>c;c++)d[c]=b[c]*a[0]+b[c+4]*a[1]+b[c+8]*a[2]+b[c+12]*a[3];
return d;
}
// mat4 functions
const lookAt4m=(a,b,d)=>{ // pos, lookAt, up
const c=new Float32Array(16);
b=normalize3(sub3(a,b));
d=normalize3(cross3(d,b));
const e=normalize3(cross3(b,d));
c[0]=d[0];c[1]=e[0];c[2]=b[0];c[3]=0;
c[4]=d[1];c[5]=e[1];c[6]=b[1];c[7]=0;
c[8]=d[2];c[9]=e[2];c[10]=b[2];c[11]=0;
c[12]=-(d[0]*a[0]+d[1]*a[1]+d[2]*a[2]);
c[13]=-(e[0]*a[0]+e[1]*a[1]+e[2]*a[2]);
c[14]=-(b[0]*a[0]+b[1]*a[1]+b[2]*a[2]);
c[15]=1;
return c;
}
const multiply4m=(a,b)=>{
const d=new Float32Array(16);
for(let c=0;16>c;c+=4)
for(let e=0;4>e;e++)
d[c+e]=b[c+0]*a[0+e]+b[c+1]*a[4+e]+b[c+2]*a[8+e]+b[c+3]*a[12+e];
return d;
}
const perspective4m=(a,b,d)=>{ // fovy, aspect. near
const c=(new Float32Array(16)).fill(0,0);
c[5]=1/Math.tan(a/2);
c[0]=c[5]/b;
c[10]=c[11]=-1;
c[14]=-2*d;
return c;
}```