Cubic space division #1

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

#polygons #3D #Escher

Created by reinder on 2018/12/2
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// Cubic space division #1. Created by Reinder Nijhoff 2018
// @reindernijhoff
//
// https://turtletoy.net/turtle/4487884a96
//

const drawOutlines = false;
const hatchDensity = .0025;
const hatchImperfection = .1;
const fogExp = 1.3;

Canvas.setpenopacity(.5);

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

const lw = .25;
const ll =  5;
const dim = 12;

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) {
                if (drawOutlines) {
                    p.addOutline(i);
                } 
                p.addHatching( .5, hatchDensity*(1+i/3)*Math.pow(dist, fogExp));
                drawPolygon(turtle, p);      
            }
        }
    }
}

function drawPolygon(turtle, p) {
    for (let j=0; j<polygonList.length; j++) {
        if(!p.boolean(polygonList[j])) {
            return;
        }
    }
    p.draw(turtle, hatchImperfection);
    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
}
Polygon.prototype.addOutline = function(s=0) {
    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.addHatching = function(a,d) {
    // todo, create a tight bounding polygon, for now fill screen
    const tp = new Polygon();
    tp.createPoly(0,0,4,200,Math.PI*.5);
    const dx = Math.sin(a)*d, dy = Math.cos(a)*d;
    const cx = Math.sin(a)*200, cy = Math.cos(a)*200;
    for (let i = .5; i<150/d; i++) {
        tp.dp.push(new LineSegment([dx*i+cy,dy*i-cx], [dx*i-cy,dy*i+cx]));
        tp.dp.push(new LineSegment([-dx*i+cy,-dy*i-cx], [-dx*i-cy,-dy*i+cx]));
    }
    tp.boolean(this, false);
    this.dp = this.dp.concat(tp.dp);
}
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]) && 
                    diff != p.inside(scale2(add2(int[j],int[j+1]),.5))) {
                    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 add2=(a,b)=>[a[0]+b[0],a[1]+b[1]];
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 add3=(a,b)=>[a[0]+b[0],a[1]+b[1],a[2]+b[2]];
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;
}