### Bezier Truchet tiles

Bezier curves are used to generate quad, double quad, double triangle or hexagon Truchet tiles.

#truchet #bezier

```// Bezier Truchet tiles. Created by Reinder Nijhoff 2019 - @reindernijhoff
//
// https://turtletoy.net/turtle/f107e05a76
//

const turtle = new Turtle();

const scale = 12;
const type = 1; // min=0, max=4, step=1 (Quad, Double Quad, Double Quad / Brick layout, Double Triangle, Hexagon)
const lineWidth = .4; // min=0.1, max=0.4, step=0.01
const curviness = .95; // min=0.0, max=1.0, step=0.01
const innerLines = 2; // min=0, max=9, step=1
const lineWGradient = .0; // min=-1.0, max=1.0, step=0.01

const generateTile = generateTileFactory(type);

function drawTile(t, tile) {
if (Math.abs(scale*tile.center[0]) > 100+scale ||
Math.abs(scale*tile.center[1]) > 100+scale) return; // early discard

const polys  = new Polygons();
const lw = lineWidth * tile.lineWidth;
// transform
const ts  = p => [scale*(p[0]+tile.center[0]),scale*(p[1]+tile.center[1])];
// vec2 helper functions
const add = (a, b) => [a[0]+b[0], a[1]+b[1]];
const sub = (a, b) => [a[0]-b[0], a[1]-b[1]];
const scl = (a, b) => [a[0]*b, a[1]*b];
const dst = (a, b) => Math.sqrt((a[0]-b[0])**2 + (a[1]-b[1])**2);
const bez = (p0, p1, p2, p3, t) => {
const k = 1 - t;
return [    k*k*k*p0[0] + 3*k*k*t*p1[0] + 3*k*t*t*p2[0] + t*t*t*p3[0],
k*k*k*p0[1] + 3*k*k*t*p1[1] + 3*k*t*t*p2[1] + t*t*t*p3[1] ];
}
// helper function: add a bezier curve to a polygon p.
const addBezier = (p, p0, d0, p1, d1, dist, asEdge, asLine) => {
// scale dist based on x and lineWidthGradient
const dist0 = dist * (ts(p0)[0]/200 * -lineWGradient + 1 - .5*Math.abs(lineWGradient));
const dist1 = dist * (ts(p1)[0]/200 * -lineWGradient + 1 - .5*Math.abs(lineWGradient));
// calculate start, end and control points for bezier
const sp = sub(p0, scl([d0[1],-d0[0]], dist0)),
curve = curviness*(dst(sp,ep)**(2/3))*tile.lineWidth,

const points = [];
for (let i=0, steps=10; i<=steps; i++) {
points.push(ts(bez(sp, sc, ec, ep, i/steps)));
}
if (asLine) {
for (let i=0, steps=10; i<steps; i++) p.addSegments(points[i],points[i+1]);
}
}
// shuffle points of tile -> this gives the random connections
const p = tile.points.sort(() => Math.random()-.5);

// create and draw a bezier-based polygon for each connection
for (let i=0; i<p.length; i+=2) {
const s = p[i+0], e = p[i+1], l = polys.create();
addBezier(l, s[0], s[1], e[0], e[1], lw, true, true);
addBezier(l, e[0], e[1], s[0], s[1], lw, true, true);
for (let j=0; j<innerLines; j++) {
addBezier(l, e[0], e[1], s[0], s[1], 2*lw*(j+1)/(innerLines+1)-lw, false, true);
}
polys.draw(t, l);
}
}

function walk(i) {
const s = (200/scale|0)+2;
const y = Math.floor(i/(s*2))-s, x = (i % (s*2)) - s;

drawTile(turtle, generateTile(x,y));

return i < s*s*4;
}

// A tile has a center and a set of points (positions + directions). The points will be used as
// start or end point of the bezier curves.
function generateTileFactory(t) {
switch(t) {
case 0: return (x,y) => { return { // quad
center: [x,y],
lineWidth: 1,
points: [[[0,.5],[0,-1]], [[0,-.5],[0,1]], [[.5,0],[-1,0]],[[-.5,0],[1,0]]]
}};
case 1: return (x,y) => { return { // double quad
center: [x,y],
lineWidth: .5,
points: [[[.25,.5],[0,-1]], [[.25,-.5],[0,1]], [[.5,.25],[-1,0]],[[-.5,.25],[1,0]],
[[-.25,.5],[0,-1]], [[-.25,-.5],[0,1]], [[.5,-.25],[-1,0]],[[-.5,-.25],[1,0]]]
}};
case 2: return (x,y) => { return { // double quad - shifted
center: [x + (y % 2 == 0 ? .5 : 0),y],
lineWidth: .5,
points: [[[.25,.5],[0,-1]], [[.25,-.5],[0,1]], [[.5,.25],[-1,0]],[[-.5,.25],[1,0]],
[[-.25,.5],[0,-1]], [[-.25,-.5],[0,1]], [[.5,-.25],[-1,0]],[[-.5,-.25],[1,0]]]
}};
case 3: return (x,y) => { // double triangle
const h = Math.sqrt(3)/4;
let points = [[[-.25,-h],[0,1]],[[.25,-h],[0,1]],
[[-1/3,-h/3],[2*h,-.5]],[[-1/6,h/3],[2*h,-.5]],
[[1/3,-h/3],[-2*h,-.5]],[[1/6,h/3],[-2*h,-.5]]];
if (x % 2 != 0) points = points.map(p => [[p[0][0],-p[0][1]],[p[1][0],-p[1][1]]]);
return {
center: [x*.5 + (y % 2 == 0 ? .5 : 0), y*h*2],
lineWidth: .35,
points
}};
case 4: return (x,y) => { // hexagon
const h = Math.sqrt(3)/4;
let points = [[[0,-h],[0,1]],[[0,h],[0,-1]],
[[-3/8,-h/2],[2*h,.5]],[[-3/8,h/2],[2*h,-.5]],
[[ 3/8,-h/2],[-2*h,.5]],[[3/8,h/2],[-2*h,-.5]]];
return {
center: [x*.75, y*2*h + ((x % 2 != 0) ? 0:h)],
lineWidth: .6,
points
}};

}
}

////////////////////////////////////////////////////////////////
// Polygon Clipping utility code - Created by Reinder Nijhoff 2019
// https://turtletoy.net/turtle/a5befa1f8d
////////////////////////////////////////////////////////////////

function Polygons() {
const polygonList = [];
const Polygon = class {
constructor() {
this.cp = [];       // clip path: array of [x,y] pairs
this.dp = [];       // 2d lines [x0,y0],[x1,y1] to draw
this.aabb = [];     // AABB bounding box
}
// add point to clip path and update bounding box
let xmin = 1e5, xmax = -1e5, ymin = 1e5, ymax = -1e5;
(this.cp = [...this.cp, ...points]).forEach( p => {
xmin = Math.min(xmin, p[0]), xmax = Math.max(xmax, p[0]);
ymin = Math.min(ymin, p[1]), ymax = Math.max(ymax, p[1]);
});
this.aabb = [(xmin+xmax)/2, (ymin+ymax)/2, (xmax-xmin)/2, (ymax-ymin)/2];
}
// add segments (each a pair of points)
points.forEach(p => this.dp.push(p));
}
for (let i = 0, l = this.cp.length; i < l; i++) {
this.dp.push(this.cp[i], this.cp[(i + 1) % l]);
}
}
draw(t) {
for (let i = 0, l = this.dp.length; i < l; i+=2) {
t.jump(this.dp[i]), t.goto(this.dp[i + 1]);
}
}
const tp = new Polygon();
tp.cp.push([-1e5,-1e5],[1e5,-1e5],[1e5,1e5],[-1e5,1e5]);
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 = 0.5; i < 150 / d; i++) {
tp.dp.push([dx * i + cy,   dy * i - cx], [dx * i - cy,   dy * i + cx]);
tp.dp.push([-dx * i + cy, -dy * i - cx], [-dx * i - cy, -dy * i + cx]);
}
tp.boolean(this, false);
this.dp = [...this.dp, ...tp.dp];
}
inside(p) {
let int = 0; // find number of i ntersection points from p to far away
for (let i = 0, l = this.cp.length; i < l; i++) {
if (this.segment_intersect(p, [0.1, -1000], this.cp[i], this.cp[(i + 1) % l])) {
int++;
}
}
return int & 1; // if even your outside
}
boolean(p, diff = true) {
// bouding box optimization by ge1doot.
if (Math.abs(this.aabb[0] - p.aabb[0]) - (p.aabb[2] + this.aabb[2]) >= 0 &&
Math.abs(this.aabb[1] - p.aabb[1]) - (p.aabb[3] + this.aabb[3]) >= 0) return this.dp.length > 0;

// polygon diff algorithm (narrow phase)
const ndp = [];
for (let i = 0, l = this.dp.length; i < l; i+=2) {
const ls0 = this.dp[i];
const ls1 = this.dp[i + 1];
// find all intersections with clip path
const int = [];
for (let j = 0, cl = p.cp.length; j < cl; j++) {
const pint = this.segment_intersect(ls0, ls1, p.cp[j], p.cp[(j + 1) % cl]);
if (pint !== false) {
int.push(pint);
}
}
if (int.length === 0) {
// 0 intersections, inside or outside?
if (diff === !p.inside(ls0)) {
ndp.push(ls0, ls1);
}
} else {
int.push(ls0, ls1);
// order intersection points on line ls.p1 to ls.p2
const cmpx = ls1[0] - ls0[0];
const cmpy = ls1[1] - ls0[1];
int.sort( (a,b) =>  (a[0] - ls0[0]) * cmpx + (a[1] - ls0[1]) * cmpy -
(b[0] - ls0[0]) * cmpx - (b[1] - ls0[1]) * cmpy);

for (let j = 0; j < int.length - 1; j++) {
if ((int[j][0] - int[j+1][0])**2 + (int[j][1] - int[j+1][1])**2 >= 0.001) {
if (diff === !p.inside([(int[j][0]+int[j+1][0])/2,(int[j][1]+int[j+1][1])/2])) {
ndp.push(int[j], int[j+1]);
}
}
}
}
}
return (this.dp = ndp).length > 0;
}
//port of http://paulbourke.net/geometry/pointlineplane/Helpers.cs
segment_intersect(l1p1, l1p2, l2p1, l2p2) {
const d   = (l2p2[1] - l2p1[1]) * (l1p2[0] - l1p1[0]) - (l2p2[0] - l2p1[0]) * (l1p2[1] - l1p1[1]);
if (d === 0) return false;
const n_a = (l2p2[0] - l2p1[0]) * (l1p1[1] - l2p1[1]) - (l2p2[1] - l2p1[1]) * (l1p1[0] - l2p1[0]);
const n_b = (l1p2[0] - l1p1[0]) * (l1p1[1] - l2p1[1]) - (l1p2[1] - l1p1[1]) * (l1p1[0] - l2p1[0]);
const ua = n_a / d;
const ub = n_b / d;
if (ua >= 0 && ua <= 1 && ub >= 0 && ub <= 1) {
return [l1p1[0] + ua * (l1p2[0] - l1p1[0]), l1p1[1] + ua * (l1p2[1] - l1p1[1])];
}
return false;
}
};
return {
list: () => polygonList,
create: () => new Polygon(),
draw: (turtle, p, addToVisList=true) => {
for (let j = 0; j < polygonList.length && p.boolean(polygonList[j]); j++);
p.draw(turtle);