3D Line Art Engine: CSG
I have ported most of "ln, The 3D Line Art Engine" by Fogleman (@FogleBird) to javascript, so it can be used on Turtletoy. Not all code is ported and not all code-paths are tested: probably there are still (a lot of) bugs left.
This Turtle is a port of the example: github.com/fogleman/ln/blob/master/examples/csg.go
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// 3D Line Art Engine: CSG. Created by Reinder Nijhoff 2019 - @reindernijhoff
// The MIT License
//
// https://turtletoy.net/turtle/1716c398e1
//
// I have ported most of "ln, The 3D Line Art Engine"[1] by Fogleman (@FogleBird) to javascript, so
// it can be used on Turtletoy. Not all code is ported and not all code-paths are tested:
// probably there are still (a lot of) bugs left.
//
// This Turtle is a port of the example: https://github.com/fogleman/ln/blob/master/examples/csg.go
//
// [1] https://github.com/fogleman/ln
//
const DRAW_HIDDEN_LINES = false;
const turtle = new Turtle();
let paths, scene;
function walk(i) {
const eye = new Vector(0, 6, 2);
const center = new Vector(0, 0, 0);
const up = new Vector(0, 0, 1);
const width = 190.0;
const height = 190.0;
const fovy = 20.0;
const znear = 0.1;
const zfar = 100.0;
const step = 0.01; // how finely to chop the paths for visibility testing
if (i==0) {
Cube.prototype.Paths = function() {
const x1 = this.Min.X, y1 = this.Min.Y, z1 = this.Min.Z;
const x2 = this.Max.X, y2 = this.Max.Y, z2 = this.Max.Z;
const paths = new Paths();
for (let i = 0; i <= 30; i++) {
const p = i / 30;
const x = x1 + (x2-x1)*p;
const y = y1 + (y2-y1)*p;
paths.Append(new Path([new Vector(x, y1, z1), new Vector(x, y1, z2)]));
paths.Append(new Path([new Vector(x, y2, z1), new Vector(x, y2, z2)]));
paths.Append(new Path([new Vector(x1, y, z1), new Vector(x1, y, z2)]));
paths.Append(new Path([new Vector(x2, y, z1), new Vector(x2, y, z2)]));
}
return paths;
}
scene = new Scene();
const m = Rotate(new Vector(0, 0, 1), 1.3);
const shape = new BooleanShape(Difference,
new BooleanShape(Intersection,
new Sphere(new Vector(0,0,0), 1),
new Cube(new Vector(-0.8, -0.8, -0.8), new Vector(0.8, 0.8, 0.8)),
),
new Cylinder(0.4, -2, 2),
new TransformedShape(new Cylinder(0.4, -2, 2), Rotate(new Vector(1, 0, 0), Radians(90))),
new TransformedShape(new Cylinder(0.4, -2, 2), Rotate(new Vector(0, 1, 0), Radians(90))),
);
scene.Add(new TransformedShape(shape, m));
}
paths = scene.Render(eye, center, up, width, height, fovy, znear, zfar, step, i);
if (paths) {
for (let k=0; k<paths.v.length; k++) {
const path = paths.v[k];
turtle.penup();
turtle.goto(path.v[0].X, path.v[0].Y);
turtle.pendown();
for (let j=1; j<path.v.length; j++) {
turtle.goto(path.v[j].X, path.v[j].Y);
}
}
return true;
} else {
return false;
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/util.go
//
const Radians = (degrees) => degrees * Math.PI / 180;
const Degrees = (radians) => radians * 180 / Math.PI;
const Median = (items) => {
let n = items.length;
if (n == 0) {
return 0;
} else if (n%2 == 1) {
return items[n/2];
} else {
const a = items[n/2-1];
const b = items[n/2];
return (a + b) / 2;
}
}
const INF = 1e9;
const EPS = 1e-9;
//
// https://github.com/fogleman/ln/blob/master/ln/axis.go
//
const AxisNone = -1;
const AxisX = 0;
const AxisY = 1;
const AxisZ = 2;
//
// https://github.com/fogleman/ln/blob/master/ln/vector.go
//
class Vector {
constructor(x, y, z) {
this.X = x; this.Y = y; this.Z = z;
}
RandomUnitVector() {
let x, y, z;
do {
x = Math.ranomd()*2 - 1;
y = Math.ranomd()*2 - 1;
z = Math.ranomd()*2 - 1;
} while (x*x+y*y+z*z > 1);
return new Vector(x, y, z).Normalize();
}
Length() { return Math.sqrt(this.X*this.X + this.Y*this.Y + this.Z*this.Z); }
Distance(b) { return this.Sub(b).Length(); }
LengthSquared() { return this.X*this.X + this.Y*this.Y + this.Z*this.Z; }
DistanceSquared(b){ return this.Sub(b).LengthSquared(); }
Dot(b) { return this.X*b.X + this.Y*b.Y + this.Z*b.Z; }
Cross(b) {
return new Vector( this.Y*b.Z - this.Z*b.Y,
this.Z*b.X - this.X*b.Z,
this.X*b.Y - this.Y*b.X); }
Normalize() { return this.MulScalar(1/this.Length()); }
Add(b) { return new Vector(this.X + b.X, this.Y + b.Y, this.Z + b.Z); }
Sub(b) { return new Vector(this.X - b.X, this.Y - b.Y, this.Z - b.Z); }
Mul(b) { return new Vector(this.X * b.X, this.Y * b.Y, this.Z * b.Z); }
Div(b) { return new Vector(this.X / b.X, this.Y / b.Y, this.Z / b.Z); }
AddScalar(b) { return new Vector(this.X + b, this.Y + b, this.Z + b); }
SubScalar(b) { return new Vector(this.X - b, this.Y - b, this.Z - b); }
MulScalar(b) { return new Vector(this.X * b, this.Y * b, this.Z * b); }
DivScalar(b) { return new Vector(this.X / b, this.Y / b, this.Z / b); }
Min(b) { return new Vector(Math.min(this.X, b.X), Math.min(this.Y, b.Y), Math.min(this.Z, b.Z)); }
Max(b) { return new Vector(Math.max(this.X, b.X), Math.max(this.Y, b.Y), Math.max(this.Z, b.Z)); }
MinAxis() {
const x = Math.abs(this.X), y = Math.abs(this.Y), z = Math.abs(this.Z);
if (x <= y && x <= z) {
return new Vector(1, 0, 0);
} else if (y <= x && y <= z) {
return new Vector(0, 1, 0);
}
return new Vector(0, 0, 1);
}
MinComponent() { return Math.min(Math.min(this.X, this.Y), this.Z); }
SegmentDistance(v, w) {
const l2 = v.DistanceSquared(w)
if (l2 == 0) {
return this.Distance(v);
}
const t = this.Sub(v).Dot(w.Sub(v)) / l2;
if (t < 0) {
return this.Distance(v);
}
if (t > 1){
return this.Distance(w);
}
return v.Add(w.Sub(v).MulScalar(t)).Distance(this);
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/matrix.go
//
class Matrix {
constructor(x00, x01, x02, x03,
x10, x11, x12, x13,
x20, x21, x22, x23,
x30, x31, x32, x33) {
this.x00=x00, this.x01=x01, this.x02=x02, this.x03=x03;
this.x10=x10, this.x11=x11, this.x12=x12, this.x13=x13;
this.x20=x20, this.x21=x21, this.x22=x22, this.x23=x23;
this.x30=x30, this.x31=x31, this.x32=x32, this.x33=x33;
}
Translate(v) {
return Translate(v).Mul(this);
}
Scale(v) {
return Scale(v).Mul(this);
}
Rotate(v, a) {
return Rotate(v, a).Mul(this);
}
Frustum(l, r, b, t, n, f) {
return Frustum(l, r, b, t, n, f).Mul(this);
}
Orthographic(l, r, b, t, n, f) {
return Orthographic(l, r, b, t, n, f).Mul(this);
}
Perspective(fovy, aspect, near, far) {
return Perspective(fovy, aspect, near, far).Mul(this);
}
Mul(b) {
const m = new Matrix();
m.x00 = this.x00*b.x00 + this.x01*b.x10 + this.x02*b.x20 + this.x03*b.x30;
m.x10 = this.x10*b.x00 + this.x11*b.x10 + this.x12*b.x20 + this.x13*b.x30;
m.x20 = this.x20*b.x00 + this.x21*b.x10 + this.x22*b.x20 + this.x23*b.x30;
m.x30 = this.x30*b.x00 + this.x31*b.x10 + this.x32*b.x20 + this.x33*b.x30;
m.x01 = this.x00*b.x01 + this.x01*b.x11 + this.x02*b.x21 + this.x03*b.x31;
m.x11 = this.x10*b.x01 + this.x11*b.x11 + this.x12*b.x21 + this.x13*b.x31;
m.x21 = this.x20*b.x01 + this.x21*b.x11 + this.x22*b.x21 + this.x23*b.x31;
m.x31 = this.x30*b.x01 + this.x31*b.x11 + this.x32*b.x21 + this.x33*b.x31;
m.x02 = this.x00*b.x02 + this.x01*b.x12 + this.x02*b.x22 + this.x03*b.x32;
m.x12 = this.x10*b.x02 + this.x11*b.x12 + this.x12*b.x22 + this.x13*b.x32;
m.x22 = this.x20*b.x02 + this.x21*b.x12 + this.x22*b.x22 + this.x23*b.x32;
m.x32 = this.x30*b.x02 + this.x31*b.x12 + this.x32*b.x22 + this.x33*b.x32;
m.x03 = this.x00*b.x03 + this.x01*b.x13 + this.x02*b.x23 + this.x03*b.x33;
m.x13 = this.x10*b.x03 + this.x11*b.x13 + this.x12*b.x23 + this.x13*b.x33;
m.x23 = this.x20*b.x03 + this.x21*b.x13 + this.x22*b.x23 + this.x23*b.x33;
m.x33 = this.x30*b.x03 + this.x31*b.x13 + this.x32*b.x23 + this.x33*b.x33;
return m;
}
MulPosition(b) {
const x = this.x00*b.X + this.x01*b.Y + this.x02*b.Z + this.x03;
const y = this.x10*b.X + this.x11*b.Y + this.x12*b.Z + this.x13;
const z = this.x20*b.X + this.x21*b.Y + this.x22*b.Z + this.x23;
return new Vector(x, y, z);
}
MulPositionW(b) {
const x = this.x00*b.X + this.x01*b.Y + this.x02*b.Z + this.x03;
const y = this.x10*b.X + this.x11*b.Y + this.x12*b.Z + this.x13;
const z = this.x20*b.X + this.x21*b.Y + this.x22*b.Z + this.x23;
const w = this.x30*b.X + this.x31*b.Y + this.x32*b.Z + this.x33;
return new Vector(x / w, y / w, z / w);
}
MulDirection(b) {
const x = this.x00*b.X + this.x01*b.Y + this.x02*b.Z;
const y = this.x10*b.X + this.x11*b.Y + this.x12*b.Z;
const z = this.x20*b.X + this.x21*b.Y + this.x22*b.Z;
return new Vector(x, y, z).Normalize();
}
MulRay(b) {
return new Ray(this.MulPosition(b.Origin), this.MulDirection(b.Direction));
}
MulBox(box) {
// http://dev.theomader.com/transform-bounding-boxes/
const r = new Vector(this.x00, this.x10, this.x20);
const u = new Vector(this.x01, this.x11, this.x21);
const b = new Vector(this.x02, this.x12, this.x22);
const t = new Vector(this.x03, this.x13, this.x23);
let xa = r.MulScalar(box.Min.X);
let xb = r.MulScalar(box.Max.X);
let ya = u.MulScalar(box.Min.Y);
let yb = u.MulScalar(box.Max.Y);
let za = b.MulScalar(box.Min.Z);
let zb = b.MulScalar(box.Max.Z);
const min = xa.Min(xb).Add(ya.Min(yb)).Add(za.Min(zb)).Add(t);
const max = xa.Max(xb).Add(ya.Max(yb)).Add(za.Max(zb)).Add(t);
return new Box(min, max);
}
Transpose() {
return new Matrix(
this.x00, this.x10, this.x20, this.x30,
this.x01, this.x11, this.x21, this.x31,
this.x02, this.x12, this.x22, this.x32,
this.x03, this.x13, this.x23, this.x33);
}
Determinant() {
return (this.x00*this.x11*this.x22*this.x33 - this.x00*this.x11*this.x23*this.x32 +
this.x00*this.x12*this.x23*this.x31 - this.x00*this.x12*this.x21*this.x33 +
this.x00*this.x13*this.x21*this.x32 - this.x00*this.x13*this.x22*this.x31 -
this.x01*this.x12*this.x23*this.x30 + this.x01*this.x12*this.x20*this.x33 -
this.x01*this.x13*this.x20*this.x32 + this.x01*this.x13*this.x22*this.x30 -
this.x01*this.x10*this.x22*this.x33 + this.x01*this.x10*this.x23*this.x32 +
this.x02*this.x13*this.x20*this.x31 - this.x02*this.x13*this.x21*this.x30 +
this.x02*this.x10*this.x21*this.x33 - this.x02*this.x10*this.x23*this.x31 +
this.x02*this.x11*this.x23*this.x30 - this.x02*this.x11*this.x20*this.x33 -
this.x03*this.x10*this.x21*this.x32 + this.x03*this.x10*this.x22*this.x31 -
this.x03*this.x11*this.x22*this.x30 + this.x03*this.x11*this.x20*this.x32 -
this.x03*this.x12*this.x20*this.x31 + this.x03*this.x12*this.x21*this.x30);
}
Inverse() {
const m = new Matrix();
const d = this.Determinant();
m.x00 = (this.x12*this.x23*this.x31 - this.x13*this.x22*this.x31 + this.x13*this.x21*this.x32 - this.x11*this.x23*this.x32 - this.x12*this.x21*this.x33 + this.x11*this.x22*this.x33) / d;
m.x01 = (this.x03*this.x22*this.x31 - this.x02*this.x23*this.x31 - this.x03*this.x21*this.x32 + this.x01*this.x23*this.x32 + this.x02*this.x21*this.x33 - this.x01*this.x22*this.x33) / d;
m.x02 = (this.x02*this.x13*this.x31 - this.x03*this.x12*this.x31 + this.x03*this.x11*this.x32 - this.x01*this.x13*this.x32 - this.x02*this.x11*this.x33 + this.x01*this.x12*this.x33) / d;
m.x03 = (this.x03*this.x12*this.x21 - this.x02*this.x13*this.x21 - this.x03*this.x11*this.x22 + this.x01*this.x13*this.x22 + this.x02*this.x11*this.x23 - this.x01*this.x12*this.x23) / d;
m.x10 = (this.x13*this.x22*this.x30 - this.x12*this.x23*this.x30 - this.x13*this.x20*this.x32 + this.x10*this.x23*this.x32 + this.x12*this.x20*this.x33 - this.x10*this.x22*this.x33) / d;
m.x11 = (this.x02*this.x23*this.x30 - this.x03*this.x22*this.x30 + this.x03*this.x20*this.x32 - this.x00*this.x23*this.x32 - this.x02*this.x20*this.x33 + this.x00*this.x22*this.x33) / d;
m.x12 = (this.x03*this.x12*this.x30 - this.x02*this.x13*this.x30 - this.x03*this.x10*this.x32 + this.x00*this.x13*this.x32 + this.x02*this.x10*this.x33 - this.x00*this.x12*this.x33) / d;
m.x13 = (this.x02*this.x13*this.x20 - this.x03*this.x12*this.x20 + this.x03*this.x10*this.x22 - this.x00*this.x13*this.x22 - this.x02*this.x10*this.x23 + this.x00*this.x12*this.x23) / d;
m.x20 = (this.x11*this.x23*this.x30 - this.x13*this.x21*this.x30 + this.x13*this.x20*this.x31 - this.x10*this.x23*this.x31 - this.x11*this.x20*this.x33 + this.x10*this.x21*this.x33) / d;
m.x21 = (this.x03*this.x21*this.x30 - this.x01*this.x23*this.x30 - this.x03*this.x20*this.x31 + this.x00*this.x23*this.x31 + this.x01*this.x20*this.x33 - this.x00*this.x21*this.x33) / d;
m.x22 = (this.x01*this.x13*this.x30 - this.x03*this.x11*this.x30 + this.x03*this.x10*this.x31 - this.x00*this.x13*this.x31 - this.x01*this.x10*this.x33 + this.x00*this.x11*this.x33) / d;
m.x23 = (this.x03*this.x11*this.x20 - this.x01*this.x13*this.x20 - this.x03*this.x10*this.x21 + this.x00*this.x13*this.x21 + this.x01*this.x10*this.x23 - this.x00*this.x11*this.x23) / d;
m.x30 = (this.x12*this.x21*this.x30 - this.x11*this.x22*this.x30 - this.x12*this.x20*this.x31 + this.x10*this.x22*this.x31 + this.x11*this.x20*this.x32 - this.x10*this.x21*this.x32) / d;
m.x31 = (this.x01*this.x22*this.x30 - this.x02*this.x21*this.x30 + this.x02*this.x20*this.x31 - this.x00*this.x22*this.x31 - this.x01*this.x20*this.x32 + this.x00*this.x21*this.x32) / d;
m.x32 = (this.x02*this.x11*this.x30 - this.x01*this.x12*this.x30 - this.x02*this.x10*this.x31 + this.x00*this.x12*this.x31 + this.x01*this.x10*this.x32 - this.x00*this.x11*this.x32) / d;
m.x33 = (this.x01*this.x12*this.x20 - this.x02*this.x11*this.x20 + this.x02*this.x10*this.x21 - this.x00*this.x12*this.x21 - this.x01*this.x10*this.x22 + this.x00*this.x11*this.x22) / d;
return m;
}
}
function Identity() {
return new Matrix(
1, 0, 0, 0,
0, 1, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1);
}
function Translate(v) {
return new Matrix(
1, 0, 0, v.X,
0, 1, 0, v.Y,
0, 0, 1, v.Z,
0, 0, 0, 1);
}
function Scale(v) {
return new Matrix(
v.X, 0, 0, 0,
0, v.Y, 0, 0,
0, 0, v.Z, 0,
0, 0, 0, 1);
}
function Rotate(v, a) {
v = v.Normalize();
const s = Math.sin(a);
const c = Math.cos(a);
const m = 1 - c;
return new Matrix(
m*v.X*v.X + c, m*v.X*v.Y + v.Z*s, m*v.Z*v.X - v.Y*s, 0,
m*v.X*v.Y - v.Z*s, m*v.Y*v.Y + c, m*v.Y*v.Z + v.X*s, 0,
m*v.Z*v.X + v.Y*s, m*v.Y*v.Z - v.X*s, m*v.Z*v.Z + c, 0,
0, 0, 0, 1);
}
function Frustum(l, r, b, t, n, f) {
const t1 = 2 * n;
const t2 = r - l;
const t3 = t - b;
const t4 = f - n;
return new Matrix(
t1 / t2, 0, (r + l) / t2, 0,
0, t1 / t3, (t + b) / t3, 0,
0, 0, (-f - n) / t4, (-t1 * f) / t4,
0, 0, -1, 0);
}
function Orthographic(l, r, b, t, n, f) {
return new Matrix(
2 / (r - l), 0, 0, -(r + l) / (r - l),
0, 2 / (t - b), 0, -(t + b) / (t - b),
0, 0, -2 / (f - n), -(f + n) / (f - n),
0, 0, 0, 1);
}
function Perspective(fovy, aspect, near, far) {
const ymax = near * Math.tan(fovy*Math.PI/360);
const xmax = ymax * aspect;
return Frustum(-xmax, xmax, -ymax, ymax, near, far);
}
function LookAt(eye, center, up) {
up = up.Normalize();
const f = center.Sub(eye).Normalize();
const s = f.Cross(up).Normalize();
const u = s.Cross(f).Normalize();
const m = new Matrix(
s.X, u.X, -f.X, eye.X,
s.Y, u.Y, -f.Y, eye.Y,
s.Z, u.Z, -f.Z, eye.Z,
0, 0, 0, 1,
);
return m.Inverse();
}
//
// https://github.com/fogleman/ln/blob/master/ln/tree.go
//
class Tree {
constructor(shapes) {
this.Box = BoxForShapes(shapes);
this.Root = new Node(shapes);
this.Root.Split(0);
}
Intersect(r) {
const i = this.Box.Intersect(r);
if (i.t2 < i.t1 || i.t2 <= 0) {
return NoHit;
}
return this.Root.Intersect(r, i.t1, i.t2);
}
}
class Node {
constructor(shapes) {
this.Axis = AxisNone
this.Point = 0;
this.Shapes = shapes;
this.Left = false;
this.Right = false;
}
Intersect(r, tmin, tmax) {
let tsplit = 0;
let leftFirst = false;
switch (this.Axis) {
case AxisNone:
return this.IntersectShapes(r);
case AxisX:
tsplit = (this.Point - r.Origin.X) / r.Direction.X;
leftFirst = (r.Origin.X < this.Point) || (r.Origin.X == this.Point && r.Direction.X <= 0);
case AxisY:
tsplit = (this.Point - r.Origin.Y) / r.Direction.Y;
leftFirst = (r.Origin.Y < this.Point) || (r.Origin.Y == this.Point && r.Direction.Y <= 0);
case AxisZ:
tsplit = (this.Point - r.Origin.Z) / r.Direction.Z;
leftFirst = (r.Origin.Z < this.Point) || (r.Origin.Z == this.Point && r.Direction.Z <= 0);
}
let first, second;
if (leftFirst) {
first = this.Left;
second = this.Right;
} else {
first = this.Right;
second = this.Left;
}
if (tsplit > tmax || tsplit <= 0) {
return first.Intersect(r, tmin, tmax);
} else if (tsplit < tmin) {
return second.Intersect(r, tmin, tmax);
} else {
const h1 = first.Intersect(r, tmin, tsplit);
if (h1.T <= tsplit) {
return h1;
}
const h2 = second.Intersect(r, tsplit, Math.min(tmax, h1.T));
if (h1.T <= h2.T) {
return h1;
} else {
return h2;
}
}
}
IntersectShapes(r) {
let hit = NoHit;
for (const _ in this.Shapes) {
const shape = this.Shapes[_];
const h = shape.Intersect(r);
if (h.T < hit.T ){
hit = h;
}
}
return hit;
}
PartitionScore(axis, point) {
let left = 0, right = 0;
for (const _ in this.Shapes) {
const shape = this.Shapes[_];
const box = shape.BoundingBox();
const p = box.Partition(axis, point);
if (p.left) {
left++;
}
if (p.right) {
right++;
}
}
if (left >= right) {
return left;
} else {
return right;
}
}
Partition(size, axis, point) {
const left = [], right = [];
for (const _ in this.Shapes) {
const shape = this.Shapes[_];
const box = shape.BoundingBox();
const p = box.Partition(axis, point);
if (p.left) {
left.push(shape);
}
if (p.right) {
right.push(shape);
}
}
return { left, right };
}
Split(depth) {
if (this.Shapes.length < 8) {
return;
}
const xs = [], ys = [], zs = [];
for (const _ in this.Shapes) {
const shape = this.Shapes[_];
const box = shape.BoundingBox();
xs.push(box.Min.X);
xs.push(box.Max.X);
ys.push(box.Min.Y);
ys.push(box.Max.Y);
zs.push(box.Min.Z);
zs.push(box.Max.Z);
}
xs.sort();
ys.sort();
zs.sort();
const mx = Median(xs), my = Median(ys), mz = Median(zs);
let best = Math.floor(this.Shapes.length * 0.85);
let bestAxis = AxisNone;
let bestPoint = 0.0;
const sx = this.PartitionScore(AxisX, mx);
if (sx < best) {
best = sx;
bestAxis = AxisX;
bestPoint = mx;
}
const sy = this.PartitionScore(AxisY, my);
if (sy < best) {
best = sy;
bestAxis = AxisY;
bestPoint = my;
}
const sz = this.PartitionScore(AxisZ, mz);
if (sz < best) {
best = sz;
bestAxis = AxisZ;
bestPoint = mz;
}
if (bestAxis == AxisNone ){
return;
}
const p = this.Partition(best, bestAxis, bestPoint)
this.Axis = bestAxis;
this.Point = bestPoint;
this.Left = new Node(p.left);
this.Right = new Node(p.right);
this.Left.Split(depth + 1);
this.Right.Split(depth + 1);
this.Shapes = false; // only needed at leaf nodes
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/shape.go
//
class TransformedShape{
constructor(shape, matrix) {
this.Shape = shape;
this.Matrix = matrix;
this.Inverse = matrix.Inverse();
}
Compile() {
this.Shape.Compile();
}
BoundingBox() {
return this.Matrix.MulBox(this.Shape.BoundingBox());
}
Contains(v, f) {
return this.Shape.Contains(this.Inverse.MulPosition(v), f);
}
Intersect(r ) {
return this.Shape.Intersect(this.Inverse.MulRay(r));
}
Paths() {
return this.Shape.Paths().Transform(this.Matrix);
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/scene.go
//
class Scene {
constructor() {
this.Shapes = [];
this.Tree = false;
}
Compile() {
for (const _ in this.Shapes) {
const shape = this.Shapes[_];
shape.Compile()
}
if (!this.Tree) {
this.Tree = new Tree(this.Shapes);
}
}
Add(shapes) {
if(Array.isArray(shapes)) {
this.Shapes = [...this.Shapes, ...shapes];
} else {
this.Shapes.push(shapes);
}
}
Intersect(r) {
return this.Tree.Intersect(r);
}
Visible(eye, point) {
const v = eye.Sub(point);
const r = new Ray(point, v.Normalize());
const hit = this.Intersect(r);
return (hit.T >= v.Length());
}
Paths(p = -1) {
if (p >= 0) {
if (p < this.Shapes.length) {
return this.Shapes[p].Paths();
} else {
return false;
}
} else {
let result = new Paths();
for (const _ in this.Shapes) {
const shape = this.Shapes[_];
result.Append(shape.Paths().v);
}
return result;
}
}
Render(eye, center, up, width, height, fovy, near, far, step, p = -1) {
const aspect = width / height;
let matrix = LookAt(eye, center, up);
matrix = matrix.Perspective(fovy, aspect, near, far);
return this.RenderWithMatrix(matrix, eye, width, height, step, p);
}
RenderWithMatrix(matrix, eye, width, height, step, p = -1) {
if (p <= 0) {
this.Compile();
}
let paths = this.Paths(p);
if (!paths) {
return false;
}
if (step > 0) {
paths = paths.Chop(step);
}
paths = paths.Filter(new ClipFilter(matrix, eye, this));
if (step > 0) {
paths = paths.Simplify(1e-6);
}
matrix = Scale( new Vector(-width / 2, -height / 2, 0));
paths = paths.Transform(matrix);
return paths;
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/ray.go
//
class Ray {
constructor(origin, direction) {
this.Origin = origin;
this.Direction = direction;
}
Position(t) {
return this.Origin.Add(this.Direction.MulScalar(t));
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/path.go
//
class Path {
constructor(v = []) {
this.v = v;
}
Append(v) {
if (Array.isArray(v)) {
this.v = this.v.concat(v);
} else {
this.v.push(v);
}
}
BoundingBox() {
let box = new Box(this.v[0], this.v[0]);
for (const _ in this.v) {
const v = this.v[_];
box = box.Extend(new Box(v, v));
}
return box
}
Transform(matrix) {
const result = new Path();
for (const _ in this.v) {
const v = this.v[_];
result.Append(matrix.MulPosition(v));
}
return result;
}
Chop(step) {
const result = new Path();
for (let i = 0; i < this.v.length-1; i++) {
const a = this.v[i];
const b = this.v[i+1];
const v = b.Sub(a);
const l = v.Length();
if (i == 0) {
result.Append(a);
}
for (let d = step;d < l; d += step) {
result.Append(a.Add(v.MulScalar(d/l)));
}
result.Append(b);
}
return result;
}
Filter(f) {
const result = new Paths();
let path = new Path();
for (const _ in this.v) {
const v = this.v[_];
const fr = f.Filter(v);
if (fr.ok || (DRAW_HIDDEN_LINES && _%8 < 4)) { // show hidden lines
path.Append(fr.w);
} else {
if (path && path.v.length > 1) {
result.Append(path);
}
path = new Path();
}
}
if (path && path.v.length > 1) {
result.Append(path);
}
return result;
}
Simplify(threshold) {
if (this.v.length < 3) {
return this;
}
const a = this.v[0];
const b = this.v[this.v.length-1];
let index = -1;
let distance = 0.0;
for (let i = 1; i < this.v.length-1; i++) {
const d = this.v[i].SegmentDistance(a, b);
if (d > distance) {
index = i;
distance = d;
}
}
if (distance > threshold) {
const r1 = new Path(this.v.slice(0, index+1)).Simplify(threshold);
const r2 = new Path(this.v.slice(index)).Simplify(threshold);
return new Path([...r1.v.slice(0, r1.v.length-1), ...r2.v]);
} else {
return new Path([a, b]);
}
}
}
class Paths {
constructor(v = []) {
this.v = v;
}
Append(v) {
if (Array.isArray(v)) {
this.v = this.v.concat(v);
} else {
this.v.push(v);
}
}
BoundingBox() {
let box = this.v[0].BoundingBox();
for (const _ in this.v) {
const path = this.v[_];
box = box.Extend(path.BoundingBox());
}
return box;
}
Transform(matrix) {
const result = new Paths();
for (const _ in this.v) {
const path = this.v[_];
result.Append(path.Transform(matrix));
}
return result;
}
Chop(step) {
const result = new Paths();
for (const _ in this.v) {
const path = this.v[_];
result.Append(path.Chop(step));
}
return result;
}
Filter(f) {
const result = new Paths();
for (const _ in this.v) {
const path = this.v[_];
result.Append(path.Filter(f).v);
}
return result;
}
Simplify(threshold) {
const result = new Paths();
for (const _ in this.v) {
const path = this.v[_];
result.Append(path.Simplify(threshold));
}
return result;
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/box.go
//
class Box {
constructor(min, max) {
this.Min = min;
this.Max = max;
}
Anchor(anchor) { return this.Min.Add(this.Size().Mul(anchor)); }
Center() { return this.Anchor(new Vector(0.5, 0.5, 0.5)); }
Size() { return this.Max.Sub(a.Min); }
Contains(b) {
return this.Min.X <= b.X && this.Max.X >= b.X &&
this.Min.Y <= b.Y && this.Max.Y >= b.Y &&
this.Min.Z <= b.Z && this.Max.Z >= b.Z;
}
Extend(b) { return new Box(this.Min.Min(b.Min), this.Max.Max(b.Max)); }
Intersect(r) {
let x1 = (this.Min.X - r.Origin.X) / r.Direction.X;
let y1 = (this.Min.Y - r.Origin.Y) / r.Direction.Y;
let z1 = (this.Min.Z - r.Origin.Z) / r.Direction.Z;
let x2 = (this.Max.X - r.Origin.X) / r.Direction.X;
let y2 = (this.Max.Y - r.Origin.Y) / r.Direction.Y;
let z2 = (this.Max.Z - r.Origin.Z) / r.Direction.Z;
if (x1 > x2) {
const tmp = x1; x1 = x2, x2 = tmp;
}
if (y1 > y2) {
const tmp = y1; y1 = y2, y2 = tmp;
}
if (z1 > z2) {
const tmp = z1; z1 = z2, z2 = tmp;
}
const t1 = Math.max(Math.max(x1, y1), z1);
const t2 = Math.min(Math.min(x2, y2), z2);
return {t1, t2};
}
Partition(axis, point) {
let left, right;
switch (axis) {
case AxisX:
left = this.Min.X <= point;
right = this.Max.X >= point;
case AxisY:
left = this.Min.Y <= point;
right = this.Max.Y >= point;
case AxisZ:
left = this.Min.Z <= point;
right = this.Max.Z >= point;
}
return {left, right};
}
}
function BoxForShapes(shapes) {
if (shapes.length == 0) {
return new Box();
}
let box = shapes[0].BoundingBox();
for (const _ in shapes) {
const shape = shapes[_];
box = box.Extend(shape.BoundingBox());
}
return box;
}
function BoxForTriangles(shapes) {
if (shapes.length == 0) {
return new Box();
}
let box = shapes[0].BoundingBox();
for (const _ in shapes) {
const shape = shapes[_];
box = box.Extend(shape.BoundingBox());
}
return box;
}
function BoxForVectors(vectors) {
if (vectors.length == 0) {
return new Box();
}
let min = vectors[0]
let max = vectors[0]
for (const _ in vectors) {
const v = vectors[_];
min = min.Min(v);
max = max.Max(v);
}
return new Box(min, max);
}
//
// https://github.com/fogleman/ln/blob/master/ln/box.go
//
class Hit {
constructor(shape, T) {
this.Shape = shape;
this.T = T;
}
Ok() {
return this.T < INF;
}
Min(b) {
if (this.T <= b.T) {
return this;
}
return b;
}
Max(b) {
if (this.T > b.T) {
return a;
}
return b;
}
}
const NoHit = new Hit(false, INF);
//
// https://github.com/fogleman/ln/blob/master/ln/filter.go
//
const ClipBox = new Box(new Vector(-1, -1, -1), new Vector(1, 1, 1));
class ClipFilter {
constructor(m, eye, scene) {
this.Matrix = m;
this.Eye = eye;
this.Scene = scene;
}
Filter(v) {
const w = this.Matrix.MulPositionW(v);
if (!this.Scene.Visible(this.Eye, v)) {
return {w, ok: false};
}
if (!ClipBox.Contains(w)) {
return {w, ok: false};
}
return {w, ok: true};
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/triangle.go
//
class Triangle {
constructor(v1, v2, v3) {
this.V1 = v1; this.V2 = v2; this.V3 = v3;
this.UpdateBoundingBox();
}
UpdateBoundingBox() {
const min = this.V1.Min(this.V2).Min(this.V3);
const max = this.V1.Max(this.V2).Max(this.V3);
this.Box = new Box(min, max);
}
Compile() {}
BoundingBox() { return this.Box; }
Contains(v, f) { return false; }
Intersect(r) {
const e1 = this.V2.Sub(this.V1);
const e2 = this.V3.Sub(this.V1);
const p =r.Direction.Cross(e2);
const det = e1.Dot(p);
if (det > -EPS && det < EPS) {
return NoHit;
}
const inv = 1 / det;
const t = r.Origin.Sub(this.V1);
const u = t.Dot(p) * inv;
if (u < 0 || u > 1) {
return NoHit;
}
const q = t.Cross(e1);
const v = r.Direction.Dot(q) * inv;
if (v < 0 || u+v > 1) {
return NoHit;
}
const d = e2.Dot(q) * inv;
if (d < EPS) {
return NoHit;
}
return new Hit(t, d);
}
Paths() { return new Paths([
new Path([this.V1, this.V2]),
new Path([this.V2, this.V3]),
new Path([this.V3, this.V1])]);
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/cube.go
//
class Cube {
constructor(min, max) {
this.Min = min;
this.Max = max;
this.Box = new Box(min, max);
}
Compile() {}
BoundingBox() {
return this.Box;
}
Contains(v, f) {
if (v.X < this.Min.X-f || v.X > this.Max.X+f) {
return false;
}
if (v.Y < this.Min.Y-f || v.Y > this.Max.Y+f) {
return false;
}
if (v.Z < this.Min.Z-f || v.Z > this.Max.Z+f) {
return false;
}
return true;
}
Intersect(r) {
let n = this.Min.Sub(r.Origin).Div(r.Direction);
let f = this.Max.Sub(r.Origin).Div(r.Direction);
const v = n.Min(f); f = n.Max(f);
const t0 = Math.max(Math.max(v.X, v.Y), v.Z);
const t1 = Math.min(Math.min(f.X, f.Y), f.Z);
if (t0 < 1e-3 && t1 > 1e-3) {
return new Hit(this, t1);
}
if (t0 >= 1e-3 && t0 < t1) {
return new Hit(this, t0);
}
return NoHit;
}
Paths() {
const x1 = this.Min.X, y1 = this.Min.Y, z1 = this.Min.Z;
const x2 = this.Max.X, y2 = this.Max.Y, z2 = this.Max.Z;
const paths = new Paths([
new Path([new Vector(x1, y1, z1), new Vector(x1, y1, z2)]),
new Path([new Vector(x1, y1, z1), new Vector(x1, y2, z1)]),
new Path([new Vector(x1, y1, z1), new Vector(x2, y1, z1)]),
new Path([new Vector(x1, y1, z2), new Vector(x1, y2, z2)]),
new Path([new Vector(x1, y1, z2), new Vector(x2, y1, z2)]),
new Path([new Vector(x1, y2, z1), new Vector(x1, y2, z2)]),
new Path([new Vector(x1, y2, z1), new Vector(x2, y2, z1)]),
new Path([new Vector(x1, y2, z2), new Vector(x2, y2, z2)]),
new Path([new Vector(x2, y1, z1), new Vector(x2, y1, z2)]),
new Path([new Vector(x2, y1, z1), new Vector(x2, y2, z1)]),
new Path([new Vector(x2, y1, z2), new Vector(x2, y2, z2)]),
new Path([new Vector(x2, y2, z1), new Vector(x2, y2, z2)]),
]);
return paths;
}
}
//
// https://github.com/fogleman/ln/blob/master/examples/csg.go
//
const Intersection = 0;
const Difference = 1;
class BooleanShape {
constructor(op, A, B) {
this.Op = op;
this.A = arguments[1];
this.B = arguments[2];
for (let i=3; i<arguments.length; i++) {
this.A = new BooleanShape(op, this.A, arguments[i]);
}
}
Compile() {}
BoundingBox() {
// TODO: fix this
const a = this.A.BoundingBox();
const b = this.B.BoundingBox();
return a.Extend(b);
}
Contains(v, f) {
f = 1e-3;
switch (this.Op) {
case Intersection:
return this.A.Contains(v, f) && this.B.Contains(v, f);
case Difference:
return this.A.Contains(v, f) && !this.B.Contains(v, -f);
}
return false;
}
Intersect(r) {
const h1 = this.A.Intersect(r);
const h2 = this.B.Intersect(r);
const h = h1.Min(h2);
const v = r.Position(h.T);
if (!h.Ok() || this.Contains(v, 0)) {
return h;
}
return this.Intersect(new Ray(r.Position(h.T+0.01), r.Direction));
}
Paths() {
let p = this.A.Paths();
p.Append(this.B.Paths().v);
return p.Chop(0.01).Filter(this);
}
Filter(w) {
return {w, ok: this.Contains(w, 0)};
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/cylinder.go
//
class Cylinder {
constructor(radius, z0, z1) {
this.Radius = radius;
this.Z0 = z0;
this.Z1 = z1;
}
Compile() {}
BoundingBox() {
const r = this.Radius;
return new Box(new Vector(-r, -r, this.Z0), new Vector(r, r, this.Z1));
}
Contains(v, f) {
const xy = new Vector(v.X, v.Y, 0);
if (xy.Length() > this.Radius+f) {
return false;
}
return v.Z >= this.Z0-f && v.Z <= this.Z1+f;
}
Intersect(ray) {
const r = this.Radius;
const o = ray.Origin;
const d = ray.Direction;
const a = d.X*d.X + d.Y*d.Y;
const b = 2*o.X*d.X + 2*o.Y*d.Y;
const c = o.X*o.X + o.Y*o.Y - r*r;
const q = b*b - 4*a*c;
if (q < 0) {
return NoHit;
}
const s = Math.sqrt(q);
let t0 = (-b + s) / (2 * a);
let t1 = (-b - s) / (2 * a);
if (t0 > t1) {
const tmp = t0;
t0 = t1; t1 = tmp;
}
const z0 = o.Z + t0*d.Z;
const z1 = o.Z + t1*d.Z;
if (t0 > 1e-6 && this.Z0 < z0 && z0 < this.Z1) {
return new Hit(this, t0);
}
if (t1 > 1e-6 && this.Z0 < z1 && z1 < this.Z1) {
return new Hit(this, t1);
}
return NoHit;
}
Paths() {
const result = new Paths();
for (let a = 0; a < 360; a += 10) {
const x = this.Radius * Math.cos(Radians(a));
const y = this.Radius * Math.sin(Radians(a));
result.Append(new Path([new Vector(x, y, this.Z0), new Vector(x, y, this.Z1)]));
}
return result;
}
}
class OutlineCylinder extends Cylinder {
constructor(eye, up, radius, z0, z1) {
super(radius, z0, z1);
this.Eye = eye;
this.Up = up;
}
Paths() {
let center = new Vector(0, 0, this.Z0);
let hyp = center.Sub(this.Eye).Length();
let opp = this.Radius;
let theta = Math.asin(opp / hyp);
let adj = opp / Math.tan(theta);
let d = Math.cos(theta) * adj;
// r := math.Sin(theta) * adj
let w = center.Sub(this.Eye).Normalize();
let u = w.Cross(this.Up).Normalize();
const c0 = this.Eye.Add(w.MulScalar(d));
const a0 = c0.Add(u.MulScalar(this.Radius * 1.01));
const b0 = c0.Add(u.MulScalar(-this.Radius * 1.01));
center = new Vector(0, 0, this.Z1);
hyp = center.Sub(this.Eye).Length();
opp = this.Radius;
theta = Math.asin(opp / hyp);
adj = opp / Math.tan(theta);
d = Math.cos(theta) * adj;
// r = math.Sin(theta) * adj
w = center.Sub(this.Eye).Normalize();
u = w.Cross(this.Up).Normalize();
const c1 = this.Eye.Add(w.MulScalar(d));
const a1 = c1.Add(u.MulScalar(this.Radius * 1.01));
const b1 = c1.Add(u.MulScalar(-this.Radius * 1.01));
const p0 = new Path(), p1 = new Path();
for (let a = 0; a < 360; a++) {
const x = this.Radius * Math.cos(Radians(a));
const y = this.Radius * Math.sin(Radians(a));
p0.Append(new Vector(x, y, this.Z0));
p1.Append(new Vector(x, y, this.Z1));
}
return new Paths([
p0,
p1,
new Path([new Vector(a0.X, a0.Y, this.Z0), new Vector(a1.X, a1.Y, this.Z1)]),
new Path([new Vector(b0.X, b0.Y, this.Z0), new Vector(b1.X, b1.Y, this.Z1)]),
]);
}
}
function NewTransformedOutlineCylinder(eye, up, v0, v1, radius) {
const d = v1.Sub(v0);
const z = d.Length();
const a = Math.acos(d.Normalize().Dot(up));
let m = Translate(v0);
if (a != 0) {
const u = d.Cross(up).Normalize();
m = Rotate(u, a).Translate(v0);
}
const c = new OutlineCylinder(m.Inverse().MulPosition(eye), up, radius, 0, z);
return new TransformedShape(c, m);
}
//
// https://github.com/fogleman/ln/blob/master/ln/sphere.go
//
class Sphere {
constructor(center, radius) {
this.Center = center;
this.Radius = radius;
const min = new Vector(center.X - radius, center.Y - radius, center.Z - radius);
const max = new Vector(center.X + radius, center.Y + radius, center.Z + radius);
this.Box = new Box(min, max);
}
Compile() {}
BoundingBox() {
return this.Box;
}
Contains(v, f) {
return v.Sub(this.Center).Length() <= this.Radius+f;
}
Intersect(r) {
const radius = this.Radius;
const to = r.Origin.Sub(this.Center);
const b = to.Dot(r.Direction);
const c = to.Dot(to) - radius*radius;
let d = b*b - c;
if (d > 0) {
d = Math.sqrt(d);
const t1 = -b - d;
if (t1 > 1e-2) {
return new Hit(this, t1);
}
const t2 = -b + d;
if (t2 > 1e-2) {
return new Hit(this, t2);
}
}
return NoHit
}
Paths3() {
const paths = new Paths();
for (let i = 0; i < 20000; i++) {
let v = RandomUnitVector();
v = v.MulScalar(this.Radius).Add(this.Center);
paths.Append(new Path([v, v]));
}
return paths;
}
Paths2() {
var equator = new Path();
for (let lng = 0; lng <= 360; lng++) {
const v = LatLngToXYZ(0, lng, this.Radius);
equator.Append(v);
}
var paths = new Paths();
for (let i = 0; i < 100; i++) {
const m = Identity();
for (let j = 0; j < 3; j++) {
const v = RandomUnitVector();
m = m.Rotate(v, Math.random()*2*Math.PI);
}
m = m.Translate(this.Center);
paths.Append(equator.Transform(m));
}
return paths;
}
Paths() {
const paths = new Paths();
const n = 5;
const o = 5;
for (let lat = -90 + o; lat <= 90-o; lat += n) {
const path = new Path();
for (let lng = 0; lng <= 360; lng++) {
const v = LatLngToXYZ(lat, lng, this.Radius).Add(this.Center);
path.Append(v);
}
paths.Append(path);
}
for (let lng = 0; lng < 360; lng += n) {
const path = new Path();
for (let lat = -90 + o; lat <= 90-o; lat++) {
const v = LatLngToXYZ(lat, lng, this.Radius).Add(this.Center);
path.Append(v);
}
paths.Append(path);
}
return paths;
}
}
function LatLngToXYZ(lat, lng, radius) {
const latr = Radians(lat);
const lngr = Radians(lng);
return new Vector( radius * Math.cos(latr) * Math.cos(lngr),
radius * Math.cos(latr) * Math.sin(lngr),
radius * Math.sin(latr) );
}
class OutlineSphere extends Sphere {
constructor(eye, up, center, radius) {
super(center, radius);
this.Eye = eye;
this.Up = up;
}
Paths() {
const path = new Path();
const center = this.Center;
const radius = this.Radius;
const hyp = center.Sub(this.Eye).Length();
const opp = radius;
const theta = Math.asin(opp / hyp);
const adj = opp / Math.tan(theta);
const d = Math.cos(theta) * adj;
const r = Math.sin(theta) * adj;
const w = center.Sub(this.Eye).Normalize()
const u = w.Cross(this.Up).Normalize()
const v = w.Cross(u).Normalize()
const c = this.Eye.Add(w.MulScalar(d))
for (let i = 0; i <= 360; i++) {
const a = Radians(i);
let p = c;
p = p.Add(u.MulScalar(Math.cos(a) * r));
p = p.Add(v.MulScalar(Math.sin(a) * r));
path.Append(p);
}
return new Paths([path]);
}
}
//
// https://github.com/fogleman/ln/blob/master/ln/function.go
//
const Above = 0;
const Below = 1;
class FunctionShape {
constructor(func, box, direction) {
this.Func = func;
this.Box = box;
this.Direction = direction;
}
Compile() {}
BoundingBox() { return this.Box; }
Contains(v, eps) {
if (this.Direction == Below) {
return v.Z < this.Func(v.X, v.Y);
} else {
return v.Z > this.Func(v.X, v.Y);
}
}
Intersect(ray) {
const step = 1.0 / 64;
const sign = this.Contains(ray.Position(step), 0);
for (let t = step; t < 10; t += step) {
const v = ray.Position(t);
if (this.Contains(v, 0) != sign && this.Box.Contains(v)) {
return new Hit(this, t);
}
}
return NoHit
}
Paths3() {
const path = new Path();
const n = 10000
for (let i = 0; i < n; i++) {
const t = i / n;
const r = 8 - Math.pow(t, 0.1)*8;
const x = Math.cos(Radians(t*2*Math.PI*3000)) * r;
const y = Math.sin(Radians(t*2*Math.PI*3000)) * r;
let z = this.Func(x, y);
z = Math.min(z, this.Box.Max.Z);
z = Math.max(z, this.Box.Min.Z);
path.Append(new Vector(x, y, z));
}
// return append(f.Paths2(), path)
return new Paths([path]);
}
Paths() {
const paths = new Paths();
const fine = 1.0 / 256;
for (let a = 0; a < 360; a += 5) {
const path = new Path();
for (let r = 0.0; r <= 8.0; r += fine) {
let x = Math.cos(Radians(a)) * r;
let y = Math.sin(Radians(a)) * r;
let z = this.Func(x, y);
const o = -Math.pow(-z, 1.4);
x = Math.cos(Radians(a)-o) * r;
y = Math.sin(Radians(a)-o) * r;
z = Math.min(z, this.Box.Max.Z);
z = Math.max(z, this.Box.Min.Z);
path.Append(new Vector(x, y, z));
}
paths.Append(path);
}
return paths;
}
Paths1() {
const paths = new Paths();
const step = 1.0 / 8;
const fine = 1.0 / 64;
for (let x = this.Box.Min.X; x <= this.Box.Max.X; x += step) {
const path = new Path();
for (let y = this.Box.Min.Y; y <= this.Box.Max.Y; y += fine) {
let z = this.Func(x, y);
z = Math.min(z, this.Box.Max.Z);
z = Math.max(z, this.Box.Min.Z);
path.Append(new Vector(x, y, z));
}
paths.Append(path);
}
for (let y = this.Box.Min.Y; y <= this.Box.Max.Y; y += step) {
const path = new Path();
for (let x = this.Box.Min.X; x <= this.Box.Max.X; x += fine) {
let z = this.Func(x, y);
z = Math.min(z, this.Box.Max.Z);
z = Math.max(z, this.Box.Min.Z);
path.Append(new Vector(x, y, z));
}
paths.Append([path]);
}
return paths
}
}