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chore: Unify math types, utils and functions (#8389)
Co-authored-by: dwelle <5153846+dwelle@users.noreply.github.com>
This commit is contained in:
Márk Tolmács
GitHub
parent
e3d1dee9d0
commit
f4dd23fc31
98 changed files with 4291 additions and 3661 deletions
70
packages/math/ga/ga.test.ts
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70
packages/math/ga/ga.test.ts
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import * as GA from "./ga";
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import { point, toString, direction, offset } from "./ga";
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import * as GAPoint from "./gapoints";
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import * as GALine from "./galines";
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import * as GATransform from "./gatransforms";
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describe("geometric algebra", () => {
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describe("points", () => {
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it("distanceToLine", () => {
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const point = GA.point(3, 3);
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const line = GALine.equation(0, 1, -1);
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expect(GAPoint.distanceToLine(point, line)).toEqual(2);
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});
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it("distanceToLine neg", () => {
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const point = GA.point(-3, -3);
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const line = GALine.equation(0, 1, -1);
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expect(GAPoint.distanceToLine(point, line)).toEqual(-4);
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});
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});
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describe("lines", () => {
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it("through", () => {
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const a = GA.point(0, 0);
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const b = GA.point(2, 0);
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expect(toString(GALine.through(a, b))).toEqual(
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toString(GALine.equation(0, 2, 0)),
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);
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});
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it("parallel", () => {
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const point = GA.point(3, 3);
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const line = GALine.equation(0, 1, -1);
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const parallel = GALine.parallel(line, 2);
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expect(GAPoint.distanceToLine(point, parallel)).toEqual(0);
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});
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});
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describe("translation", () => {
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it("points", () => {
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const start = point(2, 2);
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const move = GATransform.translation(direction(0, 1));
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const end = GATransform.apply(move, start);
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expect(toString(end)).toEqual(toString(point(2, 3)));
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});
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it("points 2", () => {
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const start = point(2, 2);
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const move = GATransform.translation(offset(3, 4));
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const end = GATransform.apply(move, start);
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expect(toString(end)).toEqual(toString(point(5, 6)));
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});
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it("lines", () => {
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const original = GALine.through(point(2, 2), point(3, 4));
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const move = GATransform.translation(offset(3, 4));
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const parallel = GATransform.apply(move, original);
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expect(toString(parallel)).toEqual(
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toString(GALine.through(point(5, 6), point(6, 8))),
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);
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});
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});
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describe("rotation", () => {
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it("points", () => {
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const start = point(2, 2);
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const pivot = point(1, 1);
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const rotate = GATransform.rotation(pivot, Math.PI / 2);
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const end = GATransform.apply(rotate, start);
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expect(toString(end)).toEqual(toString(point(2, 0)));
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});
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});
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});
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317
packages/math/ga/ga.ts
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317
packages/math/ga/ga.ts
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/**
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* This is a 2D Projective Geometric Algebra implementation.
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*
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* For wider context on geometric algebra visit see https://bivector.net.
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*
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* For this specific algebra see cheatsheet https://bivector.net/2DPGA.pdf.
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*
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* Converted from generator written by enki, with a ton of added on top.
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*
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* This library uses 8-vectors to represent points, directions and lines
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* in 2D space.
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*
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* An array `[a, b, c, d, e, f, g, h]` represents a n(8)vector:
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* a + b*e0 + c*e1 + d*e2 + e*e01 + f*e20 + g*e12 + h*e012
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*
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* See GAPoint, GALine, GADirection and GATransform modules for common
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* operations.
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*/
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export type Point = NVector;
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export type Direction = NVector;
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export type Line = NVector;
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export type Transform = NVector;
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export const point = (x: number, y: number): Point => [0, 0, 0, 0, y, x, 1, 0];
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export const origin = (): Point => [0, 0, 0, 0, 0, 0, 1, 0];
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export const direction = (x: number, y: number): Direction => {
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const norm = Math.hypot(x, y); // same as `inorm(direction(x, y))`
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return [0, 0, 0, 0, y / norm, x / norm, 0, 0];
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};
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export const offset = (x: number, y: number): Direction => [
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0,
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0,
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0,
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0,
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y,
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x,
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0,
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0,
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];
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/// This is the "implementation" part of the library
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type NVector = readonly [
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number,
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number,
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number,
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number,
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number,
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number,
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number,
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number,
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];
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// These are labels for what each number in an nvector represents
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const NVECTOR_BASE = ["1", "e0", "e1", "e2", "e01", "e20", "e12", "e012"];
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// Used to represent points, lines and transformations
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export const nvector = (value: number = 0, index: number = 0): NVector => {
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const result = [0, 0, 0, 0, 0, 0, 0, 0];
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if (index < 0 || index > 7) {
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throw new Error(`Expected \`index\` between 0 and 7, got \`${index}\``);
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}
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if (value !== 0) {
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result[index] = value;
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}
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return result as unknown as NVector;
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};
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const STRING_EPSILON = 0.000001;
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export const toString = (nvector: NVector): string => {
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const result = nvector
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.map((value, index) =>
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Math.abs(value) > STRING_EPSILON
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? value.toFixed(7).replace(/(\.|0+)$/, "") +
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(index > 0 ? NVECTOR_BASE[index] : "")
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: null,
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)
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.filter((representation) => representation != null)
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.join(" + ");
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return result === "" ? "0" : result;
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};
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// Reverse the order of the basis blades.
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export const reverse = (nvector: NVector): NVector => [
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nvector[0],
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nvector[1],
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nvector[2],
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nvector[3],
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-nvector[4],
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-nvector[5],
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-nvector[6],
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-nvector[7],
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];
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// Poincare duality operator.
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export const dual = (nvector: NVector): NVector => [
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nvector[7],
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nvector[6],
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nvector[5],
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nvector[4],
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nvector[3],
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nvector[2],
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nvector[1],
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nvector[0],
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];
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// Clifford Conjugation
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export const conjugate = (nvector: NVector): NVector => [
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nvector[0],
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-nvector[1],
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-nvector[2],
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-nvector[3],
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-nvector[4],
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-nvector[5],
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-nvector[6],
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nvector[7],
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];
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// Main involution
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export const involute = (nvector: NVector): NVector => [
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nvector[0],
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-nvector[1],
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-nvector[2],
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-nvector[3],
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nvector[4],
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nvector[5],
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nvector[6],
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-nvector[7],
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];
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// Multivector addition
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export const add = (a: NVector, b: NVector | number): NVector => {
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if (isNumber(b)) {
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return [a[0] + b, a[1], a[2], a[3], a[4], a[5], a[6], a[7]];
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}
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return [
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a[0] + b[0],
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a[1] + b[1],
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a[2] + b[2],
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a[3] + b[3],
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a[4] + b[4],
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a[5] + b[5],
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a[6] + b[6],
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a[7] + b[7],
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];
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};
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// Multivector subtraction
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export const sub = (a: NVector, b: NVector | number): NVector => {
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if (isNumber(b)) {
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return [a[0] - b, a[1], a[2], a[3], a[4], a[5], a[6], a[7]];
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}
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return [
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a[0] - b[0],
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a[1] - b[1],
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a[2] - b[2],
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a[3] - b[3],
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a[4] - b[4],
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a[5] - b[5],
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a[6] - b[6],
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a[7] - b[7],
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];
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};
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// The geometric product.
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export const mul = (a: NVector, b: NVector | number): NVector => {
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if (isNumber(b)) {
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return [
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a[0] * b,
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a[1] * b,
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a[2] * b,
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a[3] * b,
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a[4] * b,
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a[5] * b,
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a[6] * b,
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a[7] * b,
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];
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}
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return [
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mulScalar(a, b),
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b[1] * a[0] +
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b[0] * a[1] -
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b[4] * a[2] +
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b[5] * a[3] +
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b[2] * a[4] -
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b[3] * a[5] -
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b[7] * a[6] -
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b[6] * a[7],
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b[2] * a[0] + b[0] * a[2] - b[6] * a[3] + b[3] * a[6],
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b[3] * a[0] + b[6] * a[2] + b[0] * a[3] - b[2] * a[6],
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b[4] * a[0] +
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b[2] * a[1] -
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b[1] * a[2] +
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b[7] * a[3] +
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b[0] * a[4] +
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b[6] * a[5] -
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b[5] * a[6] +
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b[3] * a[7],
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b[5] * a[0] -
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b[3] * a[1] +
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b[7] * a[2] +
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b[1] * a[3] -
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b[6] * a[4] +
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b[0] * a[5] +
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b[4] * a[6] +
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b[2] * a[7],
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b[6] * a[0] + b[3] * a[2] - b[2] * a[3] + b[0] * a[6],
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b[7] * a[0] +
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b[6] * a[1] +
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b[5] * a[2] +
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b[4] * a[3] +
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b[3] * a[4] +
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b[2] * a[5] +
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b[1] * a[6] +
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b[0] * a[7],
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];
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};
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export const mulScalar = (a: NVector, b: NVector): number =>
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b[0] * a[0] + b[2] * a[2] + b[3] * a[3] - b[6] * a[6];
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// The outer/exterior/wedge product.
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export const meet = (a: NVector, b: NVector): NVector => [
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b[0] * a[0],
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b[1] * a[0] + b[0] * a[1],
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b[2] * a[0] + b[0] * a[2],
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b[3] * a[0] + b[0] * a[3],
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b[4] * a[0] + b[2] * a[1] - b[1] * a[2] + b[0] * a[4],
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b[5] * a[0] - b[3] * a[1] + b[1] * a[3] + b[0] * a[5],
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b[6] * a[0] + b[3] * a[2] - b[2] * a[3] + b[0] * a[6],
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b[7] * a[0] +
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b[6] * a[1] +
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b[5] * a[2] +
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b[4] * a[3] +
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b[3] * a[4] +
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b[2] * a[5] +
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b[1] * a[6],
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];
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// The regressive product.
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export const join = (a: NVector, b: NVector): NVector => [
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joinScalar(a, b),
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a[1] * b[7] + a[4] * b[5] - a[5] * b[4] + a[7] * b[1],
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a[2] * b[7] - a[4] * b[6] + a[6] * b[4] + a[7] * b[2],
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a[3] * b[7] + a[5] * b[6] - a[6] * b[5] + a[7] * b[3],
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a[4] * b[7] + a[7] * b[4],
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a[5] * b[7] + a[7] * b[5],
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a[6] * b[7] + a[7] * b[6],
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a[7] * b[7],
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];
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export const joinScalar = (a: NVector, b: NVector): number =>
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a[0] * b[7] +
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a[1] * b[6] +
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a[2] * b[5] +
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a[3] * b[4] +
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a[4] * b[3] +
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a[5] * b[2] +
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a[6] * b[1] +
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a[7] * b[0];
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// The inner product.
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export const dot = (a: NVector, b: NVector): NVector => [
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b[0] * a[0] + b[2] * a[2] + b[3] * a[3] - b[6] * a[6],
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b[1] * a[0] +
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b[0] * a[1] -
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b[4] * a[2] +
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b[5] * a[3] +
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b[2] * a[4] -
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b[3] * a[5] -
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b[7] * a[6] -
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b[6] * a[7],
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b[2] * a[0] + b[0] * a[2] - b[6] * a[3] + b[3] * a[6],
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b[3] * a[0] + b[6] * a[2] + b[0] * a[3] - b[2] * a[6],
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b[4] * a[0] + b[7] * a[3] + b[0] * a[4] + b[3] * a[7],
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b[5] * a[0] + b[7] * a[2] + b[0] * a[5] + b[2] * a[7],
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b[6] * a[0] + b[0] * a[6],
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b[7] * a[0] + b[0] * a[7],
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];
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export const norm = (a: NVector): number =>
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Math.sqrt(Math.abs(a[0] * a[0] - a[2] * a[2] - a[3] * a[3] + a[6] * a[6]));
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export const inorm = (a: NVector): number =>
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Math.sqrt(Math.abs(a[7] * a[7] - a[5] * a[5] - a[4] * a[4] + a[1] * a[1]));
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export const normalized = (a: NVector): NVector => {
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const n = norm(a);
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if (n === 0 || n === 1) {
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return a;
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}
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const sign = a[6] < 0 ? -1 : 1;
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return mul(a, sign / n);
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};
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export const inormalized = (a: NVector): NVector => {
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const n = inorm(a);
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if (n === 0 || n === 1) {
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return a;
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}
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return mul(a, 1 / n);
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};
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const isNumber = (a: any): a is number => typeof a === "number";
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export const E0: NVector = nvector(1, 1);
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export const E1: NVector = nvector(1, 2);
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export const E2: NVector = nvector(1, 3);
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export const E01: NVector = nvector(1, 4);
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export const E20: NVector = nvector(1, 5);
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export const E12: NVector = nvector(1, 6);
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export const E012: NVector = nvector(1, 7);
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export const I = E012;
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26
packages/math/ga/gadirections.ts
Normal file
26
packages/math/ga/gadirections.ts
Normal file
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import * as GA from "./ga";
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import type { Line, Direction, Point } from "./ga";
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/**
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* A direction is stored as an array `[0, 0, 0, 0, y, x, 0, 0]` representing
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* vector `(x, y)`.
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*/
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export const from = (point: Point): Point => [
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0,
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0,
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0,
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0,
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point[4],
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point[5],
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0,
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0,
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];
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export const fromTo = (from: Point, to: Point): Direction =>
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GA.inormalized([0, 0, 0, 0, to[4] - from[4], to[5] - from[5], 0, 0]);
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export const orthogonal = (direction: Direction): Direction =>
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GA.inormalized([0, 0, 0, 0, -direction[5], direction[4], 0, 0]);
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export const orthogonalToLine = (line: Line): Direction => GA.mul(line, GA.I);
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52
packages/math/ga/galines.ts
Normal file
52
packages/math/ga/galines.ts
Normal file
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@ -0,0 +1,52 @@
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import * as GA from "./ga";
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import type { Line, Point } from "./ga";
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/**
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* A line is stored as an array `[0, c, a, b, 0, 0, 0, 0]` representing:
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* c * e0 + a * e1 + b*e2
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*
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* This maps to a standard formula `a * x + b * y + c`.
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*
|
||||
* `(-b, a)` corresponds to a 2D vector parallel to the line. The lines
|
||||
* have a natural orientation, corresponding to that vector.
|
||||
*
|
||||
* The magnitude ("norm") of the line is `sqrt(a ^ 2 + b ^ 2)`.
|
||||
* `c / norm(line)` is the oriented distance from line to origin.
|
||||
*/
|
||||
|
||||
// Returns line with direction (x, y) through origin
|
||||
export const vector = (x: number, y: number): Line =>
|
||||
GA.normalized([0, 0, -y, x, 0, 0, 0, 0]);
|
||||
|
||||
// For equation ax + by + c = 0.
|
||||
export const equation = (a: number, b: number, c: number): Line =>
|
||||
GA.normalized([0, c, a, b, 0, 0, 0, 0]);
|
||||
|
||||
export const through = (from: Point, to: Point): Line =>
|
||||
GA.normalized(GA.join(to, from));
|
||||
|
||||
export const orthogonal = (line: Line, point: Point): Line =>
|
||||
GA.dot(line, point);
|
||||
|
||||
// Returns a line perpendicular to the line through `against` and `intersection`
|
||||
// going through `intersection`.
|
||||
export const orthogonalThrough = (against: Point, intersection: Point): Line =>
|
||||
orthogonal(through(against, intersection), intersection);
|
||||
|
||||
export const parallel = (line: Line, distance: number): Line => {
|
||||
const result = line.slice();
|
||||
result[1] -= distance;
|
||||
return result as unknown as Line;
|
||||
};
|
||||
|
||||
export const parallelThrough = (line: Line, point: Point): Line =>
|
||||
orthogonal(orthogonal(point, line), point);
|
||||
|
||||
export const distance = (line1: Line, line2: Line): number =>
|
||||
GA.inorm(GA.meet(line1, line2));
|
||||
|
||||
export const angle = (line1: Line, line2: Line): number =>
|
||||
Math.acos(GA.dot(line1, line2)[0]);
|
||||
|
||||
// The orientation of the line
|
||||
export const sign = (line: Line): number => Math.sign(line[1]);
|
||||
42
packages/math/ga/gapoints.ts
Normal file
42
packages/math/ga/gapoints.ts
Normal file
|
|
@ -0,0 +1,42 @@
|
|||
import * as GA from "./ga";
|
||||
import * as GALine from "./galines";
|
||||
import type { Point, Line } from "./ga";
|
||||
import { join } from "./ga";
|
||||
|
||||
export const from = ([x, y]: readonly [number, number]): Point => [
|
||||
0,
|
||||
0,
|
||||
0,
|
||||
0,
|
||||
y,
|
||||
x,
|
||||
1,
|
||||
0,
|
||||
];
|
||||
|
||||
export const toTuple = (point: Point): [number, number] => [point[5], point[4]];
|
||||
|
||||
export const abs = (point: Point): Point => [
|
||||
0,
|
||||
0,
|
||||
0,
|
||||
0,
|
||||
Math.abs(point[4]),
|
||||
Math.abs(point[5]),
|
||||
1,
|
||||
0,
|
||||
];
|
||||
|
||||
export const intersect = (line1: Line, line2: Line): Point =>
|
||||
GA.normalized(GA.meet(line1, line2));
|
||||
|
||||
// Projects `point` onto the `line`.
|
||||
// The returned point is the closest point on the `line` to the `point`.
|
||||
export const project = (point: Point, line: Line): Point =>
|
||||
intersect(GALine.orthogonal(line, point), line);
|
||||
|
||||
export const distance = (point1: Point, point2: Point): number =>
|
||||
GA.norm(join(point1, point2));
|
||||
|
||||
export const distanceToLine = (point: Point, line: Line): number =>
|
||||
GA.joinScalar(point, line);
|
||||
41
packages/math/ga/gatransforms.ts
Normal file
41
packages/math/ga/gatransforms.ts
Normal file
|
|
@ -0,0 +1,41 @@
|
|||
import * as GA from "./ga";
|
||||
import type { Line, Direction, Point, Transform } from "./ga";
|
||||
import * as GADirection from "./gadirections";
|
||||
|
||||
/**
|
||||
* TODO: docs
|
||||
*/
|
||||
|
||||
export const rotation = (pivot: Point, angle: number): Transform =>
|
||||
GA.add(GA.mul(pivot, Math.sin(angle / 2)), Math.cos(angle / 2));
|
||||
|
||||
export const translation = (direction: Direction): Transform => [
|
||||
1,
|
||||
0,
|
||||
0,
|
||||
0,
|
||||
-(0.5 * direction[5]),
|
||||
0.5 * direction[4],
|
||||
0,
|
||||
0,
|
||||
];
|
||||
|
||||
export const translationOrthogonal = (
|
||||
direction: Direction,
|
||||
distance: number,
|
||||
): Transform => {
|
||||
const scale = 0.5 * distance;
|
||||
return [1, 0, 0, 0, scale * direction[4], scale * direction[5], 0, 0];
|
||||
};
|
||||
|
||||
export const translationAlong = (line: Line, distance: number): Transform =>
|
||||
GA.add(GA.mul(GADirection.orthogonalToLine(line), 0.5 * distance), 1);
|
||||
|
||||
export const compose = (motor1: Transform, motor2: Transform): Transform =>
|
||||
GA.mul(motor2, motor1);
|
||||
|
||||
export const apply = (
|
||||
motor: Transform,
|
||||
nvector: Point | Direction | Line,
|
||||
): Point | Direction | Line =>
|
||||
GA.normalized(GA.mul(GA.mul(motor, nvector), GA.reverse(motor)));
|
||||
Loading…
Add table
Add a link
Reference in a new issue