/**
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* 曲线辅助模块
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*/
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import {
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create as v2Create,
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distSquare as v2DistSquare,
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VectorArray
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} from './vector';
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const mathPow = Math.pow;
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const mathSqrt = Math.sqrt;
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const EPSILON = 1e-8;
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const EPSILON_NUMERIC = 1e-4;
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const THREE_SQRT = mathSqrt(3);
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const ONE_THIRD = 1 / 3;
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// 临时变量
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const _v0 = v2Create();
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const _v1 = v2Create();
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const _v2 = v2Create();
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function isAroundZero(val: number) {
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return val > -EPSILON && val < EPSILON;
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}
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function isNotAroundZero(val: number) {
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return val > EPSILON || val < -EPSILON;
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}
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/**
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* 计算三次贝塞尔值
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*/
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export function cubicAt(p0: number, p1: number, p2: number, p3: number, t: number): number {
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const onet = 1 - t;
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return onet * onet * (onet * p0 + 3 * t * p1)
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+ t * t * (t * p3 + 3 * onet * p2);
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}
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/**
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* 计算三次贝塞尔导数值
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*/
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export function cubicDerivativeAt(p0: number, p1: number, p2: number, p3: number, t: number): number {
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const onet = 1 - t;
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return 3 * (
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((p1 - p0) * onet + 2 * (p2 - p1) * t) * onet
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+ (p3 - p2) * t * t
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);
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}
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/**
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* 计算三次贝塞尔方程根，使用盛金公式
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*/
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export function cubicRootAt(p0: number, p1: number, p2: number, p3: number, val: number, roots: number[]): number {
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// Evaluate roots of cubic functions
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const a = p3 + 3 * (p1 - p2) - p0;
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const b = 3 * (p2 - p1 * 2 + p0);
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const c = 3 * (p1 - p0);
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const d = p0 - val;
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const A = b * b - 3 * a * c;
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const B = b * c - 9 * a * d;
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const C = c * c - 3 * b * d;
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let n = 0;
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if (isAroundZero(A) && isAroundZero(B)) {
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if (isAroundZero(b)) {
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roots[0] = 0;
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}
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else {
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const t1 = -c / b; //t1, t2, t3, b is not zero
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if (t1 >= 0 && t1 <= 1) {
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roots[n++] = t1;
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}
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}
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}
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else {
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const disc = B * B - 4 * A * C;
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if (isAroundZero(disc)) {
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const K = B / A;
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const t1 = -b / a + K; // t1, a is not zero
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const t2 = -K / 2; // t2, t3
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if (t1 >= 0 && t1 <= 1) {
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roots[n++] = t1;
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}
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if (t2 >= 0 && t2 <= 1) {
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roots[n++] = t2;
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}
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}
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else if (disc > 0) {
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const discSqrt = mathSqrt(disc);
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let Y1 = A * b + 1.5 * a * (-B + discSqrt);
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let Y2 = A * b + 1.5 * a * (-B - discSqrt);
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if (Y1 < 0) {
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Y1 = -mathPow(-Y1, ONE_THIRD);
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}
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else {
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Y1 = mathPow(Y1, ONE_THIRD);
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}
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if (Y2 < 0) {
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Y2 = -mathPow(-Y2, ONE_THIRD);
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}
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else {
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Y2 = mathPow(Y2, ONE_THIRD);
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}
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const t1 = (-b - (Y1 + Y2)) / (3 * a);
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if (t1 >= 0 && t1 <= 1) {
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roots[n++] = t1;
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}
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}
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else {
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const T = (2 * A * b - 3 * a * B) / (2 * mathSqrt(A * A * A));
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const theta = Math.acos(T) / 3;
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const ASqrt = mathSqrt(A);
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const tmp = Math.cos(theta);
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const t1 = (-b - 2 * ASqrt * tmp) / (3 * a);
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const t2 = (-b + ASqrt * (tmp + THREE_SQRT * Math.sin(theta))) / (3 * a);
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const t3 = (-b + ASqrt * (tmp - THREE_SQRT * Math.sin(theta))) / (3 * a);
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if (t1 >= 0 && t1 <= 1) {
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roots[n++] = t1;
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}
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if (t2 >= 0 && t2 <= 1) {
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roots[n++] = t2;
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}
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if (t3 >= 0 && t3 <= 1) {
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roots[n++] = t3;
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}
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}
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}
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return n;
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}
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/**
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* 计算三次贝塞尔方程极限值的位置
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* @return 有效数目
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*/
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export function cubicExtrema(p0: number, p1: number, p2: number, p3: number, extrema: number[]): number {
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const b = 6 * p2 - 12 * p1 + 6 * p0;
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const a = 9 * p1 + 3 * p3 - 3 * p0 - 9 * p2;
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const c = 3 * p1 - 3 * p0;
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let n = 0;
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if (isAroundZero(a)) {
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if (isNotAroundZero(b)) {
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const t1 = -c / b;
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if (t1 >= 0 && t1 <= 1) {
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extrema[n++] = t1;
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}
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}
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}
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else {
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const disc = b * b - 4 * a * c;
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if (isAroundZero(disc)) {
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extrema[0] = -b / (2 * a);
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}
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else if (disc > 0) {
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const discSqrt = mathSqrt(disc);
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const t1 = (-b + discSqrt) / (2 * a);
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const t2 = (-b - discSqrt) / (2 * a);
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if (t1 >= 0 && t1 <= 1) {
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extrema[n++] = t1;
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}
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if (t2 >= 0 && t2 <= 1) {
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extrema[n++] = t2;
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}
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}
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}
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return n;
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}
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/**
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* 细分三次贝塞尔曲线
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*/
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export function cubicSubdivide(p0: number, p1: number, p2: number, p3: number, t: number, out: number[]) {
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const p01 = (p1 - p0) * t + p0;
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const p12 = (p2 - p1) * t + p1;
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const p23 = (p3 - p2) * t + p2;
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const p012 = (p12 - p01) * t + p01;
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const p123 = (p23 - p12) * t + p12;
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const p0123 = (p123 - p012) * t + p012;
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// Seg0
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out[0] = p0;
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out[1] = p01;
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out[2] = p012;
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out[3] = p0123;
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// Seg1
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out[4] = p0123;
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out[5] = p123;
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out[6] = p23;
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out[7] = p3;
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}
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/**
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* 投射点到三次贝塞尔曲线上，返回投射距离。
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* 投射点有可能会有一个或者多个，这里只返回其中距离最短的一个。
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*/
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export function cubicProjectPoint(
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x0: number, y0: number, x1: number, y1: number, x2: number, y2: number, x3: number, y3: number,
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x: number, y: number, out: VectorArray
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): number {
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// http://pomax.github.io/bezierinfo/#projections
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let t;
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let interval = 0.005;
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let d = Infinity;
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let prev;
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let next;
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let d1;
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let d2;
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_v0[0] = x;
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_v0[1] = y;
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// 先粗略估计一下可能的最小距离的 t 值
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// PENDING
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for (let _t = 0; _t < 1; _t += 0.05) {
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_v1[0] = cubicAt(x0, x1, x2, x3, _t);
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_v1[1] = cubicAt(y0, y1, y2, y3, _t);
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d1 = v2DistSquare(_v0, _v1);
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if (d1 < d) {
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t = _t;
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d = d1;
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}
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}
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d = Infinity;
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// At most 32 iteration
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for (let i = 0; i < 32; i++) {
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if (interval < EPSILON_NUMERIC) {
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break;
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}
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prev = t - interval;
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next = t + interval;
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// t - interval
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_v1[0] = cubicAt(x0, x1, x2, x3, prev);
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_v1[1] = cubicAt(y0, y1, y2, y3, prev);
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d1 = v2DistSquare(_v1, _v0);
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if (prev >= 0 && d1 < d) {
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t = prev;
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d = d1;
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}
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else {
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// t + interval
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_v2[0] = cubicAt(x0, x1, x2, x3, next);
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_v2[1] = cubicAt(y0, y1, y2, y3, next);
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d2 = v2DistSquare(_v2, _v0);
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if (next <= 1 && d2 < d) {
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t = next;
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d = d2;
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}
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else {
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interval *= 0.5;
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}
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}
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}
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// t
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if (out) {
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out[0] = cubicAt(x0, x1, x2, x3, t);
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out[1] = cubicAt(y0, y1, y2, y3, t);
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}
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// console.log(interval, i);
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return mathSqrt(d);
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}
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/**
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* 计算三次贝塞尔曲线长度
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*/
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export function cubicLength(
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x0: number, y0: number, x1: number, y1: number, x2: number, y2: number, x3: number, y3: number,
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iteration: number
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) {
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let px = x0;
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let py = y0;
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let d = 0;
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const step = 1 / iteration;
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for (let i = 1; i <= iteration; i++) {
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let t = i * step;
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const x = cubicAt(x0, x1, x2, x3, t);
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const y = cubicAt(y0, y1, y2, y3, t);
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const dx = x - px;
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const dy = y - py;
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d += Math.sqrt(dx * dx + dy * dy);
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px = x;
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py = y;
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}
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return d;
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}
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/**
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* 计算二次方贝塞尔值
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*/
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export function quadraticAt(p0: number, p1: number, p2: number, t: number): number {
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const onet = 1 - t;
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return onet * (onet * p0 + 2 * t * p1) + t * t * p2;
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}
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/**
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* 计算二次方贝塞尔导数值
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*/
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export function quadraticDerivativeAt(p0: number, p1: number, p2: number, t: number): number {
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return 2 * ((1 - t) * (p1 - p0) + t * (p2 - p1));
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}
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/**
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* 计算二次方贝塞尔方程根
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* @return 有效根数目
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*/
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export function quadraticRootAt(p0: number, p1: number, p2: number, val: number, roots: number[]): number {
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const a = p0 - 2 * p1 + p2;
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const b = 2 * (p1 - p0);
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const c = p0 - val;
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let n = 0;
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if (isAroundZero(a)) {
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if (isNotAroundZero(b)) {
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const t1 = -c / b;
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if (t1 >= 0 && t1 <= 1) {
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roots[n++] = t1;
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}
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}
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}
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else {
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const disc = b * b - 4 * a * c;
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if (isAroundZero(disc)) {
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const t1 = -b / (2 * a);
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if (t1 >= 0 && t1 <= 1) {
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roots[n++] = t1;
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}
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}
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else if (disc > 0) {
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const discSqrt = mathSqrt(disc);
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const t1 = (-b + discSqrt) / (2 * a);
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const t2 = (-b - discSqrt) / (2 * a);
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if (t1 >= 0 && t1 <= 1) {
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roots[n++] = t1;
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}
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if (t2 >= 0 && t2 <= 1) {
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roots[n++] = t2;
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}
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}
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}
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return n;
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}
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/**
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* 计算二次贝塞尔方程极限值
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*/
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export function quadraticExtremum(p0: number, p1: number, p2: number): number {
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const divider = p0 + p2 - 2 * p1;
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if (divider === 0) {
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// p1 is center of p0 and p2
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return 0.5;
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}
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else {
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return (p0 - p1) / divider;
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}
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}
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/**
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* 细分二次贝塞尔曲线
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*/
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export function quadraticSubdivide(p0: number, p1: number, p2: number, t: number, out: number[]) {
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const p01 = (p1 - p0) * t + p0;
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const p12 = (p2 - p1) * t + p1;
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const p012 = (p12 - p01) * t + p01;
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// Seg0
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out[0] = p0;
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out[1] = p01;
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out[2] = p012;
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// Seg1
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out[3] = p012;
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out[4] = p12;
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out[5] = p2;
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}
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/**
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* 投射点到二次贝塞尔曲线上，返回投射距离。
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* 投射点有可能会有一个或者多个，这里只返回其中距离最短的一个。
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* @param {number} x0
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* @param {number} y0
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* @param {number} x1
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* @param {number} y1
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* @param {number} x2
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* @param {number} y2
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* @param {number} x
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* @param {number} y
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* @param {Array.<number>} out 投射点
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* @return {number}
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*/
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export function quadraticProjectPoint(
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x0: number, y0: number, x1: number, y1: number, x2: number, y2: number,
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x: number, y: number, out: VectorArray
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): number {
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// http://pomax.github.io/bezierinfo/#projections
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let t: number;
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let interval = 0.005;
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let d = Infinity;
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_v0[0] = x;
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_v0[1] = y;
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// 先粗略估计一下可能的最小距离的 t 值
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// PENDING
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for (let _t = 0; _t < 1; _t += 0.05) {
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_v1[0] = quadraticAt(x0, x1, x2, _t);
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_v1[1] = quadraticAt(y0, y1, y2, _t);
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const d1 = v2DistSquare(_v0, _v1);
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if (d1 < d) {
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t = _t;
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d = d1;
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}
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}
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d = Infinity;
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// At most 32 iteration
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for (let i = 0; i < 32; i++) {
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if (interval < EPSILON_NUMERIC) {
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break;
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}
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const prev = t - interval;
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const next = t + interval;
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// t - interval
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_v1[0] = quadraticAt(x0, x1, x2, prev);
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_v1[1] = quadraticAt(y0, y1, y2, prev);
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const d1 = v2DistSquare(_v1, _v0);
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if (prev >= 0 && d1 < d) {
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t = prev;
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d = d1;
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}
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else {
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// t + interval
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_v2[0] = quadraticAt(x0, x1, x2, next);
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_v2[1] = quadraticAt(y0, y1, y2, next);
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const d2 = v2DistSquare(_v2, _v0);
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if (next <= 1 && d2 < d) {
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t = next;
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d = d2;
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}
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else {
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interval *= 0.5;
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}
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}
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}
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// t
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if (out) {
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out[0] = quadraticAt(x0, x1, x2, t);
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out[1] = quadraticAt(y0, y1, y2, t);
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}
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// console.log(interval, i);
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return mathSqrt(d);
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}
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/**
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* 计算二次贝塞尔曲线长度
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*/
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export function quadraticLength(
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x0: number, y0: number, x1: number, y1: number, x2: number, y2: number,
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iteration: number
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) {
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let px = x0;
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let py = y0;
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let d = 0;
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const step = 1 / iteration;
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for (let i = 1; i <= iteration; i++) {
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let t = i * step;
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const x = quadraticAt(x0, x1, x2, t);
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const y = quadraticAt(y0, y1, y2, t);
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const dx = x - px;
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const dy = y - py;
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d += Math.sqrt(dx * dx + dy * dy);
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px = x;
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py = y;
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}
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return d;
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}
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