如何利用Javascript生成平滑曲线详解如何利用Javascript生成平滑曲线

前言
贝塞尔曲线简介

二次贝塞尔曲线
三次贝塞尔曲线

贝塞尔曲线计算函数

拟合算法

附录：Vector2D相关的代码

总结

前言

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平滑曲线生成是一个很实用的技术
很多时候，我们都需要通过绘制一些折线，然后让计算机平滑的连接起来，
先来看下最终效果(红色为我们输入的直线，蓝色为拟合过后的曲线) 首尾可以特殊处理让图形看起来更好：）

实现思路是利用贝塞尔曲线进行拟合

贝塞尔曲线简介贝塞尔曲线（英语：Bézier curve）是计算机图形学中相当重要的参数曲线。

二次贝塞尔曲线

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二次方贝塞尔曲线的路径由给定点P0、P1、P2的函数B（t）追踪：

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三次贝塞尔曲线

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对于三次曲线，可由线性贝塞尔曲线描述的中介点Q0、Q1、Q2，和由二次曲线描述的点R0、R1所建构

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贝塞尔曲线计算函数根据上面的公式我们可有得到计算函数
二阶

/***** @param {number} p0* @param {number} p1* @param {number} p2* @param {number} t* @return {*}* @memberof Path*/bezier2P(p0: number, p1: number, p2: number, t: number) {const P0 = p0 * Math.pow(1 - t, 2); const P1 = p1 * 2 * t * (1 - t); const P2 = p2 * t * t; return P0 + P1 + P2; }/***** @param {Point} p0* @param {Point} p1* @param {Point} p2* @param {number} num* @param {number} tick* @return {*}{Point}* @memberof Path*/getBezierNowPoint2P(p0: Point,p1: Point,p2: Point,num: number,tick: number,): Point {return {x: this.bezier2P(p0.x, p1.x, p2.x, num * tick),y: this.bezier2P(p0.y, p1.y, p2.y, num * tick),}; }/*** 生成二次方贝塞尔曲线顶点数据** @param {Point} p0* @param {Point} p1* @param {Point} p2* @param {number} [num=100]* @param {number} [tick=1]* @return {*}* @memberof Path*/create2PBezier(p0: Point,p1: Point,p2: Point,num: number = 100,tick: number = 1,) {const t = tick / (num - 1); const points = []; for (let i = 0; i < num; i++) {const point = this.getBezierNowPoint2P(p0, p1, p2, i, t); points.push({x: point.x, y: point.y}); }return points; }

【如何利用Javascript生成平滑曲线详解】三阶

/*** 三次方塞尔曲线公式** @param {number} p0* @param {number} p1* @param {number} p2* @param {number} p3* @param {number} t* @return {*}* @memberof Path*/bezier3P(p0: number, p1: number, p2: number, p3: number, t: number) {const P0 = p0 * Math.pow(1 - t, 3); const P1 = 3 * p1 * t * Math.pow(1 - t, 2); const P2 = 3 * p2 * Math.pow(t, 2) * (1 - t); const P3 = p3 * Math.pow(t, 3); return P0 + P1 + P2 + P3; }/*** 获取坐标** @param {Point} p0* @param {Point} p1* @param {Point} p2* @param {Point} p3* @param {number} num* @param {number} tick* @return {*}* @memberof Path*/getBezierNowPoint3P(p0: Point,p1: Point,p2: Point,p3: Point,num: number,tick: number,) {return {x: this.bezier3P(p0.x, p1.x, p2.x, p3.x, num * tick),y: this.bezier3P(p0.y, p1.y, p2.y, p3.y, num * tick),}; }/*** 生成三次方贝塞尔曲线顶点数据** @param {Point} p0 起始点{ x : number, y : number}* @param {Point} p1 控制点1 { x : number, y : number}* @param {Point} p2 控制点2 { x : number, y : number}* @param {Point} p3 终止点{ x : number, y : number}* @param {number} [num=100]* @param {number} [tick=1]* @return {Point []}* @memberof Path*/create3PBezier(p0: Point,p1: Point,p2: Point,p3: Point,num: number = 100,tick: number = 1,) {const pointMum = num; const _tick = tick; const t = _tick / (pointMum - 1); const points = []; for (let i = 0; i < pointMum; i++) {const point = this.getBezierNowPoint3P(p0, p1, p2, p3, i, t); points.push({x: point.x, y: point.y}); }return points; }

拟合算法

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问题在于如何得到控制点，我们以比较简单的方法
取 p1-pt-p2的角平分线 c1c2垂直于该条角平分线 c2为p2的投影点取短边作为c1-pt c2-pt的长度对该长度进行缩放这个长度可以大概理解为曲线的弯曲程度

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ab线段这里简单处理只使用了二阶的曲线生成 -> 这里可以按照个人想法处理
bc线段使用abc计算出来的控制点c2和bcd计算出来的控制点c3 以此类推

/*** 生成平滑曲线所需的控制点** @param {Vector2D} p1* @param {Vector2D} pt* @param {Vector2D} p2* @param {number} [ratio=0.3]* @return {*}* @memberof Path*/createSmoothLineControlPoint(p1: Vector2D,pt: Vector2D,p2: Vector2D,ratio: number = 0.3,) {const vec1T: Vector2D = vector2dMinus(p1, pt); const vecT2: Vector2D = vector2dMinus(p1, pt); const len1: number = vec1T.length; const len2: number = vecT2.length; const v: number = len1 / len2; let delta; if (v > 1) {delta = vector2dMinus(p1,vector2dPlus(pt, vector2dMinus(p2, pt).scale(1 / v)),); } else {delta = vector2dMinus(vector2dPlus(pt, vector2dMinus(p1, pt).scale(v)),p2,); }delta = delta.scale(ratio); const control1: Point = {x: vector2dPlus(pt, delta).x,y: vector2dPlus(pt, delta).y,}; const control2: Point = {x: vector2dMinus(pt, delta).x,y: vector2dMinus(pt, delta).y,}; return {control1, control2}; }/*** 平滑曲线生成** @param {Point []} points* @param {number} ratio* @return {*}* @memberof Path*/createSmoothLine(points: Point[], ratio: number = 0.3) {const len = points.length; let resultPoints = []; const controlPoints = []; if (len < 3) return; for (let i = 0; i < len - 2; i++) {const {control1, control2} = this.createSmoothLineControlPoint(new Vector2D(points[i].x, points[i].y),new Vector2D(points[i + 1].x, points[i + 1].y),new Vector2D(points[i + 2].x, points[i + 2].y),ratio,); controlPoints.push(control1); controlPoints.push(control2); let points1; let points2; // 首端控制点只用一个if (i === 0) {points1 = this.create2PBezier(points[i], control1, points[i + 1], 50); } else {console.log(controlPoints); points1 = this.create3PBezier(points[i],controlPoints[2 * i - 1],control1,points[i + 1],50,); }// 尾端部分if (i + 2 === len - 1) {points2 = this.create2PBezier(points[i + 1],control2,points[i + 2],50,); }if (i + 2 === len - 1) {resultPoints = [...resultPoints, ...points1, ...points2]; } else {resultPoints = [...resultPoints, ...points1]; }}return resultPoints; }

案例代码

const input = [{ x: 0, y: 0 },{ x: 150, y: 150 },{ x: 300, y: 0 },{ x: 400, y: 150 },{ x: 500, y: 0 },{ x: 650, y: 150 },]const s = path.createSmoothLine(input); let ctx = document.getElementById('cv').getContext('2d'); ctx.strokeStyle = 'blue'; ctx.beginPath(); ctx.moveTo(0, 0); for (let i = 0; i < s.length; i++) {ctx.lineTo(s[i].x, s[i].y); }ctx.stroke(); ctx.beginPath(); ctx.moveTo(0, 0); for (let i = 0; i < input.length; i++) {ctx.lineTo(input[i].x, input[i].y); }ctx.strokeStyle = 'red'; ctx.stroke(); document.getElementById('btn').addEventListener('click', () => {let app = document.getElementById('app'); let index = 0; let move = () => {if (index < s.length) {app.style.left = s[index].x - 10 + 'px'; app.style.top = s[index].y - 10 + 'px'; index++; requestAnimationFrame(move)}}move()})

附录：Vector2D相关的代码

/** * * * @class Vector2D * @extends {Array} */class Vector2D extends Array {/*** Creates an instance of Vector2D.* @param {number} [x=1]* @param {number} [y=0]* @memberof Vector2D* */constructor(x: number = 1, y: number = 0) {super(); this.x = x; this.y = y; }/**** @param {number} v* @memberof Vector2D*/set x(v) {this[0] = v; }/**** @param {number} v* @memberof Vector2D*/set y(v) {this[1] = v; }/***** @readonly* @memberof Vector2D*/get x() {return this[0]; }/***** @readonly* @memberof Vector2D*/get y() {return this[1]; }/***** @readonly* @memberof Vector2D*/get length() {return Math.hypot(this.x, this.y); }/***** @readonly* @memberof Vector2D*/get dir() {return Math.atan2(this.y, this.x); }/***** @return {*}* @memberof Vector2D*/copy() {return new Vector2D(this.x, this.y); }/***** @param {*} v* @return {*}* @memberof Vector2D*/add(v) {this.x += v.x; this.y += v.y; return this; }/***** @param {*} v* @return {*}* @memberof Vector2D*/sub(v) {this.x -= v.x; this.y -= v.y; return this; }/***** @param {*} a* @return {Vector2D}* @memberof Vector2D*/scale(a) {this.x *= a; this.y *= a; return this; }/***** @param {*} rad* @return {*}* @memberof Vector2D*/rotate(rad) {const c = Math.cos(rad); const s = Math.sin(rad); const [x, y] = this; this.x = x * c + y * -s; this.y = x * s + y * c; return this; }/***** @param {*} v* @return {*}* @memberof Vector2D*/cross(v) {return this.x * v.y - v.x * this.y; }/***** @param {*} v* @return {*}* @memberof Vector2D*/dot(v) {return this.x * v.x + v.y * this.y; }/*** 归一** @return {*}* @memberof Vector2D*/normalize() {return this.scale(1 / this.length); }}/** * 向量的加法 * * @param {*} vec1 * @param {*} vec2 * @return {Vector2D} */function vector2dPlus(vec1, vec2) {return new Vector2D(vec1.x + vec2.x, vec1.y + vec2.y); }/** * 向量的减法 * * @param {*} vec1 * @param {*} vec2 * @return {Vector2D} */function vector2dMinus(vec1, vec2) {return new Vector2D(vec1.x - vec2.x, vec1.y - vec2.y); }export {Vector2D, vector2dPlus, vector2dMinus};

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