怎么利用Javascript生成平滑曲线

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前言

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

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：

三次贝塞尔曲线

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

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;  }

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

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

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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