如何利用Javascript生成平滑曲線詳解
前言
平滑曲線生成是一個很實用的技術
很多時候,我們都需要通過繪制一些折線,然後讓計算機平滑的連接起來,
先來看下最終效果(紅色為我們輸入的直線,藍色為擬合過後的曲線) 首尾可以特殊處理讓圖形看起來更好:)
實現思路是利用貝塞爾曲線進行擬合
貝塞爾曲線簡介
貝塞爾曲線(英語:Bézier curve)是計算機圖形學中相當重要的參數曲線。
二次貝塞爾曲線
二次方貝塞爾曲線的路徑由給定點P0、P1、P2的函數B(t)追蹤:
三次貝塞爾曲線
對於三次曲線,可由線性貝塞爾曲線描述的中介點Q0、Q1、Q2,和由二次曲線描述的點R0、R1所建構
貝塞爾曲線計算函數
根據上面的公式我們可有得到計算函數
二階
/** * * * @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), }; } /** * 生成二次方貝塞爾曲線頂點數據 * * @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; }
三階
/** * 三次方塞爾曲線公式 * * @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; } /** * 獲取坐標 * * @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), }; } /** * 生成三次方貝塞爾曲線頂點數據 * * @param {Point} p0 起始點 { x : number, y : number} * @param {Point} p1 控制點1 { x : number, y : number} * @param {Point} p2 控制點2 { x : number, y : number} * @param {Point} p3 終止點 { 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; }
擬合算法
問題在於如何得到控制點,我們以比較簡單的方法
取 p1-pt-p2的角平分線 c1c2垂直於該條角平分線 c2為p2的投影點取短邊作為c1-pt c2-pt的長度對該長度進行縮放 這個長度可以大概理解為曲線的彎曲程度
ab線段 這裡簡單處理 隻使用瞭二階的曲線生成 -> 🌈 這裡可以按照個人想法處理
bc線段使用abc計算出來的控制點c2和bcd計算出來的控制點c3 以此類推
/** * 生成平滑曲線所需的控制點 * * @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}; } /** * 平滑曲線生成 * * @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; // 首端控制點隻用一個 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; }
案例代碼
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() })
附錄:Vector2D相關的代碼
/** * * * @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; } /** * 歸一 * * @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); } /** * 向量的減法 * * @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};
總結
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