如何利用Javascript生成平滑曲线详解
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};
总结
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