curve.js

/**
 * 曲线辅助模块
 * @module zrender/core/curve
 * @author pissang(https://www.github.com/pissang)
 */

import {
    create as v2Create,
    distSquare as v2DistSquare
} from './vector';

var mathPow = Math.pow;
var mathSqrt = Math.sqrt;

var EPSILON = 1e-8;
var EPSILON_NUMERIC = 1e-4;

var THREE_SQRT = mathSqrt(3);
var ONE_THIRD = 1 / 3;

// 临时变量
var _v0 = v2Create();
var _v1 = v2Create();
var _v2 = v2Create();

function isAroundZero(val) {
    return val > -EPSILON && val < EPSILON;
}
function isNotAroundZero(val) {
    return val > EPSILON || val < -EPSILON;
}
/**
 * 计算三次贝塞尔值
 * @memberOf module:zrender/core/curve
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} p3
 * @param  {number} t
 * @return {number}
 */
export function cubicAt(p0, p1, p2, p3, t) {
    var onet = 1 - t;
    return onet * onet * (onet * p0 + 3 * t * p1)
            + t * t * (t * p3 + 3 * onet * p2);
}

/**
 * 计算三次贝塞尔导数值
 * @memberOf module:zrender/core/curve
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} p3
 * @param  {number} t
 * @return {number}
 */
export function cubicDerivativeAt(p0, p1, p2, p3, t) {
    var onet = 1 - t;
    return 3 * (
        ((p1 - p0) * onet + 2 * (p2 - p1) * t) * onet
        + (p3 - p2) * t * t
    );
}

/**
 * 计算三次贝塞尔方程根，使用盛金公式
 * @memberOf module:zrender/core/curve
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} p3
 * @param  {number} val
 * @param  {Array.<number>} roots
 * @return {number} 有效根数目
 */
export function cubicRootAt(p0, p1, p2, p3, val, roots) {
    // Evaluate roots of cubic functions
    var a = p3 + 3 * (p1 - p2) - p0;
    var b = 3 * (p2 - p1 * 2 + p0);
    var c = 3 * (p1  - p0);
    var d = p0 - val;

    var A = b * b - 3 * a * c;
    var B = b * c - 9 * a * d;
    var C = c * c - 3 * b * d;

    var n = 0;

    if (isAroundZero(A) && isAroundZero(B)) {
        if (isAroundZero(b)) {
            roots[0] = 0;
        }
        else {
            var t1 = -c / b;  //t1, t2, t3, b is not zero
            if (t1 >= 0 && t1 <= 1) {
                roots[n++] = t1;
            }
        }
    }
    else {
        var disc = B * B - 4 * A * C;

        if (isAroundZero(disc)) {
            var K = B / A;
            var t1 = -b / a + K;  // t1, a is not zero
            var t2 = -K / 2;  // t2, t3
            if (t1 >= 0 && t1 <= 1) {
                roots[n++] = t1;
            }
            if (t2 >= 0 && t2 <= 1) {
                roots[n++] = t2;
            }
        }
        else if (disc > 0) {
            var discSqrt = mathSqrt(disc);
            var Y1 = A * b + 1.5 * a * (-B + discSqrt);
            var Y2 = A * b + 1.5 * a * (-B - discSqrt);
            if (Y1 < 0) {
                Y1 = -mathPow(-Y1, ONE_THIRD);
            }
            else {
                Y1 = mathPow(Y1, ONE_THIRD);
            }
            if (Y2 < 0) {
                Y2 = -mathPow(-Y2, ONE_THIRD);
            }
            else {
                Y2 = mathPow(Y2, ONE_THIRD);
            }
            var t1 = (-b - (Y1 + Y2)) / (3 * a);
            if (t1 >= 0 && t1 <= 1) {
                roots[n++] = t1;
            }
        }
        else {
            var T = (2 * A * b - 3 * a * B) / (2 * mathSqrt(A * A * A));
            var theta = Math.acos(T) / 3;
            var ASqrt = mathSqrt(A);
            var tmp = Math.cos(theta);

            var t1 = (-b - 2 * ASqrt * tmp) / (3 * a);
            var t2 = (-b + ASqrt * (tmp + THREE_SQRT * Math.sin(theta))) / (3 * a);
            var t3 = (-b + ASqrt * (tmp - THREE_SQRT * Math.sin(theta))) / (3 * a);
            if (t1 >= 0 && t1 <= 1) {
                roots[n++] = t1;
            }
            if (t2 >= 0 && t2 <= 1) {
                roots[n++] = t2;
            }
            if (t3 >= 0 && t3 <= 1) {
                roots[n++] = t3;
            }
        }
    }
    return n;
}

/**
 * 计算三次贝塞尔方程极限值的位置
 * @memberOf module:zrender/core/curve
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} p3
 * @param  {Array.<number>} extrema
 * @return {number} 有效数目
 */
export function cubicExtrema(p0, p1, p2, p3, extrema) {
    var b = 6 * p2 - 12 * p1 + 6 * p0;
    var a = 9 * p1 + 3 * p3 - 3 * p0 - 9 * p2;
    var c = 3 * p1 - 3 * p0;

    var n = 0;
    if (isAroundZero(a)) {
        if (isNotAroundZero(b)) {
            var t1 = -c / b;
            if (t1 >= 0 && t1 <=1) {
                extrema[n++] = t1;
            }
        }
    }
    else {
        var disc = b * b - 4 * a * c;
        if (isAroundZero(disc)) {
            extrema[0] = -b / (2 * a);
        }
        else if (disc > 0) {
            var discSqrt = mathSqrt(disc);
            var t1 = (-b + discSqrt) / (2 * a);
            var t2 = (-b - discSqrt) / (2 * a);
            if (t1 >= 0 && t1 <= 1) {
                extrema[n++] = t1;
            }
            if (t2 >= 0 && t2 <= 1) {
                extrema[n++] = t2;
            }
        }
    }
    return n;
}

/**
 * 细分三次贝塞尔曲线
 * @memberOf module:zrender/core/curve
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} p3
 * @param  {number} t
 * @param  {Array.<number>} out
 */
export function cubicSubdivide(p0, p1, p2, p3, t, out) {
    var p01 = (p1 - p0) * t + p0;
    var p12 = (p2 - p1) * t + p1;
    var p23 = (p3 - p2) * t + p2;

    var p012 = (p12 - p01) * t + p01;
    var p123 = (p23 - p12) * t + p12;

    var p0123 = (p123 - p012) * t + p012;
    // Seg0
    out[0] = p0;
    out[1] = p01;
    out[2] = p012;
    out[3] = p0123;
    // Seg1
    out[4] = p0123;
    out[5] = p123;
    out[6] = p23;
    out[7] = p3;
}

/**
 * 投射点到三次贝塞尔曲线上，返回投射距离。
 * 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。
 * @param {number} x0
 * @param {number} y0
 * @param {number} x1
 * @param {number} y1
 * @param {number} x2
 * @param {number} y2
 * @param {number} x3
 * @param {number} y3
 * @param {number} x
 * @param {number} y
 * @param {Array.<number>} [out] 投射点
 * @return {number}
 */
export function cubicProjectPoint(
    x0, y0, x1, y1, x2, y2, x3, y3,
    x, y, out
) {
    // http://pomax.github.io/bezierinfo/#projections
    var t;
    var interval = 0.005;
    var d = Infinity;
    var prev;
    var next;
    var d1;
    var d2;

    _v0[0] = x;
    _v0[1] = y;

    // 先粗略估计一下可能的最小距离的 t 值
    // PENDING
    for (var _t = 0; _t < 1; _t += 0.05) {
        _v1[0] = cubicAt(x0, x1, x2, x3, _t);
        _v1[1] = cubicAt(y0, y1, y2, y3, _t);
        d1 = v2DistSquare(_v0, _v1);
        if (d1 < d) {
            t = _t;
            d = d1;
        }
    }
    d = Infinity;

    // At most 32 iteration
    for (var i = 0; i < 32; i++) {
        if (interval < EPSILON_NUMERIC) {
            break;
        }
        prev = t - interval;
        next = t + interval;
        // t - interval
        _v1[0] = cubicAt(x0, x1, x2, x3, prev);
        _v1[1] = cubicAt(y0, y1, y2, y3, prev);

        d1 = v2DistSquare(_v1, _v0);

        if (prev >= 0 && d1 < d) {
            t = prev;
            d = d1;
        }
        else {
            // t + interval
            _v2[0] = cubicAt(x0, x1, x2, x3, next);
            _v2[1] = cubicAt(y0, y1, y2, y3, next);
            d2 = v2DistSquare(_v2, _v0);

            if (next <= 1 && d2 < d) {
                t = next;
                d = d2;
            }
            else {
                interval *= 0.5;
            }
        }
    }
    // t
    if (out) {
        out[0] = cubicAt(x0, x1, x2, x3, t);
        out[1] = cubicAt(y0, y1, y2, y3, t);
    }
    // console.log(interval, i);
    return mathSqrt(d);
}

/**
 * 计算二次方贝塞尔值
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} t
 * @return {number}
 */
export function quadraticAt(p0, p1, p2, t) {
    var onet = 1 - t;
    return onet * (onet * p0 + 2 * t * p1) + t * t * p2;
}

/**
 * 计算二次方贝塞尔导数值
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} t
 * @return {number}
 */
export function quadraticDerivativeAt(p0, p1, p2, t) {
    return 2 * ((1 - t) * (p1 - p0) + t * (p2 - p1));
}

/**
 * 计算二次方贝塞尔方程根
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} t
 * @param  {Array.<number>} roots
 * @return {number} 有效根数目
 */
export function quadraticRootAt(p0, p1, p2, val, roots) {
    var a = p0 - 2 * p1 + p2;
    var b = 2 * (p1 - p0);
    var c = p0 - val;

    var n = 0;
    if (isAroundZero(a)) {
        if (isNotAroundZero(b)) {
            var t1 = -c / b;
            if (t1 >= 0 && t1 <= 1) {
                roots[n++] = t1;
            }
        }
    }
    else {
        var disc = b * b - 4 * a * c;
        if (isAroundZero(disc)) {
            var t1 = -b / (2 * a);
            if (t1 >= 0 && t1 <= 1) {
                roots[n++] = t1;
            }
        }
        else if (disc > 0) {
            var discSqrt = mathSqrt(disc);
            var t1 = (-b + discSqrt) / (2 * a);
            var t2 = (-b - discSqrt) / (2 * a);
            if (t1 >= 0 && t1 <= 1) {
                roots[n++] = t1;
            }
            if (t2 >= 0 && t2 <= 1) {
                roots[n++] = t2;
            }
        }
    }
    return n;
}

/**
 * 计算二次贝塞尔方程极限值
 * @memberOf module:zrender/core/curve
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @return {number}
 */
export function quadraticExtremum(p0, p1, p2) {
    var divider = p0 + p2 - 2 * p1;
    if (divider === 0) {
        // p1 is center of p0 and p2
        return 0.5;
    }
    else {
        return (p0 - p1) / divider;
    }
}

/**
 * 细分二次贝塞尔曲线
 * @memberOf module:zrender/core/curve
 * @param  {number} p0
 * @param  {number} p1
 * @param  {number} p2
 * @param  {number} t
 * @param  {Array.<number>} out
 */
export function quadraticSubdivide(p0, p1, p2, t, out) {
    var p01 = (p1 - p0) * t + p0;
    var p12 = (p2 - p1) * t + p1;
    var p012 = (p12 - p01) * t + p01;

    // Seg0
    out[0] = p0;
    out[1] = p01;
    out[2] = p012;

    // Seg1
    out[3] = p012;
    out[4] = p12;
    out[5] = p2;
}

/**
 * 投射点到二次贝塞尔曲线上，返回投射距离。
 * 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。
 * @param {number} x0
 * @param {number} y0
 * @param {number} x1
 * @param {number} y1
 * @param {number} x2
 * @param {number} y2
 * @param {number} x
 * @param {number} y
 * @param {Array.<number>} out 投射点
 * @return {number}
 */
export function quadraticProjectPoint(
    x0, y0, x1, y1, x2, y2,
    x, y, out
) {
    // http://pomax.github.io/bezierinfo/#projections
    var t;
    var interval = 0.005;
    var d = Infinity;

    _v0[0] = x;
    _v0[1] = y;

    // 先粗略估计一下可能的最小距离的 t 值
    // PENDING
    for (var _t = 0; _t < 1; _t += 0.05) {
        _v1[0] = quadraticAt(x0, x1, x2, _t);
        _v1[1] = quadraticAt(y0, y1, y2, _t);
        var d1 = v2DistSquare(_v0, _v1);
        if (d1 < d) {
            t = _t;
            d = d1;
        }
    }
    d = Infinity;

    // At most 32 iteration
    for (var i = 0; i < 32; i++) {
        if (interval < EPSILON_NUMERIC) {
            break;
        }
        var prev = t - interval;
        var next = t + interval;
        // t - interval
        _v1[0] = quadraticAt(x0, x1, x2, prev);
        _v1[1] = quadraticAt(y0, y1, y2, prev);

        var d1 = v2DistSquare(_v1, _v0);

        if (prev >= 0 && d1 < d) {
            t = prev;
            d = d1;
        }
        else {
            // t + interval
            _v2[0] = quadraticAt(x0, x1, x2, next);
            _v2[1] = quadraticAt(y0, y1, y2, next);
            var d2 = v2DistSquare(_v2, _v0);
            if (next <= 1 && d2 < d) {
                t = next;
                d = d2;
            }
            else {
                interval *= 0.5;
            }
        }
    }
    // t
    if (out) {
        out[0] = quadraticAt(x0, x1, x2, t);
        out[1] = quadraticAt(y0, y1, y2, t);
    }
    // console.log(interval, i);
    return mathSqrt(d);
}