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Changed Newton’s method to algebraic solution #57

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Changed Newton’s method to algebraic solution #57

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2 changes: 1 addition & 1 deletion src/index.d.ts
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Expand Up @@ -4,5 +4,5 @@ export = bezier;
declare function bezier(x1: number, y1: number, x2: number, y2: number): bezier.EasingFunction;

declare namespace bezier {
export interface EasingFunction { (input: number): number }
export type EasingFunction = (input: number) => number
}
154 changes: 59 additions & 95 deletions src/index.js
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Expand Up @@ -4,101 +4,65 @@
* by Gaëtan Renaudeau 2014 - 2015 – MIT License
*/

// These values are established by empiricism with tests (tradeoff: performance VS precision)
var NEWTON_ITERATIONS = 4;
var NEWTON_MIN_SLOPE = 0.001;
var SUBDIVISION_PRECISION = 0.0000001;
var SUBDIVISION_MAX_ITERATIONS = 10;

var kSplineTableSize = 11;
var kSampleStepSize = 1.0 / (kSplineTableSize - 1.0);

var float32ArraySupported = typeof Float32Array === 'function';

function A (aA1, aA2) { return 1.0 - 3.0 * aA2 + 3.0 * aA1; }
function B (aA1, aA2) { return 3.0 * aA2 - 6.0 * aA1; }
function C (aA1) { return 3.0 * aA1; }

// Returns x(t) given t, x1, and x2, or y(t) given t, y1, and y2.
function calcBezier (aT, aA1, aA2) { return ((A(aA1, aA2) * aT + B(aA1, aA2)) * aT + C(aA1)) * aT; }

// Returns dx/dt given t, x1, and x2, or dy/dt given t, y1, and y2.
function getSlope (aT, aA1, aA2) { return 3.0 * A(aA1, aA2) * aT * aT + 2.0 * B(aA1, aA2) * aT + C(aA1); }

function binarySubdivide (aX, aA, aB, mX1, mX2) {
var currentX, currentT, i = 0;
do {
currentT = aA + (aB - aA) / 2.0;
currentX = calcBezier(currentT, mX1, mX2) - aX;
if (currentX > 0.0) {
aB = currentT;
} else {
aA = currentT;
}
} while (Math.abs(currentX) > SUBDIVISION_PRECISION && ++i < SUBDIVISION_MAX_ITERATIONS);
return currentT;
}

function newtonRaphsonIterate (aX, aGuessT, mX1, mX2) {
for (var i = 0; i < NEWTON_ITERATIONS; ++i) {
var currentSlope = getSlope(aGuessT, mX1, mX2);
if (currentSlope === 0.0) {
return aGuessT;
}
var currentX = calcBezier(aGuessT, mX1, mX2) - aX;
aGuessT -= currentX / currentSlope;
}
return aGuessT;
}

function LinearEasing (x) {
return x;
function LinearEasing(x) {
return x;
}

module.exports = function bezier (mX1, mY1, mX2, mY2) {
if (!(0 <= mX1 && mX1 <= 1 && 0 <= mX2 && mX2 <= 1)) {
throw new Error('bezier x values must be in [0, 1] range');
}

if (mX1 === mY1 && mX2 === mY2) {
return LinearEasing;
}

// Precompute samples table
var sampleValues = float32ArraySupported ? new Float32Array(kSplineTableSize) : new Array(kSplineTableSize);
for (var i = 0; i < kSplineTableSize; ++i) {
sampleValues[i] = calcBezier(i * kSampleStepSize, mX1, mX2);
}

function getTForX (aX) {
var intervalStart = 0.0;
var currentSample = 1;
var lastSample = kSplineTableSize - 1;

for (; currentSample !== lastSample && sampleValues[currentSample] <= aX; ++currentSample) {
intervalStart += kSampleStepSize;
}
--currentSample;

// Interpolate to provide an initial guess for t
var dist = (aX - sampleValues[currentSample]) / (sampleValues[currentSample + 1] - sampleValues[currentSample]);
var guessForT = intervalStart + dist * kSampleStepSize;

var initialSlope = getSlope(guessForT, mX1, mX2);
if (initialSlope >= NEWTON_MIN_SLOPE) {
return newtonRaphsonIterate(aX, guessForT, mX1, mX2);
} else if (initialSlope === 0.0) {
return guessForT;
} else {
return binarySubdivide(aX, intervalStart, intervalStart + kSampleStepSize, mX1, mX2);
}
}

return function BezierEasing (x) {
// Because JavaScript number are imprecise, we should guarantee the extremes are right.
if (x === 0 || x === 1) {
return x;
}
return calcBezier(getTForX(x), mY1, mY2);
};
const { cbrt, sqrt, PI: π } = Math;

const x2t = (x, a, b, c, d) => {
const q = a + b * x;
const s = q ** 2 + c;
if (s > 0) {
const root = sqrt(s);
return cbrt(q + root) + cbrt(q - root) - d;
}
const l = cbrt(sqrt(q * q - s));
const angle = q ? Math.atan(sqrt(-s) / q) : -π / 2;
let φ;
if (b < 0) {
φ = (q > 0 ? 2 * π : π) - angle;
} else if (d < 0) {
φ = (q > 0 ? 2 * π : -3 * π) + angle;
} else {
φ = (q > 0 ? 0 : π) + angle;
}
return 2 * l * Math.cos(φ / 3) - d;
};
const Y = (t, ay, by, cy) => ((ay * t + 3 * by) * t + cy) * t;

module.exports = function bezier(mX1, mY1, mX2, mY2) {
if (!(0 <= mX1 && mX1 <= 1 && 0 <= mX2 && mX2 <= 1)) {
throw new Error("Bezier x values must be in [0, 1] range");
}

if (mX1 === mY1 && mX2 === mY2) {
return LinearEasing;
}

const a = 6 * (3 * mX1 - 3 * mX2 + 1);
const b = 6 * (mX2 - 2 * mX1);
const c = 3 * mX1;

const a2 = a * a;
const b2 = b * b;

const d = b / a;
const e = (3 * b * c) / a2 - (b2 * b) / (a2 * a);
const w1 = (2 * c) / a - b2 / a2;
const w = w1 * w1 * w1;
const o = 3 / a;

const ay = 3 * mY1 - 3 * mY2 + 1;
const by = mY2 - 2 * mY1;
const cy = 3 * mY1;

const X2T = a ? x2t : LinearEasing;

return (x) => {
if (x === 0 || x === 1) {
return x;
}
return Y(X2T(x, e, o, w, d), ay, by, cy);
};
};