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/**
* Finds the parameter t for a given x value on a cubic Bézier curve.
* Uses Newton's method first (fast), then falls back to binary search (reliable).
* Based on WebKit/Chromium's UnitBezier implementation.
*
* @param {number} targetX - The x value to find parameter t for
* @param {number} x1 - x-coordinate of first control point
* @param {number} x2 - x-coordinate of second control point
* @returns {number} The parameter t that gives the specified x value
*/
function solveCubicBezierX(targetX: number, x1: number, x2: number): number {
const epsilon = 1e-6;
// Pre-compute polynomial coefficients for performance
const cx = 3.0 * x1;
const bx = 3.0 * (x2 - x1) - cx;
const ax = 1.0 - cx - bx;
function sampleCurveX(t: number): number {
// Evaluate: ax*t³ + bx*t² + cx*t using Horner's rule
return ((ax * t + bx) * t + cx) * t;
}
function sampleCurveDerivativeX(t: number): number {
return (3.0 * ax * t + 2.0 * bx) * t + cx;
}
/**
* Newton's method - fast convergence with good initial guess
* Try this first as it's normally very fast
*/
function newtonRaphsonMethod(targetX: number): number | null {
let t = targetX;
for (let iteration = 0; iteration < 8; iteration++) {
const currentX = sampleCurveX(t);
const error = currentX - targetX;
if (Math.abs(error) < epsilon) {
return t;
}
const derivative = sampleCurveDerivativeX(t);
// Break if derivative is too small (avoid division by zero)
Iif (Math.abs(derivative) < epsilon) {
break;
}
// Newton-Raphson step: t_new = t_old - f(t_old) / f'(t_old)
t = t - error / derivative;
}
return null;
}
/**
* Binary search fallback - guaranteed to converge
* Use this when Newton's method fails
*/
function binarySearchFallback(targetX: number): number {
let lowerBound = 0.0;
let upperBound = 1.0;
let t = targetX;
while (lowerBound < upperBound) {
const currentX = sampleCurveX(t);
Iif (Math.abs(currentX - targetX) < epsilon) {
return t;
}
if (targetX > currentX) {
lowerBound = t;
} else {
upperBound = t;
}
t = (upperBound - lowerBound) * 0.5 + lowerBound;
}
return t;
}
const newtonResult = newtonRaphsonMethod(targetX);
if (newtonResult !== null) {
return newtonResult;
}
return binarySearchFallback(targetX);
}
/**
* Evaluates the Y coordinate of a cubic Bézier curve at parameter t.
* Optimized for CSS timing functions where P0=(0,0) and P3=(1,1).
* Uses pre-computed polynomial coefficients and Horner's rule for performance.
*
* @param {number} t - Parameter value along the curve, typically in [0, 1]
* @param {number} y1 - y-coordinate of first control point
* @param {number} y2 - y-coordinate of second control point
* @returns {number} The Y value of the curve at parameter t
*/
function sampleCurveY(t: number, y1: number, y2: number): number {
// Pre-compute polynomial coefficients for performance
const cy = 3.0 * y1;
const by = 3.0 * (y2 - y1) - cy;
const ay = 1.0 - cy - by;
// Evaluate: ay*t³ + by*t² + cy*t using Horner's rule
return ((ay * t + by) * t + cy) * t;
}
/**
* Cubic Bézier easing function implementation following CSS specifications.
*
* A cubic Bézier curve is defined by four points: P0, P1, P2, and P3.
* In CSS animations, P0 is fixed at (0, 0) and P3 is fixed at (1, 1).
* This function allows you to specify the intermediate control points P1 and P2.
*
* @param {number} x1 - x-coordinate of the first control point P1 (must be in [0, 1])
* @param {number} y1 - y-coordinate of the first control point P1
* @param {number} x2 - x-coordinate of the second control point P2 (must be in [0, 1])
* @param {number} y2 - y-coordinate of the second control point P2
* @returns {function} A function that takes time t ∈ [0, 1] and returns the eased value
*/
export function cubicBezier(x1: number, y1: number, x2: number, y2: number) {
Iif (x1 < 0 || x1 > 1 || x2 < 0 || x2 > 1) {
throw new Error('x1 and x2 must be in the range [0, 1]');
}
return function (timeProgress: number): number {
if (timeProgress <= 0) return 0;
Iif (timeProgress >= 1) return 1;
Iif (x1 === y1 && x2 === y2) {
return timeProgress;
}
const curveParameter = solveCubicBezierX(timeProgress, x1, x2);
return sampleCurveY(curveParameter, y1, y2);
};
}
/** Core Easing Functions */
/**
* Linear easing function - constant speed throughout the animation.
* Equivalent to CSS: cubic-bezier(0.0, 0.0, 1.0, 1.0)
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @returns {number} The linear eased value.
*/
const linearCore = cubicBezier(0.0, 0.0, 1.0, 1.0);
/**
* Ease-in easing function - starts slow and accelerates.
* Equivalent to CSS: cubic-bezier(0.42, 0, 1.0, 1.0)
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @returns {number} The eased value.
*/
const easeInCore = cubicBezier(0.42, 0, 1.0, 1.0);
/**
* Ease-out easing function - starts fast and decelerates.
* Equivalent to CSS: cubic-bezier(0, 0, 0.58, 1.0)
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @returns {number} The eased value.
*/
const easeOutCore = cubicBezier(0, 0, 0.58, 1.0);
/**
* Ease-in-out easing function - starts slow, accelerates, then decelerates.
* Equivalent to CSS: cubic-bezier(0.42, 0, 0.58, 1.0)
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @returns {number} The eased value.
*/
const easeInOutCore = cubicBezier(0.42, 0, 0.58, 1.0);
/**
* Standard ease easing function (CSS default).
* Equivalent to CSS: cubic-bezier(0.25, 0.1, 0.25, 1.0)
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @returns {number} The eased value.
*/
const easeCore = cubicBezier(0.25, 0.1, 0.25, 1.0);
/** Flexible Easing Function Factory */
/**
* Flexible factory function that creates easing functions with optional baseline and scale support.
* Provides bell-curve behavior: baseline → baseline*scale → baseline
*
* @param {Function} coreEasingFn - The core easing function to wrap
* @returns {Function} A function that accepts (timeProgress, baseline?, scale?) and provides flexible behavior
*/
function flexibleEasingFunctionFactory(coreEasingFn: (timeProgress: number) => number) {
return function (timeProgress: number, baseline: number = 0, scale?: number): number {
Iif (baseline === 0) {
return coreEasingFn(timeProgress);
}
const easedProgress = coreEasingFn(timeProgress);
const targetValue = scale ? baseline * scale : 1;
const range = targetValue - baseline;
// Create bell-curve: baseline → targetValue → baseline
const bellMultiplier = 1 - Math.abs(2 * easedProgress - 1);
return baseline + bellMultiplier * range;
};
}
/**
* Linear easing function with optional baseline and scale support.
* - No params: standard linear progression [0, 1]
* - With baseline only: bell-curve baseline → 1 → baseline
* - With baseline+scale: bell-curve baseline → baseline*scale → baseline
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @param {number} baseline - Optional baseline value (default: 0). Creates bell-curve effect.
* @param {number} scale - Optional scale multiplier (default: 1). Peak value = baseline * scale.
* @returns {number} The linear eased value.
*/
export const linear = flexibleEasingFunctionFactory(linearCore);
/**
* Standard ease easing function with optional baseline and scale support.
* - No params: standard ease progression [0, 1]
* - With baseline only: bell-curve baseline → 1 → baseline
* - With baseline+scale: bell-curve baseline → baseline*scale → baseline
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @param {number} baseline - Optional baseline value (default: 0). Creates bell-curve effect.
* @param {number} scale - Optional scale multiplier (default: 1). Peak value = baseline * scale.
* @returns {number} The eased value.
*/
export const ease = flexibleEasingFunctionFactory(easeCore);
/**
* Ease-in easing function with optional baseline and scale support.
* - No params: standard ease-in progression [0, 1]
* - With baseline only: bell-curve baseline → 1 → baseline
* - With baseline+scale: bell-curve baseline → baseline*scale → baseline
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @param {number} baseline - Optional baseline value (default: 0). Creates bell-curve effect.
* @param {number} scale - Optional scale multiplier (default: 1). Peak value = baseline * scale.
* @returns {number} The eased value.
*/
export const easeIn = flexibleEasingFunctionFactory(easeInCore);
/**
* Ease-out easing function with optional baseline and scale support.
* - No params: standard ease-out progression [0, 1]
* - With baseline only: bell-curve baseline → 1 → baseline
* - With baseline+scale: bell-curve baseline → baseline*scale → baseline
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @param {number} baseline - Optional baseline value (default: 0). Creates bell-curve effect.
* @param {number} scale - Optional scale multiplier (default: 1). Peak value = baseline * scale.
* @returns {number} The eased value.
*/
export const easeOut = flexibleEasingFunctionFactory(easeOutCore);
/**
* Ease-in-out easing function with optional baseline and scale support.
* - No params: standard ease-in-out progression [0, 1]
* - With baseline only: bell-curve baseline → 1 → baseline
* - With baseline+scale: bell-curve baseline → baseline*scale → baseline
*
* @param {number} timeProgress - The animation progress, in the range [0, 1].
* @param {number} baseline - Optional baseline value (default: 0). Creates bell-curve effect.
* @param {number} scale - Optional scale multiplier (default: 1). Peak value = baseline * scale.
* @returns {number} The eased value.
*/
export const easeInOut = flexibleEasingFunctionFactory(easeInOutCore);
/** Export interfaces */
export enum EasingFunctionEnum {
EASE = 'ease',
EASE_IN = 'ease-in',
EASE_OUT = 'ease-out',
EASE_IN_OUT = 'ease-in-out',
LINEAR = 'linear',
}
export const EasingFunctionMap = new Map([
[EasingFunctionEnum.EASE, ease],
[EasingFunctionEnum.EASE_IN, easeIn],
[EasingFunctionEnum.EASE_OUT, easeOut],
[EasingFunctionEnum.EASE_IN_OUT, easeInOut],
[EasingFunctionEnum.LINEAR, linear],
]);
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