@woosh/meep-engine/src/core/math/spline/spline3_hermite_integral.js

Pure JavaScript game engine. Fully featured and production ready.

/**
 * Calculates the integral (area under the curve) of a cubic hermite spline from 0 to t.
 *
 * @param {number} t normalized interpolation position in interval [0,1]
 * @param {number} p0 first value
 * @param {number} p1 second value
 * @param {number} m0 first tangent
 * @param {number} m1 second tangent
 * @returns {number} The definite integral from 0 to t
 *
 * @see spline3_hermite_derivative
 */
export function spline3_hermite_integral(t, p0, p1, m0, m1) {
    const t2 = t * t;
    const t3 = t2 * t;
    const t4 = t3 * t;

    // Integral of (2t^3 - 3t^2 + 1)
    // Result: 0.5t^4 - t^3 + t
    const int_p0 = 0.5 * t4 - t3 + t;

    // Integral of (t^3 - 2t^2 + t)
    // Result: 0.25t^4 - (2/3)t^3 + 0.5t^2
    const int_m0 = 0.25 * t4 - (2.0 / 3.0) * t3 + 0.5 * t2;

    // Integral of (t^3 - t^2)
    // Result: 0.25t^4 - (1/3)t^3
    const int_m1 = 0.25 * t4 - (1.0 / 3.0) * t3;

    // Integral of (3t^2 - 2t^3)
    // Result: t^3 - 0.5t^4
    const int_p1 = t3 - 0.5 * t4;

    return int_p0 * p0 + int_m0 * m0 + int_m1 * m1 + int_p1 * p1;
}