smoothstep.hpp

#ifndef PASTELMATH_SMOOTHSTEP_HPP
#define PASTELMATH_SMOOTHSTEP_HPP

#include "pastel/math/interpolation/smoothstep.h"

#include "pastel/sys/math_functions.h"

namespace Pastel
{

    template <typename Real>
    Real cubicSmoothStep(
        const Real& x)
    {
        // Let
        // h(x) = ax^3 + bx^2 + cx + d
        //
        // Require:
        // h(0) = 0
        // h(1) = 1
        // h'(0) = 0
        // h'(1) = 0
        //
        // This leads to the solution
        // h(x) = -2 * x^3 + 3 * x^2

        return (x * x * (3 - 2 * x));
    }

    template <typename PointType, typename Real>
    PointType cubicSmoothStep(
        const PointType& from,
        const PointType& to,
        const Real& time)
    {
        return linear(from, to, Pastel::cubicSmoothStep(time));
    }

    template <typename Real>
    Real quarticSmoothStep(
        const Real& xSquared)
    {
        // Let
        // h(x) = ax^4 + bx^3 + cx^2 + dx + e
        //
        // Require:
        // h(0) = 0
        // h(1) = 1
        // h'(0) = 0
        // h'(1) = 0
        // b = 0
        // d = 0
        //
        // There are 6 requirements but only
        // 5 degrees of freedom. However,
        // d = 0 is equivalent to h'(0) = 0
        // so there are actually only 5
        // requirements.
        //
        // This leads to the solution
        // h(x) = -r^4 + 2r^2

        return -xSquared * xSquared + 2 * xSquared;
    }

    template <typename PointType, typename Real>
    PointType quarticSmoothStep(
        const PointType& from,
        const PointType& to,
        const Real& time)
    {
        return linear(from, to, Pastel::quarticSmoothStep(time));
    }

    template <typename Real>
    Real quinticSmoothStep(
        const Real& x)
    {
        // Let
        // h(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f
        //
        // Requirements:
        // h(0) = 0
        // h(1) = 1
        // h'(0) = 0
        // h'(1) = 0
        // h''(0) = 0
        // h''(1) = 0
        //
        // This leads to the solution:
        // h(x) = 6 * x^5 - 15 * x^4 + 10 * x^3

        return (x * x * x *
            (10 + x * (-15 + x * 6)));
    }

    template <typename PointType, typename Real>
    PointType quinticSmoothStep(
        const PointType& from,
        const PointType& to,
        const Real& time)
    {
        return linear(from, to, Pastel::quinticSmoothStep(time));
    }

}

#endif