GteSlerpEstimate.h

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// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2017
// Distributed under the Boost Software License, Version 1.0.
// http://www.boost.org/LICENSE_1_0.txt
// http://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
// File Version: 3.0.0 (2016/06/19)

#pragma once

#include <Mathematics/GteQuaternion.h>

// The spherical linear interpolation (slerp) of unit-length quaternions
// q0 and q1 for t in [0,1] and theta in (0,pi) is
//   slerp(t,q0,q1) = [sin((1-t)*theta)*q0 + sin(theta)*q1]/sin(theta)
// where theta is the angle between q0 and q1 [cos(theta) = Dot(q0,q1)].
// This function is a parameterization of the great spherical arc between
// q0 and q1 on the unit hypersphere.  Moreover, the parameterization is
// one of normalized arclength--a particle traveling along the arc through
// time t does so with constant speed.
//
// Read the comments in GteChebyshevRatio.h regarding estimates for the
// ratio sin(t*theta)/sin(theta).
//
// When using slerp in animations involving sequences of quaternions, it is
// typical that the quaternions are preprocessed so that consecutive ones
// form an acute angle A in [0,pi/2].  Other preprocessing can help with
// performance.  See the function comments in the SLERP class.

namespace gte
{

template <typename Real>
class SLERP
{
public:
    // The angle between q0 and q1 is in [0,pi).  There are no angle
    // restrictions and nothing is precomputed.
    template <int N>
    inline static Quaternion<Real> Estimate(Real t,
        Quaternion<Real> const& q0, Quaternion<Real> const& q1);


    // The angle between q0 and q1 must be in [0,pi/2].  The suffix R is for
    // 'Restricted'.  The preprocessing code is
    //   Quaternion<Real> q[n];  // assuming initialized
    //   for (i0 = 0, i1 = 1; i1 < n; i0 = i1++)
    //   {
    //       cosA = Dot(q[i0], q[i1]);
    //       if (cosA < 0)
    //       {
    //           q[i1] = -q[i1];  // now Dot(q[i0], q[i]1) >= 0
    //       }
    //   }
    template <int N>
    inline static Quaternion<Real> EstimateR(Real t,
        Quaternion<Real> const& q0, Quaternion<Real> const& q1);


    // The angle between q0 and q1 must be in [0,pi/2].  The suffix R is for
    // 'Restricted' and the suffix P is for 'Preprocessed'.  The preprocessing
    // code is
    //   Quaternion<Real> q[n];  // assuming initialized
    //   Real cosA[n-1], omcosA[n-1];  // to be precomputed
    //   for (i0 = 0, i1 = 1; i1 < n; i0 = i1++)
    //   {
    //       cs = Dot(q[i0], q[i1]);
    //       if (cosA[i0] < 0)
    //       {
    //           q[i1] = -q[i1];
    //           cs = -cs;
    //       }
    //       cosA[n-1] = cs;  // for GeneralRP
    //       omcosA[i0] = 1 - cs;  // for EstimateRP
    //   }
    template <int N>
    inline static Quaternion<Real> EstimateRP(Real t,
        Quaternion<Real> const& q0, Quaternion<Real> const& q1, Real omcosA);

    // The angle between q0 and q1 is A and must be in [0,pi/2].  Quaternion
    // qh is slerp(1/2,q0,q1) = (q0+q1)/|q0+q1|, so the angle between q0 and
    // qh is A/2 and the angle between qh and q1 is A/2.  The preprocessing
    // code is
    //   Quaternion<Real> q[n];  // assuming initialized
    //   Quaternion<Real> qh[n-1];  // to be precomputed
    //   Real omcosAH[n-1];  // to be precomputed
    //   for (i0 = 0, i1 = 1; i1 < n; i0 = i1++)
    //   {
    //       cosA = Dot(q[i0], q[i1]);
    //       if (cosA < 0)
    //       {
    //           q[i1] = -q[i1];
    //           cosA = -cosA;
    //       }
    //       cosAH = sqrt((1 + cosA)/2);
    //       qh[i0] = (q0 + q1) / (2 * cosAH[i0]);
    //       omcosAH[i0] = 1 - cosAH;
    //   }
    template <int N>
    inline static Quaternion<Real> EstimateRPH(Real t,
        Quaternion<Real> const& q0, Quaternion<Real> const& q1,
        Quaternion<Real> const& qh, Real omcosAH);
};


template <typename Real> template <int N> inline
Quaternion<Real> SLERP<Real>::Estimate(Real t,
Quaternion<Real> const& q0, Quaternion<Real> const& q1)
{
    static_assert(1 <= N && N <= 16, "Invalid degree.");

    Real cs = Dot(q0, q1);
    Real sign;
    if (cs >= (Real)0)
    {
        sign = (Real)1;
    }
    else
    {
        cs = -cs;
        sign = (Real)-1;
    }

    Real f0, f1;
    ChebyshevRatio<Real>::template
        GetEstimate<N>(t, (Real)1 - cs, f0, f1);
    return q0 * f0 + q1 * (sign * f1);
}

template <typename Real> template <int N> inline
Quaternion<Real> SLERP<Real>::EstimateR(Real t,
Quaternion<Real> const& q0, Quaternion<Real> const& q1)
{
    static_assert(1 <= N && N <= 16, "Invalid degree.");

    Real f0, f1;
    ChebyshevRatio<Real>::template
        GetEstimate<N>(t, (Real)1 - Dot(q0, q1), f0, f1);
    return q0 * f0 + q1 * f1;
}

template <typename Real> template <int N> inline
Quaternion<Real> SLERP<Real>::EstimateRP(Real t,
Quaternion<Real> const& q0, Quaternion<Real> const& q1, Real omcosA)
{
    static_assert(1 <= N && N <= 16, "Invalid degree.");

    Real f0, f1;
    ChebyshevRatio<Real>::template
        GetEstimate<N>(t, omcosA, f0, f1);
    return q0 * f0 + q1 * f1;
}

template <typename Real> template <int N> inline
Quaternion<Real> SLERP<Real>::EstimateRPH(Real t,
Quaternion<Real> const& q0, Quaternion<Real> const& q1,
Quaternion<Real> const& qh, Real omcosAH)
{
    static_assert(1 <= N && N <= 16, "Invalid degree.");

    Real f0, f1;
    Real twoT = t * (Real)2;
    if (twoT <= (Real)1)
    {
        ChebyshevRatio<Real>::template
            GetEstimate<N>(twoT, omcosAH, f0, f1);
        return q0 * f0 + qh * f1;
    }
    else
    {
        ChebyshevRatio<Real>::template
            GetEstimate<N>(twoT - (Real)1, omcosAH, f0, f1);
        return qh * f0 + q1 * f1;
    }
}


}