IAPDemo/Libraries/cocos2d/cocos/rparticle/Math/AnimationCurve.cpp

//
//  AnimationCurve.cpp
//  cocos2d_libs
//
//  Created by 徐俊杰 on 2020/5/15.
//

#include "rparticle/Math/AnimationCurve.h"

using namespace std;

#define kOneThird (1.0F / 3.0F)
#define kMaxTan 5729577.9485111479F

NS_RRP_BEGIN

int ToInternalInfinity (int pre);
int FromInternalInfinity (int pre);

int ToInternalInfinity (int pre)
{
    if (pre == kRepeat)
        return AnimationCurve::kInternalRepeat;
    else if (pre == kPingPong)
        return AnimationCurve::kInternalPingPong;
    else
        return AnimationCurve::kInternalClamp;
}

int FromInternalInfinity (int pre)
{
    if (pre == AnimationCurve::kInternalRepeat)
        return kRepeat;
    else if (pre == AnimationCurve::kInternalPingPong)
        return kPingPong;
    else
        return kClampForever;
}

template<class T>
KeyframeTpl<T>::KeyframeTpl (float t, const T& v)
{
    time = t;
    value = v;
    inSlope = Zero<T>();
    outSlope = Zero<T>();
    
#if UNITY_EDITOR
    tangentMode = 0;
#endif
}


template<class T>
void AnimationCurveTpl<T>::InvalidateCache ()
{
    m_Cache.time = std::numeric_limits<float>::infinity ();
    m_Cache.index = 0;
    m_ClampCache.time = std::numeric_limits<float>::infinity ();
    m_ClampCache.index = 0;
}

template<class T>
pair<float, float> AnimationCurveTpl<T>::GetRange () const
{
    if (!m_Curve.empty ())
        return make_pair (m_Curve[0].time, m_Curve.back ().time);
    else
        return make_pair (std::numeric_limits<float>::infinity (), -std::numeric_limits<float>::infinity ());
}



///@TODO: Handle step curves correctly
template<class T>
void AnimationCurveTpl<T>::EvaluateWithoutCache (float curveT, T& output)const
{
    DebugAssertIf (!IsValid ());
    curveT = WrapTime (curveT);
    
    int lhsIndex, rhsIndex;
    FindIndexForSampling (m_Cache, curveT, lhsIndex, rhsIndex);
    const Keyframe& lhs = m_Curve[lhsIndex];
    const Keyframe& rhs = m_Curve[rhsIndex];
    
    float dx = rhs.time - lhs.time;
    T m1;
    T m2;
    float t;
    if (dx != 0.0F)
    {
        t = (curveT - lhs.time) / dx;
        m1 = lhs.outSlope * dx;
        m2 = rhs.inSlope * dx;
    }
    else
    {
        t = 0.0F;
        m1 = Zero<T>();
        m2 = Zero<T>();
    }
    
    output = HermiteInterpolate (t, lhs.value, m1, m2, rhs.value);
    HandleSteppedCurve(lhs, rhs, output);
    //TODO: IsFinite
//    DebugAssertIf(!IsFinite(output));
}

template<class T>
inline void EvaluateCache (const typename AnimationCurveTpl<T>::Cache& cache, float curveT, T& output)
{
    //    DebugAssertIf (curveT < cache.time - kCurveTimeEpsilon || curveT > cache.timeEnd + kCurveTimeEpsilon);
    float t = curveT - cache.time;
    output = (t * (t * (t * cache.coeff[0] + cache.coeff[1]) + cache.coeff[2])) + cache.coeff[3];
    //TODO: IsFinite
//    DebugAssertIf (!IsFinite(output));
}

void SetupStepped (float* coeff, const KeyframeTpl<float>& lhs, const KeyframeTpl<float>& rhs)
{
    // If either of the tangents in the segment are set to stepped, make the constant value equal the value of the left key
    if (lhs.outSlope == std::numeric_limits<float>::infinity() || rhs.inSlope == std::numeric_limits<float>::infinity())
    {
        coeff[0] = 0.0F;
        coeff[1] = 0.0F;
        coeff[2] = 0.0F;
        coeff[3] = lhs.value;
    }
}

void HandleSteppedCurve (const KeyframeTpl<float>& lhs, const KeyframeTpl<float>& rhs, float& value)
{
    if (lhs.outSlope == std::numeric_limits<float>::infinity() || rhs.inSlope == std::numeric_limits<float>::infinity())
        value = lhs.value;
}

void HandleSteppedTangent (const KeyframeTpl<float>& lhs, const KeyframeTpl<float>& rhs, float& tangent)
{
    if (lhs.outSlope == std::numeric_limits<float>::infinity() || rhs.inSlope == std::numeric_limits<float>::infinity())
        tangent = std::numeric_limits<float>::infinity();
}


void SetupStepped (Vector3f* coeff, const KeyframeTpl<Vector3f>& lhs, const KeyframeTpl<Vector3f>& rhs)
{
    for (int i=0;i<3;i++)
    {
        // If either of the tangents in the segment are set to stepped, make the constant value equal the value of the left key
        if (lhs.outSlope[i] == std::numeric_limits<float>::infinity() || rhs.inSlope[i] == std::numeric_limits<float>::infinity())
        {
            coeff[0][i] = 0.0F;
            coeff[1][i] = 0.0F;
            coeff[2][i] = 0.0F;
            coeff[3][i] = lhs.value[i];
        }
    }
}

void HandleSteppedCurve (const KeyframeTpl<Vector3f>& lhs, const KeyframeTpl<Vector3f>& rhs, Vector3f& value)
{
    for (int i=0;i<3;i++)
    {
        if (lhs.outSlope[i] == std::numeric_limits<float>::infinity() || rhs.inSlope[i] == std::numeric_limits<float>::infinity())
            value[i] = lhs.value[i];
    }
}

void HandleSteppedTangent (const KeyframeTpl<Vector3f>& lhs, const KeyframeTpl<Vector3f>& rhs, Vector3f& value)
{
    for (int i=0;i<3;i++)
    {
        if (lhs.outSlope[i] == std::numeric_limits<float>::infinity() || rhs.inSlope[i] == std::numeric_limits<float>::infinity())
            value[i] = std::numeric_limits<float>::infinity();
    }
}

void SetupStepped (Quaternionf* coeff, const KeyframeTpl<Quaternionf>& lhs, const KeyframeTpl<Quaternionf>& rhs)
{
    // If either of the tangents in the segment are set to stepped, make the constant value equal the value of the left key
    if (lhs.outSlope[0] == std::numeric_limits<float>::infinity() || rhs.inSlope[0] == std::numeric_limits<float>::infinity() ||
        lhs.outSlope[1] == std::numeric_limits<float>::infinity() || rhs.inSlope[1] == std::numeric_limits<float>::infinity() ||
        lhs.outSlope[2] == std::numeric_limits<float>::infinity() || rhs.inSlope[2] == std::numeric_limits<float>::infinity() ||
        lhs.outSlope[3] == std::numeric_limits<float>::infinity() || rhs.inSlope[3] == std::numeric_limits<float>::infinity() )
    {
        for (int i=0;i<4;i++)
        {
            coeff[0][i] = 0.0F;
            coeff[1][i] = 0.0F;
            coeff[2][i] = 0.0F;
            coeff[3][i] = lhs.value[i];
        }
    }
}

void HandleSteppedCurve (const KeyframeTpl<Quaternionf>& lhs, const KeyframeTpl<Quaternionf>& rhs, Quaternionf& value)
{
    if (lhs.outSlope[0] == std::numeric_limits<float>::infinity() || rhs.inSlope[0] == std::numeric_limits<float>::infinity() ||
        lhs.outSlope[1] == std::numeric_limits<float>::infinity() || rhs.inSlope[1] == std::numeric_limits<float>::infinity() ||
        lhs.outSlope[2] == std::numeric_limits<float>::infinity() || rhs.inSlope[2] == std::numeric_limits<float>::infinity() ||
        lhs.outSlope[3] == std::numeric_limits<float>::infinity() || rhs.inSlope[3] == std::numeric_limits<float>::infinity() )
    {
        value = lhs.value;
    }
}

void HandleSteppedTangent (const KeyframeTpl<Quaternionf>& lhs, const KeyframeTpl<Quaternionf>& rhs, Quaternionf& tangent)
{
    for (int i=0;i<4;i++)
    {
        if (lhs.outSlope[i] == std::numeric_limits<float>::infinity() || rhs.inSlope[i] == std::numeric_limits<float>::infinity())
            tangent[i] = std::numeric_limits<float>::infinity();
    }
}

template<class T>
void AnimationCurveTpl<T>::CalculateCacheData (Cache& cache, int lhsIndex, int rhsIndex, float timeOffset) const
{
    const Keyframe& lhs = m_Curve[lhsIndex];
    const Keyframe& rhs = m_Curve[rhsIndex];
    //    DebugAssertIf (timeOffset < -0.001F || timeOffset - 0.001F > rhs.time - lhs.time);
    cache.index = lhsIndex;
    cache.time = lhs.time + timeOffset;
    cache.timeEnd = rhs.time + timeOffset;
    cache.index = lhsIndex;
    
    float dx, length;
    T dy;
    T m1, m2, d1, d2;
    
    dx = rhs.time - lhs.time;
    dx = max(dx, 0.0001F);
    dy = rhs.value - lhs.value;
    length = 1.0F / (dx * dx);
    
    m1 = lhs.outSlope;
    m2 = rhs.inSlope;
    d1 = m1 * dx;
    d2 = m2 * dx;
    
    cache.coeff[0] = (d1 + d2 - dy - dy) * length / dx;
    cache.coeff[1] = (dy + dy + dy - d1 - d1 - d2) * length;
    cache.coeff[2] = m1;
    cache.coeff[3] = lhs.value;
    SetupStepped(cache.coeff, lhs, rhs);
    
    //TODO: IsFinite
//    DebugAssertIf(!IsFinite(cache.coeff[0]));
//    DebugAssertIf(!IsFinite(cache.coeff[1]));
//    DebugAssertIf(!IsFinite(cache.coeff[2]));
//    DebugAssertIf(!IsFinite(cache.coeff[3]));
}

// When we look for the next index, how many keyframes do we just loop ahead instead of binary searching?
#define SEARCH_AHEAD 3
///@TODO: Cleanup old code to completely get rid of this
template<class T>
int AnimationCurveTpl<T>::FindIndex (const Cache& cache, float curveT) const
{
#if SEARCH_AHEAD >= 0
    int cacheIndex = cache.index;
    if (cacheIndex != -1)
    {
        // We can not use the cache time or time end since that is in unwrapped time space!
        float time = m_Curve[cacheIndex].time;
        
        if (curveT > time)
        {
            if (cacheIndex + SEARCH_AHEAD < static_cast<int>(m_Curve.size()))
            {
                for (int i=0;i<SEARCH_AHEAD;i++)
                {
                    if (curveT < m_Curve[cacheIndex + i + 1].time)
                        return cacheIndex + i;
                }
            }
        }
        else
        {
            if (cacheIndex - SEARCH_AHEAD >= 0)
            {
                for (int i=0;i<SEARCH_AHEAD;i++)
                {
                    if (curveT > m_Curve[cacheIndex - i - 1].time)
                        return cacheIndex - i - 1;
                }
            }
        }
    }
    
#endif
    
    ///@ use cache to index into next if not possible use binary search
    const_iterator i = std::lower_bound (m_Curve.begin (), m_Curve.end (), curveT, KeyframeCompare());
    int index = distance (m_Curve.begin (), i);
    index--;
    index = min<int> (m_Curve.size () - 2, index);
    index = max<int> (0, index);
    
    return index;
}

///@TODO: Cleanup old code to completely get rid of this
template<class T>
int AnimationCurveTpl<T>::FindIndex (float curveT) const
{
    pair<float, float> range = GetRange ();
    if (curveT <= range.first || curveT >= range.second)
        return -1;
    
    const_iterator i = std::lower_bound (m_Curve.begin (), m_Curve.end (), curveT, KeyframeCompare());
    AssertIf (i == m_Curve.end ());
    int index = distance (m_Curve.begin (), i);
    index--;
    index = min<int> (m_Curve.size () - 2, index);
    index = max<int> (0, index);
    
    AssertIf (curveT < m_Curve[index].time || curveT > m_Curve[index+1].time);
    return index;
}

template<class T>
void AnimationCurveTpl<T>::FindIndexForSampling (const Cache& cache, float curveT, int& lhs, int& rhs) const
{
    AssertIf (curveT < GetRange ().first || curveT > GetRange ().second);
    int actualSize = m_Curve.size();
    const Keyframe* frames = &m_Curve[0];
    
    // Reference implementation:
    // (index is the last value that is equal to or smaller than curveT)
#if 0
    int foundIndex = 0;
    for (int i=0;i<actualSize;i++)
    {
        if (frames[i].time <= curveT)
            foundIndex = i;
    }
    
    lhs = foundIndex;
    rhs = min<int>(lhs + 1, actualSize - 1);
    AssertIf (curveT < m_Curve[lhs].time || curveT > m_Curve[rhs].time);
    AssertIf(frames[rhs].time == curveT && frames[lhs].time != curveT)
    return;
#endif
    
    
#if SEARCH_AHEAD > 0
    int cacheIndex = cache.index;
    if (cacheIndex != -1)
    {
        // We can not use the cache time or time end since that is in unwrapped time space!
        float time = m_Curve[cacheIndex].time;
        
        if (curveT > time)
        {
            for (int i=0;i<SEARCH_AHEAD;i++)
            {
                int index = cacheIndex + i;
                if (index + 1 < actualSize && frames[index + 1].time > curveT)
                {
                    lhs = index;
                    
                    rhs = min<int>(lhs + 1, actualSize - 1);
                    AssertIf (curveT < frames[lhs].time || curveT > frames[rhs].time);
                    AssertIf(frames[rhs].time == curveT && frames[lhs].time != curveT);
                    return;
                }
            }
        }
        else
        {
            for (int i=0;i<SEARCH_AHEAD;i++)
            {
                int index = cacheIndex - i;
                if (index >= 0 && curveT >= frames[index].time)
                {
                    lhs = index;
                    rhs = min<int>(lhs + 1, actualSize - 1);
                    AssertIf (curveT < frames[lhs].time || curveT > m_Curve[rhs].time);
                    AssertIf(frames[rhs].time == curveT && frames[lhs].time != curveT);
                    return;
                }
            }
        }
    }
    
#endif
    
    // Fall back to using binary search
    // upper bound (first value larger than curveT)
    int __len = actualSize;
    int __half;
    int __middle;
    int __first = 0;
    while (__len > 0)
    {
        __half = __len >> 1;
        __middle = __first + __half;
        
        if (curveT < frames[__middle].time)
            __len = __half;
        else
        {
            __first = __middle;
            ++__first;
            __len = __len - __half - 1;
        }
    }
    
    // If not within range, we pick the last element twice
    lhs = __first - 1;
    rhs = min(actualSize - 1, __first);
    
    AssertIf(lhs < 0 || lhs >= actualSize);
    AssertIf(rhs < 0 || rhs >= actualSize);
    
    AssertIf (curveT < m_Curve[lhs].time || curveT > m_Curve[rhs].time);
    AssertIf(frames[rhs].time == curveT && frames[lhs].time != curveT);
}

template<class T>
void AnimationCurveTpl<T>::SetPreInfinity (int pre)
{
    m_PreInfinity = ToInternalInfinity(pre);
    InvalidateCache ();
}

template<class T>
void AnimationCurveTpl<T>::SetPostInfinity (int post)
{
    m_PostInfinity = ToInternalInfinity(post);
    InvalidateCache ();
}

template<class T>
int AnimationCurveTpl<T>::GetPreInfinity () const
{
    return FromInternalInfinity(m_PreInfinity);
}

template<class T>
int AnimationCurveTpl<T>::GetPostInfinity () const
{
    return FromInternalInfinity(m_PostInfinity);
}

template<class T>
T AnimationCurveTpl<T>::EvaluateClamp (float curveT) const
{
    T output;
    if (curveT >= m_ClampCache.time && curveT < m_ClampCache.timeEnd)
    {
        //        AssertIf (!CompareApproximately (EvaluateCache (m_Cache, curveT), EvaluateWithoutCache (curveT), 0.001F));
        EvaluateCache<T> (m_ClampCache, curveT, output);
        return output;
    }
    else
    {
        DebugAssertIf (!IsValid ());
        
        float begTime = m_Curve[0].time;
        float endTime = m_Curve.back().time;
        
        if (curveT > endTime)
        {
            m_ClampCache.time = endTime;
            m_ClampCache.timeEnd = std::numeric_limits<float>::infinity ();
            m_ClampCache.coeff[0] = m_ClampCache.coeff[1] = m_ClampCache.coeff[2] = Zero<T>();
            m_ClampCache.coeff[3] = m_Curve[m_Curve.size()-1].value;
        }
        else if (curveT < begTime)
        {
            m_ClampCache.time = curveT - 1000.0F;
            m_ClampCache.timeEnd = begTime;
            m_ClampCache.coeff[0] = m_ClampCache.coeff[1] = m_ClampCache.coeff[2] = Zero<T>();
            m_ClampCache.coeff[3] = m_Curve[0].value;
        }
        else
        {
            int lhs, rhs;
            FindIndexForSampling (m_ClampCache, curveT, lhs, rhs);
            CalculateCacheData (m_ClampCache, lhs, rhs, 0.0F);
        }
        
        //        AssertIf (!CompareApproximately (EvaluateCache (m_Cache, curveT), EvaluateWithoutCache (curveT), 0.001F));
        EvaluateCache<T> (m_ClampCache, curveT, output);
        return output;
    }
}

template<class T>
T AnimationCurveTpl<T>::Evaluate (float curveT) const
{
    int lhs, rhs;
    T output;
    if (curveT >= m_Cache.time && curveT < m_Cache.timeEnd)
    {
        //        AssertIf (!CompareApproximately (EvaluateCache (m_Cache, curveT), EvaluateWithoutCache (curveT), 0.001F));
        EvaluateCache<T> (m_Cache, curveT, output);
        return output;
    }
    // @TODO: Optimize IsValid () away if by making the non-valid case always use the m_Cache codepath
    else if (IsValid ())
    {
        float begTime = m_Curve[0].time;
        float endTime = m_Curve.back().time;
        float wrappedTime;
        
        if (curveT >= endTime)
        {
            if (m_PostInfinity == kInternalClamp)
            {
                m_Cache.time = endTime;
                m_Cache.timeEnd = std::numeric_limits<float>::infinity ();
                m_Cache.coeff[0] = m_Cache.coeff[1] = m_Cache.coeff[2] = Zero<T>();
                m_Cache.coeff[3] = m_Curve[m_Curve.size()-1].value;
            }
            else if (m_PostInfinity == kInternalRepeat)
            {
                wrappedTime = Repeat (curveT, begTime, endTime);
                
                FindIndexForSampling (m_Cache, wrappedTime, lhs, rhs);
                CalculateCacheData (m_Cache, lhs, rhs, curveT - wrappedTime);
            }
            ///@todo optimize pingpong by making it generate a cache too
            else
            {
                EvaluateWithoutCache (curveT, output);
                return output;
            }
        }
        else if (curveT < begTime)
        {
            if (m_PreInfinity == kInternalClamp)
            {
                m_Cache.time = curveT - 1000.0F;
                m_Cache.timeEnd = begTime;
                m_Cache.coeff[0] = m_Cache.coeff[1] = m_Cache.coeff[2] = Zero<T>();
                m_Cache.coeff[3] = m_Curve[0].value;
            }
            else if (m_PreInfinity == kInternalRepeat)
            {
                wrappedTime = Repeat (curveT, begTime, endTime);
                FindIndexForSampling (m_Cache, wrappedTime, lhs, rhs);
                CalculateCacheData (m_Cache, lhs, rhs, curveT - wrappedTime);
            }
            ///@todo optimize pingpong by making it generate a cache too
            else
            {
                EvaluateWithoutCache (curveT, output);
                return output;
            }
        }
        else
        {
            FindIndexForSampling (m_Cache, curveT, lhs, rhs);
            CalculateCacheData (m_Cache, lhs, rhs, 0.0F);
        }
        
        //        AssertIf (!CompareApproximately (EvaluateCache (m_Cache, curveT), EvaluateWithoutCache (curveT), 0.001F));
        EvaluateCache<T> (m_Cache, curveT, output);
        return output;
    }
    else
    {
        if (m_Curve.size () == 1)
            return m_Curve.begin()->value;
        else
            return Zero<T> ();
    }
}

template<class T>
float AnimationCurveTpl<T>::WrapTime (float curveT) const
{
    DebugAssertIf (!IsValid ());
    
    float begTime = m_Curve[0].time;
    float endTime = m_Curve.back().time;
    
    if (curveT < begTime)
    {
        if (m_PreInfinity == kInternalClamp)
            curveT = begTime;
        else if (m_PreInfinity == kInternalPingPong)
            curveT = PingPong (curveT, begTime, endTime);
        else
            curveT = Repeat (curveT, begTime, endTime);
    }
    else if (curveT > endTime)
    {
        if (m_PostInfinity == kInternalClamp)
            curveT = endTime;
        else if (m_PostInfinity == kInternalPingPong)
            curveT = PingPong (curveT, begTime, endTime);
        else
            curveT = Repeat (curveT, begTime, endTime);
    }
    return curveT;
}

template<class T>
int AnimationCurveTpl<T>::AddKey (const Keyframe& key)
{
    InvalidateCache ();
    
    iterator i = std::lower_bound (m_Curve.begin (), m_Curve.end (), key);
    
    // is not included in container and value is not a duplicate
    if (i == end () || key < *i)
    {
        iterator ii = m_Curve.insert (i, key);
        return std::distance (m_Curve.begin (), ii);
    }
    else
        return -1;
}

template<class T>
void AnimationCurveTpl<T>::RemoveKeys (iterator begin, iterator end)
{
    InvalidateCache ();
    m_Curve.erase (begin, end);
}

void ScaleCurveValue (AnimationCurve& curve, float scale)
{
    for (int i=0;i<curve.GetKeyCount ();i++)
    {
        curve.GetKey (i).value *= scale;
        curve.GetKey (i).inSlope *= scale;
        curve.GetKey (i).outSlope *= scale;
    }
    
    curve.InvalidateCache();
}

void OffsetCurveValue (AnimationCurve& curve, float offset)
{
    for (int i=0;i<curve.GetKeyCount ();i++)
        curve.GetKey (i).value += offset;
    
    curve.InvalidateCache();
}

void ScaleCurveTime (AnimationCurve& curve, float scale)
{
    for (int i=0;i<curve.GetKeyCount ();i++)
    {
        curve.GetKey (i).time *= scale;
        curve.GetKey (i).inSlope /= scale;
        curve.GetKey (i).outSlope /= scale;
    }
    curve.InvalidateCache();
}

void OffsetCurveTime (AnimationCurve& curve, float offset)
{
    for (int i=0;i<curve.GetKeyCount ();i++)
        curve.GetKey (i).time += offset;
    curve.InvalidateCache();
}


/*
 
 Calculating tangents from a hermite spline () Realtime rendering page 56
 
 
 
 > On this first pass we're stuck with linear keyframing, because we just
 > don't have the cycles in game to go to splines. I know this would help a
 > lot, but it isn't an option.
 
 In Granny I do successive least-squares approximations do the data.  I
 take the array of samples for a given channel (ie., position) which is 2x
 oversampled or better from the art tool.  I then start by doing a
 least-square solve for a spline of arbitrary degree with knots at either
 end.  I compute the error over the spline from the original samples, and
 add knots in the areas of highest error.  Repeat and salt to taste.
 
 Since it's fairly easy to write a single solver that solves for any degree
 of spline, I use the same solver for linear keyframes as I do for
 quadratic keyframes, cubic keyframes, or even "0th order keframes", which
 is to say if you don't want to interpolate _at all_, the solver can still
 place the "stop-motion" key frames in the best locations.  But hopefully
 no one is still doing that kind of animation.
 
 If I were doing this over again (which I probably will at some point), I
 would probably use some kind of weird waveletty scheme now.  I decided not
 to do that originally, because I had about a month to do the entire
 spline/solver/reduction thing the first time, and it had to be highly
 optimized.  So I didn't want to have to learn wavelets and spline solvers
 at the same time.  Next time I'll spend some time on wavelets and probably
 use some sort of hierachical reduction scheme instead, primarily so you
 can change how densely your keyframes are placed at run-time, and so you
 can get hard edges easier (motions which are intended to have sharp
 discontinuities aren't handled well at all by my current scheme).
 
 - Casey
 
 
 
 --
 
 > Is anybody looking at the way errors add up?  eg. if you do some
 > reduction on the hip, and the thigh, and the ankle bone channels, the
 > result is a foot that moves quite a bit differently than it should.
 
 This has been on my list for a long time.  I don't think it's a simple
 case of error analysis though.  I think it's more a case for using
 discrete skeletal changes or integrated IK, because hey, if what you care
 about is having a particular thing stay in one place, then it seems to me
 that the best way to compress that is by just saying "this stays in one
 place", rather than trying to spend a lot of data on the joint curves
 necessary to make that so.
 
 > Also, is anybody doing things like using IK in the reducer, to make sure
 > that even in the reduced version the feet stay in exactly the same spot?
 
 That doesn't work with splines, unfortunately, because the splines are
 continuous, and you will always have the feet slipping as a result (if not
 at the keyframes, then in between for sure).  So you end up with the
 problem that the IK reducer would need to shove a metric assload of keys
 into the streams, which defeats the compression.
 
 From Ian:
 
 > You said you are doing this on linear data as well. I can see that this
 > would work, but do you find you get a fairly minimal result? I can
 > envisage cases where you'd get many unneeded keyframes from initial
 > 'breaks' at points of large error.
 
 I'm not sure what you mean by this.  The error is controllable, so you say
 how much you are willing to accept.  You get a minimal spline for the
 error that you ask for, but no more - obviously in the areas of high
 error, it has to add more keys, but that's what you want.  The objective
 of compression, at least in my opinion, is not to remove detail, but
 rather to more efficiently store places where there is less detail.
 
 > I think I may be missing the point. Do you do least squares on the
 > already fitted cures (ie, a chi squared test) then curve fit separately,
 > or do you use least squares to fit your actual spline to the data?
 
 The latter.  The incremental knot addition sets up the t_n's of an
 arbitrary degree spline.  You have a matrix A that has sample-count rows
 and knot-count columns.  The vector you're looking for is x, the spline
 vector, which has knot-count rows.  You're producing b, the vector of
 samples, which has sample-count rows.  So it's Ax = b, but A is
 rectangular.  You solve via A^T Ax = A^Tb, a simple least squares problem.
 You can solve it any way you like.  I chose a straightforward
 implementation, because there's no numerical difficulties with these
 things.
 
 The version that comes with Granny is a highly optimized A^T Ax = A^Tb
 solver, which constructs A^T A and A^T b directly and sparsely (so it is
 O(n) in the number of samples, instead of O(n^3)).  The A^T A is band
 diagonal, because of the sparsity pattern of A, so I use a sparse banded
 cholesky solver on the back end.  It's _extremely_ fast.  In fact, the
 stupid part of Granny's solver is actually the other part (the error
 analysis + knot addition), because it adds very few knots per cycle, so on
 a really long animation it can call the least-squares Ax = b solver many
 hundreds of times for a single animation, and it _still_ never takes more
 than 10 seconds or so even on animations with many hundred frames.  So
 really, the right place to fix currently is the stupid knot addition
 alogirithm, which really should be improved quite a bit.
 
 > I have not used least squared before and from the reading I have done
 > (numerical recipes, last night), it seem that the equation of the
 > line/spline are inherent in the method and it needs to be reformulated
 > to use a different line/spline. (The version I read was for fitting a
 > straight line to a data-set, which I know won't work for quaternions
 > slerps anyway)
 
 Quaternion lerps are great, and work great with splines.  I use them
 throughout all of Granny - there is no slerping anywhere.  Slerping is not
 a very useful thing in a run-time engine, in my opinion, unless you are
 dealing with orientations that are greater the 90 degrees apart.  It
 allows me to use the same solver for position, rotation, AND scale/shear,
 with no modifications.
 
 - Casey
 
 
 -----
 > When is it best to use quaternions and when to use normal matrix
 > rotations?
 
 In Granny, I use quaternions for everything except the final composite
 phase where rotations (and everything else) are multiplied by their
 parents to produce the final world-space results.  Since we support
 scale/shear, orientation, and position, it tends to be fastest at the
 composition stage to convert the orientation from quaternion to matrix,
 and then do all the matrix concatenation together.  I'm not sure I would
 do the same if I didn't support scale/shear.  It might be a bit more fun
 and efficient to go ahead and do the whole pipe in quaternion and only
 convert to matrices at the very end (maybe somebody else has played with
 that and can comment?)
 
 > And for character animation with moving joints quaternions is usually
 > used, but why?
 
 There's lots of nice things about quaternions:
 
 1) You can treat the linearly if you want to (so you can blend animations
 quickly and easily)
 
 2) You can treat them non-linearly if you want to (so you can do exact
 geodesic interpolation at a constant speed)
 
 3) You can convert them to a matrix with no transcendentals
 
 4) They are compact, and can be easily converted to a 3-element
 representation when necessary (ie., any one of the four values can be
 easily re-generated from the other 3)
 
 5) They cover 720 degrees of rotation, not 360 (if you do a lot of work
 with character animation, you will appreciate this!), and the
 neighborhooding operator is extremely simple and fast (inner product and a
 negation)
 
 6) They are easy to visualize (they're not much more complicated than
 angle/axis) and manipulate (ie., the axis of rotation is obvious, you can
 transform the axis of rotation without changing the amount of rotation,
 and even just use a regular 3-vector transform, etc.)
 
 7) You can easily transform vectors with them without converting to a
 matrix if you want to
 
 8) They are easy to renormalize, and they can never become "unorthogonal"
 
 9) No gimbal lock
 
 10) There is a simple, fast, and relevant distance metric (the inner
 product)
 
 > Is it just by trial and error that we decide when we have to use
 > quarternions?
 
 Well, I can't speak for everyone, but I have been extremely careful in my
 selection of quaternions.  With Granny 1 I did quaternions to get some
 practice with them, but when I did 2 I did a lot of research and wrote
 code to visualize the actions of various rotational representations and
 operations, and I can very confidently say there's no better choice for
 general character animation than quaternions, at least among the
 representations that I know (euler, angle/axis, matrix, quaternion, exp
 map).  There are some other mappings that I've come across since I worked
 everything out (like the Rational Map), that I have not experimented with,
 because so far quaternions have been working superbly so I haven't had
 much cause to go hunting.
 
 > "Quarternions have the ability to have smooth interpollated rotations",
 > I've made smooth interpollated rotations with normal matrix rotations
 > though.
 
 Well, it's not so much what you can do as how easy it is to do it.
 Quaternions can be interpolated directly as a smooth geodesic (via slerp)
 or via a nonlinear geodesic (lerp).  The latter is particularly powerful
 because it distributes - it's a linear operator.  So you can literally
 plug it in to anything that would work, like splines, multi-point blends,
 etc., and it will "just work" (well, that's a bit of an exaggeration,
 because there is a neighborhooding concern, but it's always easy to do).
 
 > "They take less room, 4 elements versus 9 and some operations
 > are cheaper in terms of CPU cycles". I accept this reason except someone
 > said that quarternions use more CPU cycles, so I'm not sure who to
 > believe.
 
 It depends what you're doing.  Quaternions take more CPU cycles to
 transform a vector than a matrix does.  But quaternions are MUCH less
 expensive for lots of other operations, like, for example, splining.
 This is why it's very useful to use quaternions for everything except the
 final stage of your pipe, where matrices are more appropriate.
 
 > "Quarternions are not susceptible to gimbal lock. Gimbal lock shows its
 > face when two axes point in the same direction." So if no axes face in
 > the same direction then this is not a reason to use quarternions.
 > ...
 > Am I right in saying that if no axes face the same direction it is
 > possible to represent all rotations with normal matrix rotations?
 
 Matrix rotations are not subject to gimbal lock.  That's Euler angles.
 
 - Casey
 
 
 Least squares is very simple.  Given a set of data points (2x oversampled in caseys case) and a knot vector, compute the coefficients for the control points of the spline that minimize squared error.  This involves solving a linear system where each row corresponds to a data point (di - they are uniformly space so each point has a corresponding ti which is in the domain of the curve) and each column corresponds to a control point for the curve (cj the unkowns).  So matrix element Aij is Bj(ti) where Bj is the basis function for the jth control point.  The right hand side is just a vector of the di and the x's are the unkown control points.  Sine you are compressing things there will be many more rows then columns in this linear system so you don't have to worry about over fitting problems (which you would if there were more DOF...)
 
 -Peter-Pike
 
 -----Original Message-----
 From: Ian Elsley [mailto:ielsley@kushgames.com]
 Sent: Tue 12/17/2002 10:35 AM
 To: gdalgorithms-list@lists.sourceforge.net
 Cc:
 Subject: RE: [Algorithms] Re: Keyframe reduction
 
 
 
 Casey
 
 I think I may be missing the point. Do you do least squares on the
 already fitted cures (ie, a chi squared test) then curve fit separately,
 or do you use least squares to fit your actual spline to the data?
 
 I have not used least squared before and from the reading I have done
 (numerical recipes, last night), it seem that the equation of the
 line/spline are inherent in the method and it needs to be reformulated
 to use a different line/spline. (The version I read was for fitting a
 straight line to a data-set, which I know won't work for quaternions
 slerps anyway)
 
 I see the elegance of this method and would love to work it out, but
 I've got a few blanks here.
 
 Any pointers?
 
 Thanks in advance,
 
 Ian
 
 
 
 
 
 
 q
 q' = -----
 |q|
 
 For renormalizing, since you are very close to 1, I usually use the
 tangent-line approximation, which was suggested by Checker a long long
 time ago for 3D vectors and works just peachy for 4D ones as well:
 
 inline void NormalizeCloseToOne4(float *Dest)
 {
 float const Sum = (Dest[0] * Dest[0] +
 Dest[1] * Dest[1] +
 Dest[2] * Dest[2] +
 Dest[3] * Dest[3]);
 
 float const ApproximateOneOverRoot = (3.0f - Sum) * 0.5f;
 
 Dest[0] *= ApproximateOneOverRoot;
 Dest[1] *= ApproximateOneOverRoot;
 Dest[2] *= ApproximateOneOverRoot;
 Dest[3] *= ApproximateOneOverRoot;
 }
 
 
 */



#define INSTANTIATE(T) \
template EXPORT_COREMODULE std::pair<float, float> AnimationCurveTpl<T>::GetRange() const; \
template EXPORT_COREMODULE T AnimationCurveTpl<T>::Evaluate (float curveT) const; \
template EXPORT_COREMODULE void AnimationCurveTpl<T>::RemoveKeys (iterator begin, iterator end);\
template EXPORT_COREMODULE int AnimationCurveTpl<T>::AddKey (const Keyframe& key);\
template EXPORT_COREMODULE int AnimationCurveTpl<T>::FindIndex (float time) const; \
template EXPORT_COREMODULE int AnimationCurveTpl<T>::FindIndex (const Cache& cache, float time) const; \
template EXPORT_COREMODULE void AnimationCurveTpl<T>::InvalidateCache (); \
template EXPORT_COREMODULE void AnimationCurveTpl<T>::SetPreInfinity (int mode); \
template EXPORT_COREMODULE void AnimationCurveTpl<T>::SetPostInfinity (int mode); \
template EXPORT_COREMODULE int AnimationCurveTpl<T>::GetPreInfinity () const; \
template EXPORT_COREMODULE int AnimationCurveTpl<T>::GetPostInfinity () const; \
template EXPORT_COREMODULE KeyframeTpl<T>::KeyframeTpl (float time, const T& value);

INSTANTIATE(float)
INSTANTIATE(Vector3f)
INSTANTIATE(Quaternionf)

template EXPORT_COREMODULE void AnimationCurveTpl<float>::CalculateCacheData (Cache& cache, int lhs, int rhs, float timeOffset) const;
template EXPORT_COREMODULE float AnimationCurveTpl<float>::EvaluateClamp (float curveT) const;
template EXPORT_COREMODULE Quaternionf AnimationCurveTpl<Quaternionf>::EvaluateClamp (float curveT) const;
template EXPORT_COREMODULE Vector3f AnimationCurveTpl<Vector3f>::EvaluateClamp (float curveT) const;

NS_RRP_END