/// <summary>
/// Attempts to find a slightly better parameterization for u on the given curve.
/// </summary>
protected void Reparameterize(int first, int last, CubicBezier curve)
{
#if OPTIMIZED_REPARAMETERIZE
using var _ = ReparameterizeMarker.Auto();
unsafe
{
using (_pts.ViewAsNativeArray(out var pts))
using (_u.ViewAsNativeArray(out var u))
OptimizedHelpers.Reparameterize(first, last, curve, (VECTOR *)pts.GetUnsafePtr(), (FLOAT *)u.GetUnsafePtr());
}
#else
using var _ = ReparameterizeMarker.Auto();
List <VECTOR> pts = _pts;
List <FLOAT> u = _u;
int nPts = last - first;
for (int i = 1; i < nPts; i++)
{
VECTOR p = pts[first + i];
FLOAT t = u[i];
FLOAT ti = 1 - t;
// Control vertices for Q'
VECTOR qp0 = (curve.p1 - curve.p0) * 3;
VECTOR qp1 = (curve.p2 - curve.p1) * 3;
VECTOR qp2 = (curve.p3 - curve.p2) * 3;
// Control vertices for Q''
VECTOR qpp0 = (qp1 - qp0) * 2;
VECTOR qpp1 = (qp2 - qp1) * 2;
// Evaluate Q(t), Q'(t), and Q''(t)
VECTOR p0 = curve.Sample(t);
VECTOR p1 = ((ti * ti) * qp0) + ((2 * ti * t) * qp1) + ((t * t) * qp2);
VECTOR p2 = (ti * qpp0) + (t * qpp1);
// these are the actual fitting calculations using http://en.wikipedia.org/wiki/Newton%27s_method
// We can't just use .X and .Y because Unity uses lower-case "x" and "y".
FLOAT num = ((VectorHelper.GetX(p0) - VectorHelper.GetX(p)) * VectorHelper.GetX(p1)) + ((VectorHelper.GetY(p0) - VectorHelper.GetY(p)) * VectorHelper.GetY(p1));
FLOAT den = (VectorHelper.GetX(p1) * VectorHelper.GetX(p1)) + (VectorHelper.GetY(p1) * VectorHelper.GetY(p1)) + ((VectorHelper.GetX(p0) - VectorHelper.GetX(p)) * VectorHelper.GetX(p2)) + ((VectorHelper.GetY(p0) - VectorHelper.GetY(p)) * VectorHelper.GetY(p2));
FLOAT newU = t - num / den;
if (Math.Abs(den) > EPSILON && newU >= 0 && newU <= 1)
{
u[i] = newU;
}
}
#endif
}
/// <summary>
/// Computes the maximum squared distance from a point to the curve using the current parameterization.
/// </summary>
protected FLOAT FindMaxSquaredError(int first, int last, CubicBezier curve, out int split)
{
#if OPTIMIZED_FINDMAXSQUAREDERROR
using var _ = FindMaxSquaredErrorMarker.Auto();
unsafe
{
using (_pts.ViewAsNativeArray(out var pts))
using (_u.ViewAsNativeArray(out var u))
{
OptimizedHelpers.FindMaxSquaredError(first, last, curve, out split, (VECTOR *)pts.GetUnsafePtr(), (FLOAT *)u.GetUnsafePtr(), out var max);
return(max);
}
}
#else
using var _ = FindMaxSquaredErrorMarker.Auto();
List <VECTOR> pts = _pts;
List <FLOAT> u = _u;
int s = (last - first + 1) / 2;
int nPts = last - first + 1;
FLOAT max = 0;
for (int i = 1; i < nPts; i++)
{
VECTOR v0 = pts[first + i];
VECTOR v1 = curve.Sample(u[i]);
FLOAT d = VectorHelper.DistanceSquared(v0, v1);
if (d > max)
{
max = d;
s = i;
}
}
// split at point of maximum error
split = s + first;
if (split <= first)
{
split = first + 1;
}
if (split >= last)
{
split = last - 1;
}
return(max);
#endif
}
/// <summary>
/// Gets the tangent for the start of the cure.
/// </summary>
public VECTOR GetLeftTangent(int last)
{
#if OPTIMIZED_GETLEFTTANGENT
using var _ = GetLeftTangentMarker.Auto();
unsafe
{
using (_pts.ViewAsNativeArray(out var pts))
using (_arclen.ViewAsNativeArray(out var arclen))
{
OptimizedHelpers.GetLeftTangent(last, (VECTOR *)pts.GetUnsafePtr(), pts.Length, (FLOAT *)arclen.GetUnsafePtr(), arclen.Length, out var tanL);
return(tanL);
}
}
#else
using var _ = GetLeftTangentMarker.Auto();
List <VECTOR> pts = _pts;
List <FLOAT> arclen = _arclen;
FLOAT totalLen = arclen[arclen.Count - 1];
VECTOR p0 = pts[0];
VECTOR tanL = VectorHelper.Normalize(pts[1] - p0);
VECTOR total = tanL;
FLOAT weightTotal = 1;
last = Math.Min(END_TANGENT_N_PTS, last - 1);
for (int i = 2; i <= last; i++)
{
FLOAT ti = 1 - (arclen[i] / totalLen);
FLOAT weight = ti * ti * ti;
VECTOR v = VectorHelper.Normalize(pts[i] - p0);
total += v * weight;
weightTotal += weight;
}
// if the vectors add up to zero (ie going opposite directions), there's no way to normalize them
if (VectorHelper.Length(total) > EPSILON)
{
tanL = VectorHelper.Normalize(total / weightTotal);
}
return(tanL);
#endif
}
/// <summary>
/// Gets the tangent for the the end of the curve.
/// </summary>
public VECTOR GetRightTangent(int first)
{
#if OPTIMIZED_GETRIGHTTANGENT
using var _ = GetRightTangentMarker.Auto();
unsafe
{
using (_pts.ViewAsNativeArray(out var pts))
using (_arclen.ViewAsNativeArray(out var arclen))
{
OptimizedHelpers.GetRightTangent(first, (VECTOR *)pts.GetUnsafePtr(), pts.Length, (FLOAT *)arclen.GetUnsafePtr(), arclen.Length, out var tanR);
return(tanR);
}
}
#else
using var _ = GetRightTangentMarker.Auto();
List <VECTOR> pts = _pts;
List <FLOAT> arclen = _arclen;
FLOAT totalLen = arclen[arclen.Count - 1];
VECTOR p3 = pts[pts.Count - 1];
VECTOR tanR = VectorHelper.Normalize(pts[pts.Count - 2] - p3);
VECTOR total = tanR;
FLOAT weightTotal = 1;
first = Math.Max(pts.Count - (END_TANGENT_N_PTS + 1), first + 1);
for (int i = pts.Count - 3; i >= first; i--)
{
FLOAT t = arclen[i] / totalLen;
FLOAT weight = t * t * t;
VECTOR v = VectorHelper.Normalize(pts[i] - p3);
total += v * weight;
weightTotal += weight;
}
if (VectorHelper.Length(total) > EPSILON)
{
tanR = VectorHelper.Normalize(total / weightTotal);
}
return(tanR);
#endif
}
/// <summary>
/// Generates a bezier curve for the segment using a least-squares approximation. for the derivation of this and why it works,
/// see http://read.pudn.com/downloads141/ebook/610086/Graphics_Gems_I.pdf page 626 and beyond. tl;dr: math.
/// </summary>
protected CubicBezier GenerateBezier(int first, int last, VECTOR tanL, VECTOR tanR)
{
#if OPTIMIZED_GENERATEBEZIERSIMD
using var _ = GenerateBezierMarker.Auto();
unsafe
{
using (_pts.ViewAsNativeArray(out var pts))
using (_u.ViewAsNativeArray(out var u))
{
OptimizedHelpers.GenerateBezier(first, last, tanL, tanR, (VECTOR *)pts.GetUnsafePtr(), (FLOAT *)u.GetUnsafePtr(), out var bezier);
return(bezier);
}
}
#elif OPTIMIZED_GENERATEBEZIER
using var _ = GenerateBezierMarker.Auto();
unsafe
{
using (_pts.ViewAsNativeArray(out var pts))
using (_u.ViewAsNativeArray(out var u))
{
OptimizedHelpers.OptimizedGenerateBezier(first, last, tanL, tanR, (VECTOR *)pts.GetUnsafePtr(), (FLOAT *)u.GetUnsafePtr(), out var bezier);
return(bezier);
}
}
#else
using var _ = GenerateBezierMarker.Auto();
List <VECTOR> pts = _pts;
List <FLOAT> u = _u;
int nPts = last - first + 1;
VECTOR p0 = pts[first], p3 = pts[last]; // first and last points of curve are actual points on data
FLOAT c00 = 0, c01 = 0, c11 = 0, x0 = 0, x1 = 0; // matrix members -- both C[0,1] and C[1,0] are the same, stored in c01
for (int i = 1; i < nPts; i++)
{
// Calculate cubic bezier multipliers
FLOAT t = u[i];
FLOAT ti = 1 - t;
FLOAT t0 = ti * ti * ti;
FLOAT t1 = 3 * ti * ti * t;
FLOAT t2 = 3 * ti * t * t;
FLOAT t3 = t * t * t;
// For X matrix; moving this up here since profiling shows it's better up here (maybe a0/a1 not in registers vs only v not in regs)
VECTOR s = (p0 * t0) + (p0 * t1) + (p3 * t2) + (p3 * t3); // NOTE: this would be Q(t) if p1=p0 and p2=p3
VECTOR v = pts[first + i] - s;
// C matrix
VECTOR a0 = tanL * t1;
VECTOR a1 = tanR * t2;
c00 += VectorHelper.Dot(a0, a0);
c01 += VectorHelper.Dot(a0, a1);
c11 += VectorHelper.Dot(a1, a1);
// X matrix
x0 += VectorHelper.Dot(a0, v);
x1 += VectorHelper.Dot(a1, v);
}
// determinents of X and C matrices
FLOAT det_C0_C1 = c00 * c11 - c01 * c01;
FLOAT det_C0_X = c00 * x1 - c01 * x0;
FLOAT det_X_C1 = x0 * c11 - x1 * c01;
FLOAT alphaL = det_X_C1 / det_C0_C1;
FLOAT alphaR = det_C0_X / det_C0_C1;
// if alpha is negative, zero, or very small (or we can't trust it since C matrix is small), fall back to Wu/Barsky heuristic
FLOAT linDist = VectorHelper.Distance(p0, p3);
FLOAT epsilon2 = EPSILON * linDist;
if (Math.Abs(det_C0_C1) < EPSILON || alphaL < epsilon2 || alphaR < epsilon2)
{
FLOAT alpha = linDist / 3;
VECTOR p1 = (tanL * alpha) + p0;
VECTOR p2 = (tanR * alpha) + p3;
return(new CubicBezier(p0, p1, p2, p3));
}
else
{
VECTOR p1 = (tanL * alphaL) + p0;
VECTOR p2 = (tanR * alphaR) + p3;
return(new CubicBezier(p0, p1, p2, p3));
}
#endif
}