| private Dot ( System.Windows.Vector a, System.Windows.Vector b ) : System.Double | ||
| a | System.Windows.Vector | |
| b | System.Windows.Vector | |
| return | System.Double | |
/// <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)
{
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));
}
}
