[Bf-blender-cvs] [9b7561f16ad] sculpt-dev: sculpt-dev: Implement arc-length derivatives for BLI_arc_spline.hh
Joseph Eagar
noreply at git.blender.org
Wed Oct 19 10:57:31 CEST 2022
Commit: 9b7561f16ad9f440ca8a28413898eb40d5aecaf8
Author: Joseph Eagar
Date: Wed Oct 19 01:57:10 2022 -0700
Branches: sculpt-dev
https://developer.blender.org/rB9b7561f16ad9f440ca8a28413898eb40d5aecaf8
sculpt-dev: Implement arc-length derivatives for BLI_arc_spline.hh
===================================================================
M source/blender/blenlib/BLI_arc_spline.hh
===================================================================
diff --git a/source/blender/blenlib/BLI_arc_spline.hh b/source/blender/blenlib/BLI_arc_spline.hh
index 4bc1ba9db21..8f88c281b5c 100644
--- a/source/blender/blenlib/BLI_arc_spline.hh
+++ b/source/blender/blenlib/BLI_arc_spline.hh
@@ -57,6 +57,9 @@ darc := sqrt(dx**2 + dy**2);
arcstep := darc*dt + 0.5*df(darc, t)*dt*dt;
+d2x := df(dx / darc, t);
+d2y := df(dy / darc, t);
+
gentran
begin
declare <<
@@ -72,6 +75,8 @@ cubic;
dcubic;
icubic;
arcstep;
+d2x;
+d2y;
off fort;
*/
@@ -270,7 +275,7 @@ template<typename Float, int axes = 2, int table_size = 512> class CubicBezier {
return r;
}
- Vector derivative(Float s)
+ Vector derivative(Float s, bool exact = true)
{
Float t = arc_to_t(s);
Vector r;
@@ -279,6 +284,15 @@ template<typename Float, int axes = 2, int table_size = 512> class CubicBezier {
r[i] = dcubic(ps[0][i], ps[1][i], ps[2][i], ps[3][i], t) * length;
}
+ /* Real arc length parameterized tangent has unit length. */
+ if (exact) {
+ Float len = sqrt(_dot(r, r));
+
+ if (len > 0.00001) {
+ r = r / len;
+ }
+ }
+
return r;
}
@@ -287,8 +301,63 @@ template<typename Float, int axes = 2, int table_size = 512> class CubicBezier {
Float t = arc_to_t(s);
Vector r;
- for (int i = 0; i < axes; i++) {
- r[i] = d2cubic(ps[0][i], ps[1][i], ps[2][i], ps[3][i], t) * length;
+ Float dx = dcubic(ps[0][0], ps[1][0], ps[2][0], ps[3][0], t);
+ Float d2x = dcubic(ps[0][0], ps[1][0], ps[2][0], ps[3][0], t);
+ Float dy = dcubic(ps[0][1], ps[1][1], ps[2][1], ps[3][1], t);
+ Float d2y = dcubic(ps[0][1], ps[1][1], ps[2][1], ps[3][1], t);
+
+ /*
+ comment: arc length second derivative;
+
+ operator x, y, z, dx, dy, dz, d2x, d2y, d2z;
+ forall t let df(x(t), t) = dx(t);
+ forall t let df(y(t), t) = dy(t);
+ forall t let df(z(t), t) = dz(t);
+ forall t let df(dx(t), t) = d2x(t);
+ forall t let df(dy(t), t) = d2y(t);
+ forall t let df(dz(t), t) = d2z(t);
+
+ comment: arc length first derivative is just the normalized tangent;
+
+ comment: 2d case;
+
+ dlen := sqrt(df(x(t), t)**2 + df(y(t), t)**2);
+
+ df(df(x(t), t) / dlen, t);
+ df(df(y(t), t) / dlen, t);
+
+ comment: 3d case;
+
+ dlen := sqrt(df(x(t), t)**2 + df(y(t), t)**2 + df(z(t), t)**2);
+
+ comment: final derivatives;
+
+ df(df(x(t), t) / dlen, t);
+ df(df(y(t), t) / dlen, t);
+ df(df(z(t), t) / dlen, t);
+ */
+ if constexpr (axes == 2) {
+ /* Basically the 2d perpidicular normalized tangent multiplied by the curvature. */
+
+ Float div = sqrt(dx * dx + dy * dy) * (dx * dx + dy * dy);
+
+ r[0] = ((d2x * dy - d2y * dx) * dy) / div;
+ r[1] = (-(d2x * dy - d2y * dx) * dx) / div;
+ }
+ else if (constexpr(axes == 3)) {
+ Float dz = dcubic(ps[0][2], ps[1][2], ps[2][2], ps[3][2], t);
+ Float d2z = d2cubic(ps[0][2], ps[1][2], ps[2][2], ps[3][2], t);
+
+ Float div = sqrt(dx * dx + dy * dy + dz * dz) * (dy * dy + dz * dz + dx * dx);
+
+ r[0] = (d2x * dy * dy + d2x * dz * dz - d2y * dx * dy - d2z * dx * dz) / div;
+ r[1] = (-(d2x * dx * dy - d2y * dx * dx - d2y * dz * dz + d2z * dy * dz)) / div;
+ r[2] = (-(d2x * dx * dz + d2y * dy * dz - d2z * dx * dx - d2z * dy * dy)) / div;
+ }
+ else {
+ for (int i = 0; i < axes; i++) {
+ r[i] = d2cubic(ps[0][i], ps[1][i], ps[2][i], ps[3][i], t) * length;
+ }
}
return r;
@@ -296,25 +365,16 @@ template<typename Float, int axes = 2, int table_size = 512> class CubicBezier {
Float curvature(Float s)
{
- /* Get signed curvature if in 2d. */
+ Vector dv2 = derivative2(s);
+
if constexpr (axes == 2) {
- Vector dv1 = derivative(s);
- Vector dv2 = derivative2(s);
+ Vector dv = derivative(s, true);
- return (dv1[0] * dv2[1] - dv1[1] * dv2[0]) /
- powf(dv1[0] * dv1[0] + dv1[1] * dv1[1], 3.0 / 2.0);
+ /* Calculate signed curvature. Remember that dv is normalized. */
+ return dv[0] * dv2[1] - dv[1] * dv2[0];
}
- else { /* Otherwise use magnitude of second derivative (this works because we are arc-length
- parameterized). */
- Vector dv2 = derivative2(s);
- Float len = 0.0;
- for (int i = 0; i < axes; i++) {
- len += dv2[i] * dv2[i];
- }
-
- return sqrt(len);
- }
+ return sqrt(_dot(dv2, dv2));
}
private:
@@ -338,6 +398,17 @@ template<typename Float, int axes = 2, int table_size = 512> class CubicBezier {
return -6.0 * (k1 * t - k1 - 3.0 * k2 * t + 2.0 * k2 + 3.0 * k3 * t - k3 - k4 * t);
}
+ Float _dot(Vector a, Vector b)
+ {
+ Float sum = 0.0;
+
+ for (int i = 0; i < axes; i++) {
+ sum += a[i] * b[i];
+ }
+
+ return sum;
+ }
+
Float clamp_s(Float s)
{
s = s < 0.0 ? 0.0 : s;
@@ -458,7 +529,7 @@ template<typename Float, int axes = 2> class BezierSpline {
return seg->bezier.evaluate(s - seg->start);
}
- Vector derivative(Float s)
+ Vector derivative(Float s, bool exact = true)
{
if (segments.size() == 0) {
return Vector();
@@ -467,7 +538,7 @@ template<typename Float, int axes = 2> class BezierSpline {
s = clamp_s(s);
Segment *seg = get_segment(s);
- return seg->bezier.derivative(s - seg->start);
+ return seg->bezier.derivative(s - seg->start, exact);
}
Vector derivative2(Float s)
@@ -508,7 +579,7 @@ template<typename Float, int axes = 2> class BezierSpline {
for (int i = 0; i < steps + 1; i++, s += ds, lastp = b, lastdv = dvb) {
b = evaluate(s);
- dvb = derivative(s);
+ dvb = derivative(s, false); /* We don't need real normalized derivative here. */
if (i == 0) {
continue;
@@ -549,7 +620,7 @@ template<typename Float, int axes = 2> class BezierSpline {
const int binary_steps = 10;
for (int j = 0; j < binary_steps; j++) {
- Vector dvmid = derivative(mid);
+ Vector dvmid = derivative(mid, false);
Vector vecmid = evaluate(mid) - p;
Float sign_mid = _dot(vecmid, dvmid);
@@ -580,7 +651,7 @@ template<typename Float, int axes = 2> class BezierSpline {
mindis = _dot(vec, vec);
}
- r_tan = derivative(mins);
+ r_tan = derivative(mins, true);
r_s = mins;
r_dis = sqrtf(mindis);
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