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diff --git a/Godeps/_workspace/src/code.google.com/p/draw2d/draw2d/curve/cubic_float64_others.go b/Godeps/_workspace/src/code.google.com/p/draw2d/draw2d/curve/cubic_float64_others.go
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index 000000000..a888b22a1
--- /dev/null
+++ b/Godeps/_workspace/src/code.google.com/p/draw2d/draw2d/curve/cubic_float64_others.go
@@ -0,0 +1,696 @@
+// Copyright 2010 The draw2d Authors. All rights reserved.
+// created: 17/05/2011 by Laurent Le Goff
+package curve
+
+import (
+ "math"
+)
+
+const (
+ CurveCollinearityEpsilon = 1e-30
+ CurveAngleToleranceEpsilon = 0.01
+)
+
+//mu ranges from 0 to 1, start to end of curve
+func (c *CubicCurveFloat64) ArbitraryPoint(mu float64) (x, y float64) {
+
+ mum1 := 1 - mu
+ mum13 := mum1 * mum1 * mum1
+ mu3 := mu * mu * mu
+
+ x = mum13*c[0] + 3*mu*mum1*mum1*c[2] + 3*mu*mu*mum1*c[4] + mu3*c[6]
+ y = mum13*c[1] + 3*mu*mum1*mum1*c[3] + 3*mu*mu*mum1*c[5] + mu3*c[7]
+ return
+}
+
+func (c *CubicCurveFloat64) SubdivideAt(c1, c2 *CubicCurveFloat64, t float64) (x23, y23 float64) {
+ inv_t := (1 - t)
+ c1[0], c1[1] = c[0], c[1]
+ c2[6], c2[7] = c[6], c[7]
+
+ c1[2] = inv_t*c[0] + t*c[2]
+ c1[3] = inv_t*c[1] + t*c[3]
+
+ x23 = inv_t*c[2] + t*c[4]
+ y23 = inv_t*c[3] + t*c[5]
+
+ c2[4] = inv_t*c[4] + t*c[6]
+ c2[5] = inv_t*c[5] + t*c[7]
+
+ c1[4] = inv_t*c1[2] + t*x23
+ c1[5] = inv_t*c1[3] + t*y23
+
+ c2[2] = inv_t*x23 + t*c2[4]
+ c2[3] = inv_t*y23 + t*c2[5]
+
+ c1[6] = inv_t*c1[4] + t*c2[2]
+ c1[7] = inv_t*c1[5] + t*c2[3]
+
+ c2[0], c2[1] = c1[6], c1[7]
+ return
+}
+
+func (c *CubicCurveFloat64) EstimateDistance() float64 {
+ dx1 := c[2] - c[0]
+ dy1 := c[3] - c[1]
+ dx2 := c[4] - c[2]
+ dy2 := c[5] - c[3]
+ dx3 := c[6] - c[4]
+ dy3 := c[7] - c[5]
+ return math.Sqrt(dx1*dx1+dy1*dy1) + math.Sqrt(dx2*dx2+dy2*dy2) + math.Sqrt(dx3*dx3+dy3*dy3)
+}
+
+// subdivide the curve in straight lines using line approximation and Casteljau recursive subdivision
+func (c *CubicCurveFloat64) SegmentRec(t LineTracer, flattening_threshold float64) {
+ c.segmentRec(t, flattening_threshold)
+ t.LineTo(c[6], c[7])
+}
+
+func (c *CubicCurveFloat64) segmentRec(t LineTracer, flattening_threshold float64) {
+ var c1, c2 CubicCurveFloat64
+ c.Subdivide(&c1, &c2)
+
+ // Try to approximate the full cubic curve by a single straight line
+ //------------------
+ dx := c[6] - c[0]
+ dy := c[7] - c[1]
+
+ d2 := math.Abs(((c[2]-c[6])*dy - (c[3]-c[7])*dx))
+ d3 := math.Abs(((c[4]-c[6])*dy - (c[5]-c[7])*dx))
+
+ if (d2+d3)*(d2+d3) < flattening_threshold*(dx*dx+dy*dy) {
+ t.LineTo(c[6], c[7])
+ return
+ }
+ // Continue subdivision
+ //----------------------
+ c1.segmentRec(t, flattening_threshold)
+ c2.segmentRec(t, flattening_threshold)
+}
+
+/*
+ The function has the following parameters:
+ approximationScale :
+ Eventually determines the approximation accuracy. In practice we need to transform points from the World coordinate system to the Screen one.
+ It always has some scaling coefficient.
+ The curves are usually processed in the World coordinates, while the approximation accuracy should be eventually in pixels.
+ Usually it looks as follows:
+ curved.approximationScale(transform.scale());
+ where transform is the affine matrix that includes all the transformations, including viewport and zoom.
+ angleTolerance :
+ You set it in radians.
+ The less this value is the more accurate will be the approximation at sharp turns.
+ But 0 means that we don't consider angle conditions at all.
+ cuspLimit :
+ An angle in radians.
+ If 0, only the real cusps will have bevel cuts.
+ If more than 0, it will restrict the sharpness.
+ The more this value is the less sharp turns will be cut.
+ Typically it should not exceed 10-15 degrees.
+*/
+func (c *CubicCurveFloat64) AdaptiveSegmentRec(t LineTracer, approximationScale, angleTolerance, cuspLimit float64) {
+ cuspLimit = computeCuspLimit(cuspLimit)
+ distanceToleranceSquare := 0.5 / approximationScale
+ distanceToleranceSquare = distanceToleranceSquare * distanceToleranceSquare
+ c.adaptiveSegmentRec(t, 0, distanceToleranceSquare, angleTolerance, cuspLimit)
+ t.LineTo(c[6], c[7])
+}
+
+func computeCuspLimit(v float64) (r float64) {
+ if v == 0.0 {
+ r = 0.0
+ } else {
+ r = math.Pi - v
+ }
+ return
+}
+
+func squareDistance(x1, y1, x2, y2 float64) float64 {
+ dx := x2 - x1
+ dy := y2 - y1
+ return dx*dx + dy*dy
+}
+
+/**
+ * http://www.antigrain.com/research/adaptive_bezier/index.html
+ */
+func (c *CubicCurveFloat64) adaptiveSegmentRec(t LineTracer, level int, distanceToleranceSquare, angleTolerance, cuspLimit float64) {
+ if level > CurveRecursionLimit {
+ return
+ }
+ var c1, c2 CubicCurveFloat64
+ x23, y23 := c.Subdivide(&c1, &c2)
+
+ // Try to approximate the full cubic curve by a single straight line
+ //------------------
+ dx := c[6] - c[0]
+ dy := c[7] - c[1]
+
+ d2 := math.Abs(((c[2]-c[6])*dy - (c[3]-c[7])*dx))
+ d3 := math.Abs(((c[4]-c[6])*dy - (c[5]-c[7])*dx))
+ switch {
+ case d2 <= CurveCollinearityEpsilon && d3 <= CurveCollinearityEpsilon:
+ // All collinear OR p1==p4
+ //----------------------
+ k := dx*dx + dy*dy
+ if k == 0 {
+ d2 = squareDistance(c[0], c[1], c[2], c[3])
+ d3 = squareDistance(c[6], c[7], c[4], c[5])
+ } else {
+ k = 1 / k
+ da1 := c[2] - c[0]
+ da2 := c[3] - c[1]
+ d2 = k * (da1*dx + da2*dy)
+ da1 = c[4] - c[0]
+ da2 = c[5] - c[1]
+ d3 = k * (da1*dx + da2*dy)
+ if d2 > 0 && d2 < 1 && d3 > 0 && d3 < 1 {
+ // Simple collinear case, 1---2---3---4
+ // We can leave just two endpoints
+ return
+ }
+ if d2 <= 0 {
+ d2 = squareDistance(c[2], c[3], c[0], c[1])
+ } else if d2 >= 1 {
+ d2 = squareDistance(c[2], c[3], c[6], c[7])
+ } else {
+ d2 = squareDistance(c[2], c[3], c[0]+d2*dx, c[1]+d2*dy)
+ }
+
+ if d3 <= 0 {
+ d3 = squareDistance(c[4], c[5], c[0], c[1])
+ } else if d3 >= 1 {
+ d3 = squareDistance(c[4], c[5], c[6], c[7])
+ } else {
+ d3 = squareDistance(c[4], c[5], c[0]+d3*dx, c[1]+d3*dy)
+ }
+ }
+ if d2 > d3 {
+ if d2 < distanceToleranceSquare {
+ t.LineTo(c[2], c[3])
+ return
+ }
+ } else {
+ if d3 < distanceToleranceSquare {
+ t.LineTo(c[4], c[5])
+ return
+ }
+ }
+
+ case d2 <= CurveCollinearityEpsilon && d3 > CurveCollinearityEpsilon:
+ // p1,p2,p4 are collinear, p3 is significant
+ //----------------------
+ if d3*d3 <= distanceToleranceSquare*(dx*dx+dy*dy) {
+ if angleTolerance < CurveAngleToleranceEpsilon {
+ t.LineTo(x23, y23)
+ return
+ }
+
+ // Angle Condition
+ //----------------------
+ da1 := math.Abs(math.Atan2(c[7]-c[5], c[6]-c[4]) - math.Atan2(c[5]-c[3], c[4]-c[2]))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c[2], c[3])
+ t.LineTo(c[4], c[5])
+ return
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c[4], c[5])
+ return
+ }
+ }
+ }
+
+ case d2 > CurveCollinearityEpsilon && d3 <= CurveCollinearityEpsilon:
+ // p1,p3,p4 are collinear, p2 is significant
+ //----------------------
+ if d2*d2 <= distanceToleranceSquare*(dx*dx+dy*dy) {
+ if angleTolerance < CurveAngleToleranceEpsilon {
+ t.LineTo(x23, y23)
+ return
+ }
+
+ // Angle Condition
+ //----------------------
+ da1 := math.Abs(math.Atan2(c[5]-c[3], c[4]-c[2]) - math.Atan2(c[3]-c[1], c[2]-c[0]))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c[2], c[3])
+ t.LineTo(c[4], c[5])
+ return
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c[2], c[3])
+ return
+ }
+ }
+ }
+
+ case d2 > CurveCollinearityEpsilon && d3 > CurveCollinearityEpsilon:
+ // Regular case
+ //-----------------
+ if (d2+d3)*(d2+d3) <= distanceToleranceSquare*(dx*dx+dy*dy) {
+ // If the curvature doesn't exceed the distanceTolerance value
+ // we tend to finish subdivisions.
+ //----------------------
+ if angleTolerance < CurveAngleToleranceEpsilon {
+ t.LineTo(x23, y23)
+ return
+ }
+
+ // Angle & Cusp Condition
+ //----------------------
+ k := math.Atan2(c[5]-c[3], c[4]-c[2])
+ da1 := math.Abs(k - math.Atan2(c[3]-c[1], c[2]-c[0]))
+ da2 := math.Abs(math.Atan2(c[7]-c[5], c[6]-c[4]) - k)
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+ if da2 >= math.Pi {
+ da2 = 2*math.Pi - da2
+ }
+
+ if da1+da2 < angleTolerance {
+ // Finally we can stop the recursion
+ //----------------------
+ t.LineTo(x23, y23)
+ return
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c[2], c[3])
+ return
+ }
+
+ if da2 > cuspLimit {
+ t.LineTo(c[4], c[5])
+ return
+ }
+ }
+ }
+ }
+
+ // Continue subdivision
+ //----------------------
+ c1.adaptiveSegmentRec(t, level+1, distanceToleranceSquare, angleTolerance, cuspLimit)
+ c2.adaptiveSegmentRec(t, level+1, distanceToleranceSquare, angleTolerance, cuspLimit)
+
+}
+
+func (curve *CubicCurveFloat64) AdaptiveSegment(t LineTracer, approximationScale, angleTolerance, cuspLimit float64) {
+ cuspLimit = computeCuspLimit(cuspLimit)
+ distanceToleranceSquare := 0.5 / approximationScale
+ distanceToleranceSquare = distanceToleranceSquare * distanceToleranceSquare
+
+ var curves [CurveRecursionLimit]CubicCurveFloat64
+ curves[0] = *curve
+ i := 0
+ // current curve
+ var c *CubicCurveFloat64
+ var c1, c2 CubicCurveFloat64
+ var dx, dy, d2, d3, k, x23, y23 float64
+ for i >= 0 {
+ c = &curves[i]
+ x23, y23 = c.Subdivide(&c1, &c2)
+
+ // Try to approximate the full cubic curve by a single straight line
+ //------------------
+ dx = c[6] - c[0]
+ dy = c[7] - c[1]
+
+ d2 = math.Abs(((c[2]-c[6])*dy - (c[3]-c[7])*dx))
+ d3 = math.Abs(((c[4]-c[6])*dy - (c[5]-c[7])*dx))
+ switch {
+ case i == len(curves)-1:
+ t.LineTo(c[6], c[7])
+ i--
+ continue
+ case d2 <= CurveCollinearityEpsilon && d3 <= CurveCollinearityEpsilon:
+ // All collinear OR p1==p4
+ //----------------------
+ k = dx*dx + dy*dy
+ if k == 0 {
+ d2 = squareDistance(c[0], c[1], c[2], c[3])
+ d3 = squareDistance(c[6], c[7], c[4], c[5])
+ } else {
+ k = 1 / k
+ da1 := c[2] - c[0]
+ da2 := c[3] - c[1]
+ d2 = k * (da1*dx + da2*dy)
+ da1 = c[4] - c[0]
+ da2 = c[5] - c[1]
+ d3 = k * (da1*dx + da2*dy)
+ if d2 > 0 && d2 < 1 && d3 > 0 && d3 < 1 {
+ // Simple collinear case, 1---2---3---4
+ // We can leave just two endpoints
+ i--
+ continue
+ }
+ if d2 <= 0 {
+ d2 = squareDistance(c[2], c[3], c[0], c[1])
+ } else if d2 >= 1 {
+ d2 = squareDistance(c[2], c[3], c[6], c[7])
+ } else {
+ d2 = squareDistance(c[2], c[3], c[0]+d2*dx, c[1]+d2*dy)
+ }
+
+ if d3 <= 0 {
+ d3 = squareDistance(c[4], c[5], c[0], c[1])
+ } else if d3 >= 1 {
+ d3 = squareDistance(c[4], c[5], c[6], c[7])
+ } else {
+ d3 = squareDistance(c[4], c[5], c[0]+d3*dx, c[1]+d3*dy)
+ }
+ }
+ if d2 > d3 {
+ if d2 < distanceToleranceSquare {
+ t.LineTo(c[2], c[3])
+ i--
+ continue
+ }
+ } else {
+ if d3 < distanceToleranceSquare {
+ t.LineTo(c[4], c[5])
+ i--
+ continue
+ }
+ }
+
+ case d2 <= CurveCollinearityEpsilon && d3 > CurveCollinearityEpsilon:
+ // p1,p2,p4 are collinear, p3 is significant
+ //----------------------
+ if d3*d3 <= distanceToleranceSquare*(dx*dx+dy*dy) {
+ if angleTolerance < CurveAngleToleranceEpsilon {
+ t.LineTo(x23, y23)
+ i--
+ continue
+ }
+
+ // Angle Condition
+ //----------------------
+ da1 := math.Abs(math.Atan2(c[7]-c[5], c[6]-c[4]) - math.Atan2(c[5]-c[3], c[4]-c[2]))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c[2], c[3])
+ t.LineTo(c[4], c[5])
+ i--
+ continue
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c[4], c[5])
+ i--
+ continue
+ }
+ }
+ }
+
+ case d2 > CurveCollinearityEpsilon && d3 <= CurveCollinearityEpsilon:
+ // p1,p3,p4 are collinear, p2 is significant
+ //----------------------
+ if d2*d2 <= distanceToleranceSquare*(dx*dx+dy*dy) {
+ if angleTolerance < CurveAngleToleranceEpsilon {
+ t.LineTo(x23, y23)
+ i--
+ continue
+ }
+
+ // Angle Condition
+ //----------------------
+ da1 := math.Abs(math.Atan2(c[5]-c[3], c[4]-c[2]) - math.Atan2(c[3]-c[1], c[2]-c[0]))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c[2], c[3])
+ t.LineTo(c[4], c[5])
+ i--
+ continue
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c[2], c[3])
+ i--
+ continue
+ }
+ }
+ }
+
+ case d2 > CurveCollinearityEpsilon && d3 > CurveCollinearityEpsilon:
+ // Regular case
+ //-----------------
+ if (d2+d3)*(d2+d3) <= distanceToleranceSquare*(dx*dx+dy*dy) {
+ // If the curvature doesn't exceed the distanceTolerance value
+ // we tend to finish subdivisions.
+ //----------------------
+ if angleTolerance < CurveAngleToleranceEpsilon {
+ t.LineTo(x23, y23)
+ i--
+ continue
+ }
+
+ // Angle & Cusp Condition
+ //----------------------
+ k := math.Atan2(c[5]-c[3], c[4]-c[2])
+ da1 := math.Abs(k - math.Atan2(c[3]-c[1], c[2]-c[0]))
+ da2 := math.Abs(math.Atan2(c[7]-c[5], c[6]-c[4]) - k)
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+ if da2 >= math.Pi {
+ da2 = 2*math.Pi - da2
+ }
+
+ if da1+da2 < angleTolerance {
+ // Finally we can stop the recursion
+ //----------------------
+ t.LineTo(x23, y23)
+ i--
+ continue
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c[2], c[3])
+ i--
+ continue
+ }
+
+ if da2 > cuspLimit {
+ t.LineTo(c[4], c[5])
+ i--
+ continue
+ }
+ }
+ }
+ }
+
+ // Continue subdivision
+ //----------------------
+ curves[i+1], curves[i] = c1, c2
+ i++
+ }
+ t.LineTo(curve[6], curve[7])
+}
+
+/********************** Ahmad thesis *******************/
+
+/**************************************************************************************
+* This code is the implementation of the Parabolic Approximation (PA). Although *
+* it uses recursive subdivision as a safe net for the failing cases, this is an *
+* iterative routine and reduces considerably the number of vertices (point) *
+* generation. *
+**************************************************************************************/
+
+func (c *CubicCurveFloat64) ParabolicSegment(t LineTracer, flattening_threshold float64) {
+ estimatedIFP := c.numberOfInflectionPoints()
+ if estimatedIFP == 0 {
+ // If no inflection points then apply PA on the full Bezier segment.
+ c.doParabolicApproximation(t, flattening_threshold)
+ return
+ }
+ // If one or more inflection point then we will have to subdivide the curve
+ numOfIfP, t1, t2 := c.findInflectionPoints()
+ if numOfIfP == 2 {
+ // Case when 2 inflection points then divide at the smallest one first
+ var sub1, tmp1, sub2, sub3 CubicCurveFloat64
+ c.SubdivideAt(&sub1, &tmp1, t1)
+ // Now find the second inflection point in the second curve an subdivide
+ numOfIfP, t1, t2 = tmp1.findInflectionPoints()
+ if numOfIfP == 2 {
+ tmp1.SubdivideAt(&sub2, &sub3, t2)
+ } else if numOfIfP == 1 {
+ tmp1.SubdivideAt(&sub2, &sub3, t1)
+ } else {
+ return
+ }
+ // Use PA for first subsegment
+ sub1.doParabolicApproximation(t, flattening_threshold)
+ // Use RS for the second (middle) subsegment
+ sub2.Segment(t, flattening_threshold)
+ // Drop the last point in the array will be added by the PA in third subsegment
+ //noOfPoints--;
+ // Use PA for the third curve
+ sub3.doParabolicApproximation(t, flattening_threshold)
+ } else if numOfIfP == 1 {
+ // Case where there is one inflection point, subdivide once and use PA on
+ // both subsegments
+ var sub1, sub2 CubicCurveFloat64
+ c.SubdivideAt(&sub1, &sub2, t1)
+ sub1.doParabolicApproximation(t, flattening_threshold)
+ //noOfPoints--;
+ sub2.doParabolicApproximation(t, flattening_threshold)
+ } else {
+ // Case where there is no inflection USA PA directly
+ c.doParabolicApproximation(t, flattening_threshold)
+ }
+}
+
+// Find the third control point deviation form the axis
+func (c *CubicCurveFloat64) thirdControlPointDeviation() float64 {
+ dx := c[2] - c[0]
+ dy := c[3] - c[1]
+ l2 := dx*dx + dy*dy
+ if l2 == 0 {
+ return 0
+ }
+ l := math.Sqrt(l2)
+ r := (c[3] - c[1]) / l
+ s := (c[0] - c[2]) / l
+ u := (c[2]*c[1] - c[0]*c[3]) / l
+ return math.Abs(r*c[4] + s*c[5] + u)
+}
+
+// Find the number of inflection point
+func (c *CubicCurveFloat64) numberOfInflectionPoints() int {
+ dx21 := (c[2] - c[0])
+ dy21 := (c[3] - c[1])
+ dx32 := (c[4] - c[2])
+ dy32 := (c[5] - c[3])
+ dx43 := (c[6] - c[4])
+ dy43 := (c[7] - c[5])
+ if ((dx21*dy32 - dy21*dx32) * (dx32*dy43 - dy32*dx43)) < 0 {
+ return 1 // One inflection point
+ } else if ((dx21*dy32 - dy21*dx32) * (dx21*dy43 - dy21*dx43)) > 0 {
+ return 0 // No inflection point
+ } else {
+ // Most cases no inflection point
+ b1 := (dx21*dx32 + dy21*dy32) > 0
+ b2 := (dx32*dx43 + dy32*dy43) > 0
+ if b1 || b2 && !(b1 && b2) { // xor!!
+ return 0
+ }
+ }
+ return -1 // cases where there in zero or two inflection points
+}
+
+// This is the main function where all the work is done
+func (curve *CubicCurveFloat64) doParabolicApproximation(tracer LineTracer, flattening_threshold float64) {
+ var c *CubicCurveFloat64
+ c = curve
+ var d, t, dx, dy, d2, d3 float64
+ for {
+ dx = c[6] - c[0]
+ dy = c[7] - c[1]
+
+ d2 = math.Abs(((c[2]-c[6])*dy - (c[3]-c[7])*dx))
+ d3 = math.Abs(((c[4]-c[6])*dy - (c[5]-c[7])*dx))
+
+ if (d2+d3)*(d2+d3) < flattening_threshold*(dx*dx+dy*dy) {
+ // If the subsegment deviation satisfy the flatness then store the last
+ // point and stop
+ tracer.LineTo(c[6], c[7])
+ break
+ }
+ // Find the third control point deviation and the t values for subdivision
+ d = c.thirdControlPointDeviation()
+ t = 2 * math.Sqrt(flattening_threshold/d/3)
+ if t > 1 {
+ // Case where the t value calculated is invalid so using RS
+ c.Segment(tracer, flattening_threshold)
+ break
+ }
+ // Valid t value to subdivide at that calculated value
+ var b1, b2 CubicCurveFloat64
+ c.SubdivideAt(&b1, &b2, t)
+ // First subsegment should have its deviation equal to flatness
+ dx = b1[6] - b1[0]
+ dy = b1[7] - b1[1]
+
+ d2 = math.Abs(((b1[2]-b1[6])*dy - (b1[3]-b1[7])*dx))
+ d3 = math.Abs(((b1[4]-b1[6])*dy - (b1[5]-b1[7])*dx))
+
+ if (d2+d3)*(d2+d3) > flattening_threshold*(dx*dx+dy*dy) {
+ // if not then use RS to handle any mathematical errors
+ b1.Segment(tracer, flattening_threshold)
+ } else {
+ tracer.LineTo(b1[6], b1[7])
+ }
+ // repeat the process for the left over subsegment.
+ c = &b2
+ }
+}
+
+// Find the actual inflection points and return the number of inflection points found
+// if 2 inflection points found, the first one returned will be with smaller t value.
+func (curve *CubicCurveFloat64) findInflectionPoints() (int, firstIfp, secondIfp float64) {
+ // For Cubic Bezier curve with equation P=a*t^3 + b*t^2 + c*t + d
+ // slope of the curve dP/dt = 3*a*t^2 + 2*b*t + c
+ // a = (float)(-bez.p1 + 3*bez.p2 - 3*bez.p3 + bez.p4);
+ // b = (float)(3*bez.p1 - 6*bez.p2 + 3*bez.p3);
+ // c = (float)(-3*bez.p1 + 3*bez.p2);
+ ax := (-curve[0] + 3*curve[2] - 3*curve[4] + curve[6])
+ bx := (3*curve[0] - 6*curve[2] + 3*curve[4])
+ cx := (-3*curve[0] + 3*curve[2])
+ ay := (-curve[1] + 3*curve[3] - 3*curve[5] + curve[7])
+ by := (3*curve[1] - 6*curve[3] + 3*curve[5])
+ cy := (-3*curve[1] + 3*curve[3])
+ a := (3 * (ay*bx - ax*by))
+ b := (3 * (ay*cx - ax*cy))
+ c := (by*cx - bx*cy)
+ r2 := (b*b - 4*a*c)
+ firstIfp = 0.0
+ secondIfp = 0.0
+ if r2 >= 0.0 && a != 0.0 {
+ r := math.Sqrt(r2)
+ firstIfp = ((-b + r) / (2 * a))
+ secondIfp = ((-b - r) / (2 * a))
+ if (firstIfp > 0.0 && firstIfp < 1.0) && (secondIfp > 0.0 && secondIfp < 1.0) {
+ if firstIfp > secondIfp {
+ tmp := firstIfp
+ firstIfp = secondIfp
+ secondIfp = tmp
+ }
+ if secondIfp-firstIfp > 0.00001 {
+ return 2, firstIfp, secondIfp
+ } else {
+ return 1, firstIfp, secondIfp
+ }
+ } else if firstIfp > 0.0 && firstIfp < 1.0 {
+ return 1, firstIfp, secondIfp
+ } else if secondIfp > 0.0 && secondIfp < 1.0 {
+ firstIfp = secondIfp
+ return 1, firstIfp, secondIfp
+ }
+ return 0, firstIfp, secondIfp
+ }
+ return 0, firstIfp, secondIfp
+}
generated by cgit v1.2.3-1-g7c22 (git 2.25.1) at 2024-06-11 19:34:43 +0000