diff --git a/draw2d/curve/curve_float64.go b/draw2d/curve/curve_float64.go
index 40abf8d..ba7eb33 100644
--- a/draw2d/curve/curve_float64.go
+++ b/draw2d/curve/curve_float64.go
@@ -6,14 +6,21 @@ import (
"math"
)
-var (
- flattening_threshold float64 = 0.25
+const (
+ CurveRecursionLimit = 32
+ CurveCollinearityEpsilon = 1e-30
+ CurveAngleToleranceEpsilon = 0.01
)
type CubicCurveFloat64 struct {
X1, Y1, X2, Y2, X3, Y3, X4, Y4 float64
}
+type LineTracer interface {
+ LineTo(x, y float64)
+}
+
+
//mu ranges from 0 to 1, start to end of curve
func (c *CubicCurveFloat64) ArbitraryPoint(mu float64) (x, y float64) {
@@ -26,7 +33,7 @@ func (c *CubicCurveFloat64) ArbitraryPoint(mu float64) (x, y float64) {
return
}
-func (c *CubicCurveFloat64) SubdivideAt(c1, c2 *CubicCurveFloat64, t float64) {
+func (c *CubicCurveFloat64) SubdivideAt(c1, c2 *CubicCurveFloat64, t float64) (x23, y23 float64) {
inv_t := (1 - t)
c1.X1, c1.Y1 = c.X1, c.Y1
c2.X4, c2.Y4 = c.X4, c.Y4
@@ -34,8 +41,8 @@ func (c *CubicCurveFloat64) SubdivideAt(c1, c2 *CubicCurveFloat64, t float64) {
c1.X2 = inv_t*c.X1 + t*c.X2
c1.Y2 = inv_t*c.Y1 + t*c.Y2
- x23 := inv_t*c.X2 + t*c.X3
- y23 := inv_t*c.Y2 + t*c.Y3
+ x23 = inv_t*c.X2 + t*c.X3
+ y23 = inv_t*c.Y2 + t*c.Y3
c2.X3 = inv_t*c.X3 + t*c.X4
c2.Y3 = inv_t*c.Y3 + t*c.Y4
@@ -50,17 +57,18 @@ func (c *CubicCurveFloat64) SubdivideAt(c1, c2 *CubicCurveFloat64, t float64) {
c1.Y4 = inv_t*c1.Y3 + t*c2.Y2
c2.X1, c2.Y1 = c1.X4, c1.Y4
+ return
}
-func (c *CubicCurveFloat64) Subdivide(c1, c2 *CubicCurveFloat64) {
+func (c *CubicCurveFloat64) Subdivide(c1, c2 *CubicCurveFloat64) (x23, y23 float64) {
// Calculate all the mid-points of the line segments
//----------------------
c1.X1, c1.Y1 = c.X1, c.Y1
c2.X4, c2.Y4 = c.X4, c.Y4
c1.X2 = (c.X1 + c.X2) / 2
c1.Y2 = (c.Y1 + c.Y2) / 2
- x23 := (c.X2 + c.X3) / 2
- y23 := (c.Y2 + c.Y3) / 2
+ x23 = (c.X2 + c.X3) / 2
+ y23 = (c.Y2 + c.Y3) / 2
c2.X3 = (c.X3 + c.X4) / 2
c2.Y3 = (c.Y3 + c.Y4) / 2
c1.X3 = (c1.X2 + x23) / 2
@@ -70,6 +78,7 @@ func (c *CubicCurveFloat64) Subdivide(c1, c2 *CubicCurveFloat64) {
c1.X4 = (c1.X3 + c2.X2) / 2
c1.Y4 = (c1.Y3 + c2.Y2) / 2
c2.X1, c2.Y1 = c1.X4, c1.Y4
+ return
}
func (c *CubicCurveFloat64) EstimateDistance() float64 {
@@ -83,19 +92,14 @@ func (c *CubicCurveFloat64) EstimateDistance() float64 {
}
// subdivide the curve in straight lines using line approximation and Casteljau recursive subdivision
-func (c *CubicCurveFloat64) SegmentRec(segments []float64) []float64 {
+func (c *CubicCurveFloat64) SegmentRec(t LineTracer, flattening_threshold float64) {
// reinit segments
- segments = segments[0 : len(segments)+2]
- segments[len(segments)-2] = c.X1
- segments[len(segments)-1] = c.Y1
- segments = c.segmentRec(segments)
- segments = segments[0 : len(segments)+2]
- segments[len(segments)-2] = c.X4
- segments[len(segments)-1] = c.Y4
- return segments
+ t.LineTo(c.X1, c.Y1)
+ c.segmentRec(t, flattening_threshold)
+ t.LineTo(c.X4, c.Y4)
}
-func (c *CubicCurveFloat64) segmentRec(segments []float64) []float64 {
+func (c *CubicCurveFloat64) segmentRec(t LineTracer, flattening_threshold float64) {
var c1, c2 CubicCurveFloat64
c.Subdivide(&c1, &c2)
@@ -108,25 +112,20 @@ func (c *CubicCurveFloat64) segmentRec(segments []float64) []float64 {
d3 := math.Fabs(((c.X3-c.X4)*dy - (c.Y3-c.Y4)*dx))
if (d2+d3)*(d2+d3) < flattening_threshold*(dx*dx+dy*dy) {
- segments = segments[0 : len(segments)+2]
- segments[len(segments)-2] = c2.X4
- segments[len(segments)-1] = c2.Y4
- return segments
+ t.LineTo(c.X4, c.Y4)
+ return
}
// Continue subdivision
//----------------------
- segments = c1.segmentRec(segments)
- segments = c2.segmentRec(segments)
- return segments
+ c1.segmentRec(t, flattening_threshold)
+ c2.segmentRec(t, flattening_threshold)
}
-func (curve *CubicCurveFloat64) Segment(segments []float64) []float64 {
+func (curve *CubicCurveFloat64) Segment(t LineTracer, flattening_threshold float64) {
// Add the first point
- segments = segments[0 : len(segments)+2]
- segments[len(segments)-2] = curve.X1
- segments[len(segments)-1] = curve.Y1
+ t.LineTo(curve.X1, curve.Y1)
- var curves [32]CubicCurveFloat64
+ var curves [CurveRecursionLimit]CubicCurveFloat64
curves[0] = *curve
i := 0
// current curve
@@ -141,9 +140,7 @@ func (curve *CubicCurveFloat64) Segment(segments []float64) []float64 {
d3 = math.Fabs(((c.X3-c.X4)*dy - (c.Y3-c.Y4)*dx))
if (d2+d3)*(d2+d3) < flattening_threshold*(dx*dx+dy*dy) || i == len(curves)-1 {
- segments = segments[0 : len(segments)+2]
- segments[len(segments)-2] = c.X4
- segments[len(segments)-1] = c.Y4
+ t.LineTo(c.X4, c.Y4)
i--
} else {
// second half of bezier go lower onto the stack
@@ -151,5 +148,618 @@ func (curve *CubicCurveFloat64) Segment(segments []float64) []float64 {
i++
}
}
- return segments
+}
+
+/*
+ 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
+ t.LineTo(c.X1, c.Y1)
+ c.adaptiveSegmentRec(t, 0, distanceToleranceSquare, angleTolerance, cuspLimit)
+ t.LineTo(c.X4, c.Y4)
+}
+
+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.X4 - c.X1
+ dy := c.Y4 - c.Y1
+
+ d2 := math.Fabs(((c.X2-c.X4)*dy - (c.Y2-c.Y4)*dx))
+ d3 := math.Fabs(((c.X3-c.X4)*dy - (c.Y3-c.Y4)*dx))
+ switch {
+ case d2 <= CurveCollinearityEpsilon && d3 <= CurveCollinearityEpsilon:
+ // All collinear OR p1==p4
+ //----------------------
+ k := dx*dx + dy*dy
+ if k == 0 {
+ d2 = squareDistance(c.X1, c.Y1, c.X2, c.Y2)
+ d3 = squareDistance(c.X4, c.Y4, c.X3, c.Y3)
+ } else {
+ k = 1 / k
+ da1 := c.X2 - c.X1
+ da2 := c.Y2 - c.Y1
+ d2 = k * (da1*dx + da2*dy)
+ da1 = c.X3 - c.X1
+ da2 = c.Y3 - c.Y1
+ 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.X2, c.Y2, c.X1, c.Y1)
+ } else if d2 >= 1 {
+ d2 = squareDistance(c.X2, c.Y2, c.X4, c.Y4)
+ } else {
+ d2 = squareDistance(c.X2, c.Y2, c.X1+d2*dx, c.Y1+d2*dy)
+ }
+
+ if d3 <= 0 {
+ d3 = squareDistance(c.X3, c.Y3, c.X1, c.Y1)
+ } else if d3 >= 1 {
+ d3 = squareDistance(c.X3, c.Y3, c.X4, c.Y4)
+ } else {
+ d3 = squareDistance(c.X3, c.Y3, c.X1+d3*dx, c.Y1+d3*dy)
+ }
+ }
+ if d2 > d3 {
+ if d2 < distanceToleranceSquare {
+ t.LineTo(c.X2, c.Y2)
+ return
+ }
+ } else {
+ if d3 < distanceToleranceSquare {
+ t.LineTo(c.X3, c.Y3)
+ 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.Fabs(math.Atan2(c.Y4-c.Y3, c.X4-c.X3) - math.Atan2(c.Y3-c.Y2, c.X3-c.X2))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c.X2, c.Y2)
+ t.LineTo(c.X3, c.Y3)
+ return
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c.X3, c.Y3)
+ 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.Fabs(math.Atan2(c.Y3-c.Y2, c.X3-c.X2) - math.Atan2(c.Y2-c.Y1, c.X2-c.X1))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c.X2, c.Y2)
+ t.LineTo(c.X3, c.Y3)
+ return
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c.X2, c.Y2)
+ 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.Y3-c.Y2, c.X3-c.X2)
+ da1 := math.Fabs(k - math.Atan2(c.Y2-c.Y1, c.X2-c.X1))
+ da2 := math.Fabs(math.Atan2(c.Y4-c.Y3, c.X4-c.X3) - 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.X2, c.Y2)
+ return
+ }
+
+ if da2 > cuspLimit {
+ t.LineTo(c.X3, c.Y3)
+ 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) {
+ // Add the first point
+ t.LineTo(curve.X1, curve.Y1)
+ 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.X4 - c.X1
+ dy = c.Y4 - c.Y1
+
+ d2 = math.Fabs(((c.X2-c.X4)*dy - (c.Y2-c.Y4)*dx))
+ d3 = math.Fabs(((c.X3-c.X4)*dy - (c.Y3-c.Y4)*dx))
+ switch {
+ case i == len(curves)-1:
+ t.LineTo(c.X4, c.Y4)
+ i--
+ continue
+ case d2 <= CurveCollinearityEpsilon && d3 <= CurveCollinearityEpsilon:
+ // All collinear OR p1==p4
+ //----------------------
+ k = dx*dx + dy*dy
+ if k == 0 {
+ d2 = squareDistance(c.X1, c.Y1, c.X2, c.Y2)
+ d3 = squareDistance(c.X4, c.Y4, c.X3, c.Y3)
+ } else {
+ k = 1 / k
+ da1 := c.X2 - c.X1
+ da2 := c.Y2 - c.Y1
+ d2 = k * (da1*dx + da2*dy)
+ da1 = c.X3 - c.X1
+ da2 = c.Y3 - c.Y1
+ 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.X2, c.Y2, c.X1, c.Y1)
+ } else if d2 >= 1 {
+ d2 = squareDistance(c.X2, c.Y2, c.X4, c.Y4)
+ } else {
+ d2 = squareDistance(c.X2, c.Y2, c.X1+d2*dx, c.Y1+d2*dy)
+ }
+
+ if d3 <= 0 {
+ d3 = squareDistance(c.X3, c.Y3, c.X1, c.Y1)
+ } else if d3 >= 1 {
+ d3 = squareDistance(c.X3, c.Y3, c.X4, c.Y4)
+ } else {
+ d3 = squareDistance(c.X3, c.Y3, c.X1+d3*dx, c.Y1+d3*dy)
+ }
+ }
+ if d2 > d3 {
+ if d2 < distanceToleranceSquare {
+ t.LineTo(c.X2, c.Y2)
+ i--
+ continue
+ }
+ } else {
+ if d3 < distanceToleranceSquare {
+ t.LineTo(c.X3, c.Y3)
+ 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.Fabs(math.Atan2(c.Y4-c.Y3, c.X4-c.X3) - math.Atan2(c.Y3-c.Y2, c.X3-c.X2))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c.X2, c.Y2)
+ t.LineTo(c.X3, c.Y3)
+ i--
+ continue
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c.X3, c.Y3)
+ 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.Fabs(math.Atan2(c.Y3-c.Y2, c.X3-c.X2) - math.Atan2(c.Y2-c.Y1, c.X2-c.X1))
+ if da1 >= math.Pi {
+ da1 = 2*math.Pi - da1
+ }
+
+ if da1 < angleTolerance {
+ t.LineTo(c.X2, c.Y2)
+ t.LineTo(c.X3, c.Y3)
+ i--
+ continue
+ }
+
+ if cuspLimit != 0.0 {
+ if da1 > cuspLimit {
+ t.LineTo(c.X2, c.Y2)
+ 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.Y3-c.Y2, c.X3-c.X2)
+ da1 := math.Fabs(k - math.Atan2(c.Y2-c.Y1, c.X2-c.X1))
+ da2 := math.Fabs(math.Atan2(c.Y4-c.Y3, c.X4-c.X3) - 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.X2, c.Y2)
+ i--
+ continue
+ }
+
+ if da2 > cuspLimit {
+ t.LineTo(c.X3, c.Y3)
+ i--
+ continue
+ }
+ }
+ }
+ }
+
+ // Continue subdivision
+ //----------------------
+ curves[i+1], curves[i] = c1, c2
+ i++
+ }
+ t.LineTo(curve.X4, curve.Y4)
+}
+
+
+/********************** 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) {
+ t.LineTo(c.X1, c.Y1)
+ 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.X2 - c.X1
+ dy := c.Y2 - c.Y1
+ l2 := dx*dx + dy*dy
+ if l2 == 0 {
+ return 0
+ }
+ l := math.Sqrt(l2)
+ r := (c.Y2 - c.Y1) / l
+ s := (c.X1 - c.X2) / l
+ u := (c.X2*c.Y1 - c.X1*c.Y2) / l
+ return math.Fabs(r*c.X3 + s*c.Y3 + u)
+}
+
+// Find the number of inflection point
+func (c *CubicCurveFloat64) numberOfInflectionPoints() int {
+ dx21 := (c.X2 - c.X1)
+ dy21 := (c.Y2 - c.Y1)
+ dx32 := (c.X3 - c.X2)
+ dy32 := (c.Y3 - c.Y2)
+ dx43 := (c.X4 - c.X3)
+ dy43 := (c.Y4 - c.Y3)
+ 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.X4 - c.X1
+ dy = c.Y4 - c.Y1
+
+ d2 = math.Fabs(((c.X2-c.X4)*dy - (c.Y2-c.Y4)*dx))
+ d3 = math.Fabs(((c.X3-c.X4)*dy - (c.Y3-c.Y4)*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.X4, c.Y4)
+ 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.X4 - b1.X1
+ dy = b1.Y4 - b1.Y1
+
+ d2 = math.Fabs(((b1.X2-b1.X4)*dy - (b1.Y2-b1.Y4)*dx))
+ d3 = math.Fabs(((b1.X3-b1.X4)*dy - (b1.Y3-b1.Y4)*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.X4, b1.Y4)
+ }
+ // 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.X1 + 3*curve.X2 - 3*curve.X3 + curve.X4)
+ bx := (3*curve.X1 - 6*curve.X2 + 3*curve.X3)
+ cx := (-3*curve.X1 + 3*curve.X2)
+ ay := (-curve.Y1 + 3*curve.Y2 - 3*curve.Y3 + curve.Y4)
+ by := (3*curve.Y1 - 6*curve.Y2 + 3*curve.Y3)
+ cy := (-3*curve.Y1 + 3*curve.Y2)
+ 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
}
diff --git a/draw2d/curve/curve_test.go b/draw2d/curve/curve_test.go
index 951f4e6..c43fbf9 100644
--- a/draw2d/curve/curve_test.go
+++ b/draw2d/curve/curve_test.go
@@ -14,13 +14,33 @@ import (
var (
- testsFloat64 = []CubicCurveFloat64{
+ flattening_threshold float64 = 0.25
+ testsFloat64 = []CubicCurveFloat64{
CubicCurveFloat64{100, 100, 200, 100, 100, 200, 200, 200},
CubicCurveFloat64{100, 100, 300, 200, 200, 200, 300, 100},
CubicCurveFloat64{100, 100, 0, 300, 200, 0, 300, 300},
+ CubicCurveFloat64{150, 290, 10, 10, 290, 10, 150, 290},
+ CubicCurveFloat64{10, 290, 10, 10, 290, 10, 290, 290},
+ CubicCurveFloat64{100, 290, 290, 10, 10, 10, 200, 290},
}
)
+type Path struct {
+ points []float64
+}
+
+func (p *Path) LineTo(x, y float64) {
+ if len(p.points)+2 > cap(p.points) {
+ points := make([]float64, len(p.points)+2, len(p.points)+32)
+ copy(points, p.points)
+ p.points = points
+ } else {
+ p.points = p.points[0 : len(p.points)+2]
+ }
+ p.points[len(p.points)-2] = x
+ p.points[len(p.points)-1] = y
+}
+
func init() {
f, err := os.Create("_test.html")
if err != nil {
@@ -31,7 +51,7 @@ func init() {
log.Printf("Create html viewer")
f.Write([]byte(""))
for i := 0; i < len(testsFloat64); i++ {
- f.Write([]byte(fmt.Sprintf("", i, i)))
+ f.Write([]byte(fmt.Sprintf("\n", i, i, i, i, i)))
}
f.Write([]byte(""))
@@ -69,55 +89,91 @@ func drawPoints(img draw.Image, c image.Color, s ...float64) image.Image {
img.Set(x-1, y, c)
img.Set(x-1, y+1, c)
img.Set(x-1, y-1, c)
-
+
}
return img
}
func TestCubicCurveRec(t *testing.T) {
for i, curve := range testsFloat64 {
- d := curve.EstimateDistance()
- log.Printf("Distance estimation: %f\n", d)
- numSegments := int(d * 0.25)
- log.Printf("Max segments estimation: %d\n", numSegments)
- s := make([]float64, 0, numSegments)
- s = curve.SegmentRec(s)
+ var p Path
+ curve.SegmentRec(&p, flattening_threshold)
img := image.NewNRGBA(300, 300)
raster.PolylineBresenham(img, image.NRGBAColor{0xff, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
- raster.PolylineBresenham(img, image.Black, s...)
- drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
- drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, s...)
+ raster.PolylineBresenham(img, image.Black, p.points...)
+ //drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, p.points...)
savepng(fmt.Sprintf("_testRec%d.png", i), img)
- log.Printf("Num of points: %d\n", len(s))
+ log.Printf("Num of points: %d\n", len(p.points))
}
+ fmt.Println()
}
func TestCubicCurve(t *testing.T) {
for i, curve := range testsFloat64 {
- d := curve.EstimateDistance()
- log.Printf("Distance estimation: %f\n", d)
- numSegments := int(d * 0.25)
- log.Printf("Max segments estimation: %d\n", numSegments)
- s := make([]float64, 0, numSegments)
- s = curve.Segment(s)
+ var p Path
+ curve.Segment(&p, flattening_threshold)
img := image.NewNRGBA(300, 300)
raster.PolylineBresenham(img, image.NRGBAColor{0xff, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
- raster.PolylineBresenham(img, image.Black, s...)
- drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
- drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, s...)
+ raster.PolylineBresenham(img, image.Black, p.points...)
+ //drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, p.points...)
savepng(fmt.Sprintf("_test%d.png", i), img)
- log.Printf("Num of points: %d\n", len(s))
+ log.Printf("Num of points: %d\n", len(p.points))
}
+ fmt.Println()
}
+func TestCubicCurveAdaptiveRec(t *testing.T) {
+ for i, curve := range testsFloat64 {
+ var p Path
+ curve.AdaptiveSegmentRec(&p, 1, 0, 0)
+ img := image.NewNRGBA(300, 300)
+ raster.PolylineBresenham(img, image.NRGBAColor{0xff, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ raster.PolylineBresenham(img, image.Black, p.points...)
+ //drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, p.points...)
+ savepng(fmt.Sprintf("_testAdaptiveRec%d.png", i), img)
+ log.Printf("Num of points: %d\n", len(p.points))
+ }
+ fmt.Println()
+}
+
+func TestCubicCurveAdaptive(t *testing.T) {
+ for i, curve := range testsFloat64 {
+ var p Path
+ curve.AdaptiveSegment(&p, 1, 0, 0)
+ img := image.NewNRGBA(300, 300)
+ raster.PolylineBresenham(img, image.NRGBAColor{0xff, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ raster.PolylineBresenham(img, image.Black, p.points...)
+ //drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, p.points...)
+ savepng(fmt.Sprintf("_testAdaptive%d.png", i), img)
+ log.Printf("Num of points: %d\n", len(p.points))
+ }
+ fmt.Println()
+}
+
+func TestCubicCurveParabolic(t *testing.T) {
+ for i, curve := range testsFloat64 {
+ var p Path
+ curve.ParabolicSegment(&p, flattening_threshold)
+ img := image.NewNRGBA(300, 300)
+ raster.PolylineBresenham(img, image.NRGBAColor{0xff, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ raster.PolylineBresenham(img, image.Black, p.points...)
+ //drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, curve.X1, curve.Y1, curve.X2, curve.Y2, curve.X3, curve.Y3, curve.X4, curve.Y4)
+ drawPoints(img, image.NRGBAColor{0, 0, 0, 0xff}, p.points...)
+ savepng(fmt.Sprintf("_testParabolic%d.png", i), img)
+ log.Printf("Num of points: %d\n", len(p.points))
+ }
+ fmt.Println()
+}
func BenchmarkCubicCurveRec(b *testing.B) {
for i := 0; i < b.N; i++ {
for _, curve := range testsFloat64 {
- d := curve.EstimateDistance()
- numSegments := int(d * 0.25)
- s := make([]float64, 0, numSegments)
- curve.SegmentRec(s)
+ p := Path{make([]float64, 0, 32)}
+ curve.SegmentRec(&p, flattening_threshold)
}
}
}
@@ -125,10 +181,35 @@ func BenchmarkCubicCurveRec(b *testing.B) {
func BenchmarkCubicCurve(b *testing.B) {
for i := 0; i < b.N; i++ {
for _, curve := range testsFloat64 {
- d := curve.EstimateDistance()
- numSegments := int(d * 0.25)
- s := make([]float64, 0, numSegments)
- curve.Segment(s)
+ p := Path{make([]float64, 0, 32)}
+ curve.Segment(&p, flattening_threshold)
+ }
+ }
+}
+
+func BenchmarkCubicCurveAdaptiveRec(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ for _, curve := range testsFloat64 {
+ p := Path{make([]float64, 0, 32)}
+ curve.AdaptiveSegmentRec(&p, 1, 0, 0)
+ }
+ }
+}
+
+func BenchmarkCubicCurveAdaptive(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ for _, curve := range testsFloat64 {
+ p := Path{make([]float64, 0, 32)}
+ curve.AdaptiveSegment(&p, 1, 0, 0)
+ }
+ }
+}
+
+func BenchmarkCubicCurveParabolic(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ for _, curve := range testsFloat64 {
+ p := Path{make([]float64, 0, 32)}
+ curve.ParabolicSegment(&p, flattening_threshold)
}
}
}
\n
\n
\n