public static Intersection QuadraticBezierLineSegmentIntersection0(
double p1X, double p1Y,
double p2X, double p2Y,
double p3X, double p3Y,
double a1X, double a1Y,
double a2X, double a2Y,
double epsilon = Epsilon)
{
var min = MinPointTests.MinPoint(a1X, a1Y, a2X, a2Y);
var max = MaxPointTests.MaxPoint(a1X, a1Y, a2X, a2Y);
var result = new Intersection(IntersectionStates.NoIntersection);
var a = new Vector2D(p2X, p2Y) * (-2);
var c2 = new Vector2D(p1X, p1Y) + (a + new Vector2D(p3X, p3Y));
a = new Vector2D(p1X, p1Y) * (-2);
var b = new Vector2D(p2X, p2Y) * 2;
var c1 = a + b;
var c0 = new Point2D(p1X, p1Y);
var n = new Point2D(a1Y - a2Y, a2X - a1X);
var cl = (a1X * a2Y) - (a2X * a1Y);
var roots = QuadraticRootsTests.QuadraticRoots(
DotProduct2Vector2DTests.DotProduct2D(n.X, n.Y, c2.I, c2.J),
DotProduct2Vector2DTests.DotProduct2D(n.X, n.Y, c1.I, c1.J),
DotProduct2Vector2DTests.DotProduct2D(n.X, n.Y, c0.X + cl, c0.Y + cl),
epsilon);
for (var i = 0; i < roots.Count; i++)
{
var t = roots[i];
if (0 <= t && t <= 1)
{
var p4 = InterpolateLinear2DTests.LinearInterpolate2D(t, p1X, p1Y, p2X, p2Y);
var p5 = InterpolateLinear2DTests.LinearInterpolate2D(t, p2X, p2Y, p3X, p3Y);
var p6 = InterpolateLinear2DTests.LinearInterpolate2D(t, p4.X, p4.Y, p5.X, p5.Y);
if (a1X == a2X)
{
if (min.Y <= p6.Y && p6.Y <= max.Y)
{
result.State = IntersectionStates.Intersection;
result.AppendPoint(p6);
}
}
else if (a1Y == a2Y)
{
if (min.X <= p6.X && p6.X <= max.X)
{
result.State = IntersectionStates.Intersection;
result.AppendPoint(p6);
}
}
else if (GreaterThanOrEqualTests.GreaterThanOrEqual(p6.X, p6.Y, min.X, min.Y) && LessThanOrEqualTests.LessThanOrEqual(p6.X, p6.Y, max.X, max.Y))
{
result.State = IntersectionStates.Intersection;
result.AppendPoint(p6);
}
}
}
return(result);
}
public static IList <double> CubicRootsStephenRSchmitt(double a, double b, double c, double d, double epsilon = Epsilon)
{
// If a is 0 the polynomial is quadratic.
if (a is 0d)
{
return(QuadraticRootsTests.QuadraticRoots(b, c, d, epsilon));
}
var ba = b / a;
var ca = c / a;
var da = d / a;
var q = ((3d * ca) - (ba * ba)) / 9d;
var r = (-(2d * ba * ba * ba) + (9d * ba * ca) - (27d * da)) / 54d;
var offset = ba * OneThird;
// Polynomial discriminant
var discriminant = (r * r) + (q * q * q);
// ToDo: May need to switch from a hash set to a list for scan-beams.
var results = new HashSet <double>();
if (Abs(discriminant) <= epsilon)
{
discriminant = 0d;
}
if (discriminant == 0d)
{
var t = Sign(r) * Cbrt(Abs(r));
// Real root.
results.Add(-offset + (t + t));
// Real part of complex root.
results.Add(-offset - ((t + t) * OneHalf));
}
if (discriminant > 0)
{
var e = Sqrt(discriminant);
var s = Sign(r + e) * Cbrt(Abs(r + e));
var t = Sign(r - e) * Cbrt(Abs(r - e));
// Real root.
results.Add(-offset + (s + t));
// Complex part of root pair.
var Im = Abs(Sqrt3 * (s - t) * OneHalf);
if (Im == 0d)
{
// Real part of complex root.
results.Add(-offset - ((s + t) * OneHalf));
}
}
else if (discriminant < 0)
{
// Distinct real roots.
var th = Acos(r / Sqrt(-q * q * q));
var sq = Sqrt(-q);
results.Add((2d * sq * Cos(th * OneThird)) - offset);
results.Add((2d * sq * Cos((th + Tau) * OneThird)) - offset);
results.Add((2d * sq * Cos((th + (4d * PI)) * OneThird)) - offset);
}
return(results.ToList());
}
public static IList <double> CubicRootsSwitch(double a, double b, double c, double d, double epsilon = Epsilon)
{
if (a is 0d)
{
return(QuadraticRootsTests.QuadraticRoots(b, c, d, epsilon));
}
var ba = b / a;
var ca = c / a;
var da = d / a;
var q = ((3d * ca) - (ba * ba)) * OneThird; // / 9d;
var r = ((2d * ba * ba * ba) - (9d * ba * ca) + (27d * da)) * OneTwentySeventh; // / 54d;
var offset = ba * OneThird;
var discriminant = (r * r * OneQuarter) + (q * q * q * OneTwentySeventh);
var halfB = OneHalf * r;
if (Abs(discriminant) <= epsilon)
{
discriminant = 0d;
}
switch (discriminant)
{
case 0:
{
var f = halfB >= 0d ? -Cbrt(halfB) : Cbrt(-halfB);
return(new double[] {
(2d * f) - offset,
-f - offset
}.ToList());
}
case double v when v > 0d:
{
var e = Sqrt(v);
var tmp = -halfB + e;
var root = tmp >= 0 ? Cbrt(tmp) : -Cbrt(-tmp);
tmp = -halfB - e;
if (tmp >= 0)
{
root += Cbrt(tmp);
}
else
{
root -= Cbrt(-tmp);
}
return(new double[] { root - offset }.ToList());
//var s = Sign(r + e) * Cbrt(Abs(r + e));
//var t = Sign(r - e) * Cbrt(Abs(r - e));
//var im = Abs(Sqrt(3d) * (s - t) * OneHalf);
//return im == 0d ?
// new double[] {
// -offset + (s + t)
// }.ToList() :
// new double[] {
// -offset + (s + t),
// -offset - ((s + t) * OneHalf)
// }.ToList();
}
default:
{
var distance = Sqrt(-q * OneThird);
var angle = Atan2(Sqrt(-discriminant), -halfB) * OneThird;
var cos = Cos(angle);
var sin = Sin(angle);
return(new double[] {
(2d * distance * cos) - offset,
(-distance * (cos + (Sqrt3 * sin))) - offset,
(-distance * (cos - (Sqrt3 * sin))) - offset
}.ToList());
}
}
}
public static IList <double> CubicRootsKevinLinDev(double a, double b, double c, double d, double epsilon = Epsilon)
{
// If a is 0 the polynomial is quadratic.
if (a is 0d)
{
return(QuadraticRootsTests.QuadraticRoots(b, c, d, epsilon));
}
var ba = b / a;
var ca = c / a;
var da = d / a;
var q = ((3d * ca) - (ba * ba)) * OneThird;
var r = ((2d * ba * ba * ba) - (9d * ca * ba) + (27d * da)) * OneTwentySeventh;
var offset = ba * OneThird;
var discriminant = (r * r * OneQuarter) + (q * q * q * OneTwentySeventh);
var halfR = OneHalf * r;
//var ZEROepsilon = ZeroErrorEstimate();
if (Abs(discriminant) <= epsilon)//ZEROepsilon)
{
discriminant = 0d;
}
var results = new HashSet <double>();
if (discriminant > 0d)
{
var e = Sqrt(discriminant);
var tmp = -halfR + e;
var root = tmp >= 0 ? Cbrt(tmp) : -Cbrt(-tmp);
tmp = -halfR - e;
if (tmp >= 0)
{
root += Cbrt(tmp);
}
else
{
root -= Cbrt(-tmp);
}
results.Add(root - offset);
}
else if (discriminant < 0d)
{
var distance = Sqrt(-q * OneThird);
var angle = Atan2(Sqrt(-discriminant), -halfR) * OneThird;
//var (c, s) = ();
var cos = Cos(angle);
var sin = Sin(angle);
results.Add((2d * distance * cos) - offset);
results.Add((-distance * (cos + (Sqrt3 * sin))) - offset);
results.Add((-distance * (cos - (Sqrt3 * sin))) - offset);
}
else
{
var tmp = halfR >= 0d ? -Cbrt(halfR) : Cbrt(-halfR);
results.Add((2d * tmp) - offset);
// really should return next root twice, but we return only one
results.Add(-tmp - offset);
}
return(results.ToList());
}
public static Intersection QuadraticBezierSegmentCubicBezierSegmentIntersection2(
double a1X, double a1Y, double a2X, double a2Y, double a3X, double a3Y,
double b1X, double b1Y, double b2X, double b2Y, double b3X, double b3Y, double b4X, double b4Y,
double epsilon = Epsilon)
{
var result = new Intersection(IntersectionStates.NoIntersection);
// ToDo: Break early if the AABB bounding box of the curve does not intersect.
var c12 = new Vector2D(a1X - (a2X * 2) + a3X, a1Y - (a2Y * 2) + a3Y);
var c11 = new Vector2D(2 * (a2X - a1X), 2 * (a2Y - a1Y));
var c23 = new Vector2D(b4X - (b3X * 3) + (b2X * 3) - (b1X * 1), b4Y - (b3Y * 3) + (b2Y * 3) - (b1Y * 1));
var c22 = new Vector2D(3 * (b3X - (b2X * 2) + b1X), 3 * (b3Y - (b2Y * 2) + b1Y));
var c21 = new Vector2D(3 * (b2X - b1X), 3 * (b2Y - b1Y));
var c10x2 = a1X * a1X;
var c10y2 = a1Y * a1Y;
var c11x2 = c11.I * c11.I;
var c11y2 = c11.J * c11.J;
var c12x2 = c12.I * c12.I;
var c12y2 = c12.J * c12.J;
var c20x2 = b1X * b1X;
var c20y2 = b1Y * b1Y;
var c21x2 = c21.I * c21.I;
var c21y2 = c21.J * c21.J;
var c22x2 = c22.I * c22.I;
var c22y2 = c22.J * c22.J;
var c23x2 = c23.I * c23.I;
var c23y2 = c23.J * c23.J;
var roots = new Polynomial(
/* t^0 */ (-2 * c12.I * c12.J * c22.I * c23.J) - (2 * c12.I * c12.J * c22.J * c23.I) + (2 * c12y2 * c22.I * c23.I) + (2 * c12x2 * c22.J * c23.J),
/* t^1 */ (-2 * c12.I * c12.J * c23.I * c23.J) + (c12x2 * c23y2) + (c12y2 * c23x2),
/* t^2 */ (-2 * c12.I * c21.I * c12.J * c23.J) - (2 * c12.I * c12.J * c21.J * c23.I) - (2 * c12.I * c12.J * c22.I * c22.J) + (2 * c21.I * c12y2 * c23.I) + (c12y2 * c22x2) + (c12x2 * ((2 * c21.J * c23.J) + c22y2)),
/* t^3 */ (2 * a1X * c12.I * c12.J * c23.J) + (2 * a1Y * c12.I * c12.J * c23.I) + (c11.I * c11.J * c12.I * c23.J) + (c11.I * c11.J * c12.J * c23.I) - (2 * b1X * c12.I * c12.J * c23.J) - (2 * c12.I * b1Y * c12.J * c23.I) - (2 * c12.I * c21.I * c12.J * c22.J) - (2 * c12.I * c12.J * c21.J * c22.I) - (2 * a1X * c12y2 * c23.I) - (2 * a1Y * c12x2 * c23.J) + (2 * b1X * c12y2 * c23.I) + (2 * c21.I * c12y2 * c22.I) - (c11y2 * c12.I * c23.I) - (c11x2 * c12.J * c23.J) + (c12x2 * ((2 * b1Y * c23.J) + (2 * c21.J * c22.J))),
/* t^4 */ (2 * a1X * c12.I * c12.J * c22.J) + (2 * a1Y * c12.I * c12.J * c22.I) + (c11.I * c11.J * c12.I * c22.J) + (c11.I * c11.J * c12.J * c22.I) - (2 * b1X * c12.I * c12.J * c22.J) - (2 * c12.I * b1Y * c12.J * c22.I) - (2 * c12.I * c21.I * c12.J * c21.J) - (2 * a1X * c12y2 * c22.I) - (2 * a1Y * c12x2 * c22.J) + (2 * b1X * c12y2 * c22.I) - (c11y2 * c12.I * c22.I) - (c11x2 * c12.J * c22.J) + (c21x2 * c12y2) + (c12x2 * ((2 * b1Y * c22.J) + c21y2)),
/* t^5 */ (2 * a1X * c12.I * c12.J * c21.J) + (2 * a1Y * c12.I * c21.I * c12.J) + (c11.I * c11.J * c12.I * c21.J) + (c11.I * c11.J * c21.I * c12.J) - (2 * b1X * c12.I * c12.J * c21.J) - (2 * c12.I * b1Y * c21.I * c12.J) - (2 * a1X * c21.I * c12y2) - (2 * a1Y * c12x2 * c21.J) + (2 * b1X * c21.I * c12y2) - (c11y2 * c12.I * c21.I) - (c11x2 * c12.J * c21.J) + (2 * c12x2 * b1Y * c21.J),
/* t^6 */ (-2 * a1X * a1Y * c12.I * c12.J) - (a1X * c11.I * c11.J * c12.J) - (a1Y * c11.I * c11.J * c12.I) + (2 * a1X * c12.I * b1Y * c12.J) + (2 * a1Y * b1X * c12.I * c12.J) + (c11.I * b1X * c11.J * c12.J) + (c11.I * c11.J * c12.I * b1Y) - (2 * b1X * c12.I * b1Y * c12.J) - (2 * a1X * b1X * c12y2) + (a1X * c11y2 * c12.I) + (a1Y * c11x2 * c12.J) - (2 * a1Y * c12x2 * b1Y) - (b1X * c11y2 * c12.I) - (c11x2 * b1Y * c12.J) + (c10x2 * c12y2) + (c10y2 * c12x2) + (c20x2 * c12y2) + (c12x2 * c20y2)
).RootsInInterval(0, 1);
foreach (var s in roots)
{
var point = new Point2D((c23.I * s * s * s) + (c22.I * s * s) + (c21.I * s) + b1X, (c23.J * s * s * s) + (c22.J * s * s) + (c21.J * s) + b1Y);
var xRoots = QuadraticRootsTests.QuadraticRoots(
/* c */ c12.I,
/* t^1 */ c11.I,
/* t^2 */ a1X - point.X,
epsilon);
var yRoots = QuadraticRootsTests.QuadraticRoots(
/* c */ c12.J,
/* t^1 */ c11.J,
/* t^2 */ a1Y - point.Y,
epsilon);
if (xRoots.Count > 0 && yRoots.Count > 0)
{
foreach (var xRoot in xRoots)
{
if (0 <= xRoot && xRoot <= 1)
{
foreach (var yRoot in yRoots)
{
var t = xRoot - yRoot;
if ((t >= 0 ? t : -t) < epsilon)
{
result.Items.Add(point);
goto checkRoots;
}
}
}
}
checkRoots :;
}
}
if (result.Items.Count > 0)
{
result.State = IntersectionStates.Intersection;
}
return(result);
}
public static Intersection QuadraticBezierSegmentCubicBezierSegmentIntersection1(
double a1X, double a1Y, double a2X, double a2Y, double a3X, double a3Y,
double b1X, double b1Y, double b2X, double b2Y, double b3X, double b3Y, double b4X, double b4Y,
double epsilon = Epsilon)
{
var a = new Vector2D(a2X, a2Y) * -2;
var c12 = new Vector2D(a1X, a1Y) + a + new Vector2D(a3X, a3Y);
a = new Vector2D(a1X, a1Y) * -2;
var b = new Vector2D(a2X, a2Y) * 2;
var c11 = a + b;
var c10 = new Vector2D(a1X, a1Y);
a = new Vector2D(b1X, b1Y) * -1;
b = new Vector2D(b2X, b2Y) * 3;
var c = new Vector2D(b3X, b3Y) * -3;
var d = a + b + c + new Vector2D(b4X, b4Y);
var c23 = new Vector2D(d.I, d.J);
a = new Vector2D(b1X, b1Y) * 3;
b = new Vector2D(b2X, b2Y) * -6;
c = new Vector2D(b3X, b3Y) * 3;
d = a + b + c;
var c22 = new Vector2D(d.I, d.J);
a = new Vector2D(b1X, b1Y) * -3;
b = new Vector2D(b2X, b2Y) * 3;
c = a + b;
var c21 = new Vector2D(c.I, c.J);
var c20 = new Vector2D(b1X, b1Y);
var c10x2 = c10.I * c10.I;
var c10y2 = c10.J * c10.J;
var c11x2 = c11.I * c11.I;
var c11y2 = c11.J * c11.J;
var c12x2 = c12.I * c12.I;
var c12y2 = c12.J * c12.J;
var c20x2 = c20.I * c20.I;
var c20y2 = c20.J * c20.J;
var c21x2 = c21.I * c21.I;
var c21y2 = c21.J * c21.J;
var c22x2 = c22.I * c22.I;
var c22y2 = c22.J * c22.J;
var c23x2 = c23.I * c23.I;
var c23y2 = c23.J * c23.J;
var roots = new Polynomial(
(-2 * c12.I * c12.J * c22.I * c23.J) - (2 * c12.I * c12.J * c22.J * c23.I) + (2 * c12y2 * c22.I * c23.I) + (2 * c12x2 * c22.J * c23.J),
(-2 * c12.I * c12.J * c23.I * c23.J) + (c12x2 * c23y2) + (c12y2 * c23x2),
(-2 * c12.I * c21.I * c12.J * c23.J) - (2 * c12.I * c12.J * c21.J * c23.I) - (2 * c12.I * c12.J * c22.I * c22.J) + (2 * c21.I * c12y2 * c23.I) + (c12y2 * c22x2) + (c12x2 * ((2 * c21.J * c23.J) + c22y2)),
(2 * c10.I * c12.I * c12.J * c23.J) + (2 * c10.J * c12.I * c12.J * c23.I) + (c11.I * c11.J * c12.I * c23.J) + (c11.I * c11.J * c12.J * c23.I) - (2 * c20.I * c12.I * c12.J * c23.J) - (2 * c12.I * c20.J * c12.J * c23.I) - (2 * c12.I * c21.I * c12.J * c22.J) - (2 * c12.I * c12.J * c21.J * c22.I) - (2 * c10.I * c12y2 * c23.I) - (2 * c10.J * c12x2 * c23.J) + (2 * c20.I * c12y2 * c23.I) + (2 * c21.I * c12y2 * c22.I) - (c11y2 * c12.I * c23.I) - (c11x2 * c12.J * c23.J) + (c12x2 * ((2 * c20.J * c23.J) + (2 * c21.J * c22.J))),
(2 * c10.I * c12.I * c12.J * c22.J) + (2 * c10.J * c12.I * c12.J * c22.I) + (c11.I * c11.J * c12.I * c22.J) + (c11.I * c11.J * c12.J * c22.I) - (2 * c20.I * c12.I * c12.J * c22.J) - (2 * c12.I * c20.J * c12.J * c22.I) - (2 * c12.I * c21.I * c12.J * c21.J) - (2 * c10.I * c12y2 * c22.I) - (2 * c10.J * c12x2 * c22.J) + (2 * c20.I * c12y2 * c22.I) - (c11y2 * c12.I * c22.I) - (c11x2 * c12.J * c22.J) + (c21x2 * c12y2) + (c12x2 * ((2 * c20.J * c22.J) + c21y2)),
(2 * c10.I * c12.I * c12.J * c21.J) + (2 * c10.J * c12.I * c21.I * c12.J) + (c11.I * c11.J * c12.I * c21.J) + (c11.I * c11.J * c21.I * c12.J) - (2 * c20.I * c12.I * c12.J * c21.J) - (2 * c12.I * c20.J * c21.I * c12.J) - (2 * c10.I * c21.I * c12y2) - (2 * c10.J * c12x2 * c21.J) + (2 * c20.I * c21.I * c12y2) - (c11y2 * c12.I * c21.I) - (c11x2 * c12.J * c21.J) + (2 * c12x2 * c20.J * c21.J),
(-2 * c10.I * c10.J * c12.I * c12.J) - (c10.I * c11.I * c11.J * c12.J) - (c10.J * c11.I * c11.J * c12.I) + (2 * c10.I * c12.I * c20.J * c12.J) + (2 * c10.J * c20.I * c12.I * c12.J) + (c11.I * c20.I * c11.J * c12.J) + (c11.I * c11.J * c12.I * c20.J) - (2 * c20.I * c12.I * c20.J * c12.J) - (2 * c10.I * c20.I * c12y2) + (c10.I * c11y2 * c12.I) + (c10.J * c11x2 * c12.J) - (2 * c10.J * c12x2 * c20.J) - (c20.I * c11y2 * c12.I) - (c11x2 * c20.J * c12.J) + (c10x2 * c12y2) + (c10y2 * c12x2) + (c20x2 * c12y2) + (c12x2 * c20y2)
).RootsInInterval();
var result = new Intersection(IntersectionStates.NoIntersection);
for (var i = 0; i < roots.Length; i++)
{
var s = roots[i];
var xRoots = QuadraticRootsTests.QuadraticRoots(
c12.I,
c11.I,
c10.I - c20.I - (s * c21.I) - (s * s * c22.I) - (s * s * s * c23.I),
epsilon);
var yRoots = QuadraticRootsTests.QuadraticRoots(
c12.J,
c11.J,
c10.J - c20.J - (s * c21.J) - (s * s * c22.J) - (s * s * s * c23.J),
epsilon);
if (xRoots.Count > 0 && yRoots.Count > 0)
{
for (var j = 0; j < xRoots.Count; j++)
{
var xRoot = xRoots[j];
if (0 <= xRoot && xRoot <= 1)
{
for (var k = 0; k < yRoots.Count; k++)
{
if (Abs(xRoot - yRoots[k]) < epsilon)
{
result.Items.Add(((Point2D)c23 * s * s * s) + ((c22 * s * s) + ((c21 * s) + c20)));
goto checkRoots;
}
}
}
}
checkRoots :;
}
}
if (result.Items.Count > 0)
{
result.State = IntersectionStates.Intersection;
}
return(result);
}
public static Intersection QuadraticBezierSegmentQuadraticBezierSegmentIntersection00(
double a1X, double a1Y, double a2X, double a2Y, double a3X, double a3Y,
double b1X, double b1Y, double b2X, double b2Y, double b3X, double b3Y,
double epsilon = Epsilon)
{
// Initialize the intersection.
var result = new Intersection(IntersectionStates.NoIntersection);
// ToDo: Break early if the AABB of the ends and handles do not intersect.
// ToDo: Break early if the AABB of the curve does not intersect.
// Parametric matrix form of the Bézier curve
var xCoeffA = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(a1X, a2X, a3X);
var yCoeffA = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(a1Y, a2Y, a3Y);
var xCoeffB = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(b1X, b2X, b3X);
var yCoeffB = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(b1Y, b2Y, b3Y);
IList <double> roots = new List <double>();
if (yCoeffA.C == 0)
{
var v0 = xCoeffA.C * (yCoeffA.A - yCoeffB.A);
var v1 = v0 - (xCoeffA.B * yCoeffA.B);
var v2 = v0 + v1;
var v3 = yCoeffA.B * yCoeffA.B;
roots = QuarticRootsTests.QuarticRoots(
/* t^4 */ xCoeffA.C * yCoeffB.C * yCoeffB.C,
/* t^3 */ 2 * xCoeffA.C * yCoeffB.B * yCoeffB.C,
/* t^2 */ (xCoeffA.C * yCoeffB.B * yCoeffB.B) - (xCoeffB.C * v3) - (yCoeffB.C * v0) - (yCoeffB.C * v1),
/* t^1 */ (-xCoeffB.B * v3) - (yCoeffB.B * v0) - (yCoeffB.B * v1),
/* C^0 */ ((xCoeffA.A - xCoeffB.A) * v3) + ((yCoeffA.A - yCoeffB.A) * v1),
epsilon);
}
else
{
var v0 = (xCoeffA.C * yCoeffB.C) - (yCoeffA.C * xCoeffB.C);
var v1 = (xCoeffA.C * yCoeffB.B) - (xCoeffB.B * yCoeffA.C);
var v2 = (xCoeffA.B * yCoeffA.C) - (yCoeffA.B * xCoeffA.C);
var v3 = yCoeffA.A - yCoeffB.A;
var v4 = (yCoeffA.C * (xCoeffA.A - xCoeffB.A)) - (xCoeffA.C * v3);
var v5 = (-yCoeffA.B * v2) + (yCoeffA.C * v4);
var v6 = v2 * v2;
roots = QuarticRootsTests.QuarticRoots(
/* t^4 */ v0 * v0,
/* t^3 */ 2 * v0 * v1,
/* t^2 */ ((-yCoeffB.C * v6) + (yCoeffA.C * v1 * v1) + (yCoeffA.C * v0 * v4) + (v0 * v5)) / yCoeffA.C,
/* t^1 */ ((-yCoeffB.B * v6) + (yCoeffA.C * v1 * v4) + (v1 * v5)) / yCoeffA.C,
/* C^0 */ ((v3 * v6) + (v4 * v5)) / yCoeffA.C,
epsilon);
}
foreach (var s in roots)
{
var point = new Point2D(
(xCoeffB.C * s * s) + (xCoeffB.B * s) + xCoeffB.A,
(yCoeffB.C * s * s) + (yCoeffB.B * s) + yCoeffB.A);
if (s >= 0 && s <= 1)
{
var xRoots = QuadraticRootsTests.QuadraticRoots(
/* t^2 */ -xCoeffA.C,
/* t^1 */ -xCoeffA.B,
/* C^0 */ -xCoeffA.A + point.X,
epsilon);
var yRoots = QuadraticRootsTests.QuadraticRoots(
/* t^2 */ -yCoeffA.C,
/* t^1 */ -yCoeffA.B,
/* C^0 */ -yCoeffA.A + point.Y,
epsilon);
if (xRoots.Count > 0 && yRoots.Count > 0)
{
// Find the nearest matching x and y roots in the ranges 0 < x < 1; 0 < y < 1.
foreach (var xRoot in xRoots)
{
if (xRoot >= 0 && xRoot <= 1)
{
foreach (var yRoot in yRoots)
{
var t = xRoot - yRoot;
if ((t >= 0 ? t : -t) < epsilon)
{
result.AppendPoint(point);
goto checkRoots; // Break through two levels of foreach loops to exit early. Using goto for performance.
}
}
}
}
checkRoots :;
}
}
}
if (result.Items.Count > 0)
{
result.State = IntersectionStates.Intersection;
}
return(result);
}
public static Intersection QuadraticBezierSegmentQuadraticBezierSegmentIntersection3(
double a1X, double a1Y, double a2X, double a2Y, double a3X, double a3Y,
double b1X, double b1Y, double b2X, double b2Y, double b3X, double b3Y,
double epsilon = Epsilon)
{
var result = new Intersection(IntersectionStates.NoIntersection);
// ToDo: Break early if the AABB bounding box of the curve does not intersect.
var c12 = new Vector2D(a1X - (a2X * 2) + a3X, a1Y - (a2Y * 2) + a3Y);
var c11 = new Vector2D(2 * (a2X - a1X), 2 * (a2Y - a1Y));
var c22 = new Vector2D(b1X - (b2X * 2) + b3X, b1Y - (b2Y * 2) + b3Y);
var c21 = new Vector2D(2 * (b2X - b1X), 2 * (b2Y - b1Y));
var a = (c12.I * c11.J) - (c11.I * c12.J);
var b = (c22.I * c11.J) - (c11.I * c22.J);
var c = (c21.I * c11.J) - (c11.I * c21.J);
var d = (c11.I * (a1Y - b1Y)) + (c11.J * (b1X - a1X));
var e = (c22.I * c12.J) - (c12.I * c22.J);
var f = (c21.I * c12.J) - (c12.I * c21.J);
var g = (c12.I * (a1Y - b1Y)) + (c12.J * (b1X - a1X));
var roots = QuarticRootsTests.QuarticRoots(
/* C */ -e * e,
/* t^1 */ -2 * e * f,
/* t^2 */ (a * b) - (f * f) - (2 * e * g),
/* t^3 */ (a * c) - (2 * f * g),
/* t^4 */ (a * d) - (g * g),
epsilon);
foreach (var s in roots)
{
var point = new Point2D((c22.I * s * s) + (c21.I * s) + b1X, (c22.J * s * s) + (c21.J * s) + b1Y);
if (0 <= s && s <= 1)
{
var xRoots = QuadraticRootsTests.QuadraticRoots(
/* C */ -c12.I,
/* t^1 */ -c11.I,
/* t^2 */ -a1X + point.X,
epsilon);
var yRoots = QuadraticRootsTests.QuadraticRoots(
/* C */ -c12.J,
/* t^1 */ -c11.J,
/* t^2 */ -a1Y + point.Y,
epsilon);
if (xRoots.Count > 0 && yRoots.Count > 0)
{
// Find the nearest matching x and y roots in the ranges 0 < x < 1; 0 < y < 1.
foreach (var xRoot in xRoots)
{
if (xRoot >= 0 && xRoot <= 1)
{
foreach (var yRoot in yRoots)
{
var t = xRoot - yRoot;
if ((t >= 0 ? t : -t) < epsilon)
{
result.Items.Add(point);
goto checkRoots; // Break through two levels of foreach loops. Using goto for performance.
}
}
}
}
checkRoots :;
}
}
}
if (result.Items.Count > 0)
{
result.State = IntersectionStates.Intersection;
}
return(result);
}
public static Intersection QuadraticBezierSegmentQuadraticBezierSegmentIntersection2(
double a1X, double a1Y, double a2X, double a2Y, double a3X, double a3Y,
double b1X, double b1Y, double b2X, double b2Y, double b3X, double b3Y,
double epsilon = Epsilon)
{
var va = new Vector2D(a2X, a2Y) * -2;
var c12 = new Vector2D(a1X, a1Y) + va + new Vector2D(a3X, a3Y);
va = new Vector2D(a1X, a1Y) * -2;
var vb = new Vector2D(a2X, a2Y) * 2;
var c11 = va + vb;
var c10 = new Vector2D(a1X, a1Y);
va = new Vector2D(b2X, b2Y) * -2;
var c22 = new Vector2D(b1X, b1Y) + va + new Vector2D(b3X, b3Y);
va = new Vector2D(b1X, b1Y) * -2;
vb = new Vector2D(b2X, b2Y) * 2;
var c21 = va + vb;
var c20 = new Vector2D(b1X, b1Y);
var a = (c12.I * c11.J) - (c11.I * c12.J);
var b = (c22.I * c11.J) - (c11.I * c22.J);
var c = (c21.I * c11.J) - (c11.I * c21.J);
var d = (c11.I * (c10.J - c20.J)) + (c11.J * (-c10.I + c20.I));
var e = (c22.I * c12.J) - (c12.I * c22.J);
var f = (c21.I * c12.J) - (c12.I * c21.J);
var g = (c12.I * (c10.J - c20.J)) + (c12.J * (-c10.I + c20.I));
var roots = QuarticRootsTests.QuarticRoots(
-e * e,
-2 * e * f,
(a * b) - (f * f) - (2 * e * g),
(a * c) - (2 * f * g),
(a * d) - (g * g),
epsilon);
var result = new Intersection(IntersectionStates.NoIntersection);
for (var i = 0; i < roots.Count; i++)
{
var s = roots[i];
if (0 <= s && s <= 1)
{
var xRoots = QuadraticRootsTests.QuadraticRoots(
-c12.I,
-c11.I,
-c10.I + c20.I + (s * c21.I) + (s * s * c22.I),
epsilon);
var yRoots = QuadraticRootsTests.QuadraticRoots(
-c12.J,
-c11.J,
-c10.J + c20.J + (s * c21.J) + (s * s * c22.J),
epsilon);
if (xRoots.Count > 0 && yRoots.Count > 0)
{
for (var j = 0; j < xRoots.Count; j++)
{
var xRoot = xRoots[j];
if (0 <= xRoot && xRoot <= 1)
{
for (var k = 0; k < yRoots.Count; k++)
{
if (Abs(xRoot - yRoots[k]) < epsilon)
{
result.Items.Add(((Point2D)c22 * s * s) + ((c21 * s) + c20));
goto checkRoots;
}
}
}
}
checkRoots :;
}
}
}
if (result.Items.Count > 0)
{
result.State = IntersectionStates.Intersection;
}
return(result);
}
public static Intersection QuadraticBezierSegmentQuadraticBezierSegmentIntersection1(
double a1X, double a1Y, double a2X, double a2Y, double a3X, double a3Y,
double b1X, double b1Y, double b2X, double b2Y, double b3X, double b3Y,
double epsilon = Epsilon)
{
var result = new Intersection(IntersectionStates.NoIntersection);
// ToDo: Break early if the AABB bounding box of the curve does not intersect.
// ToDo: Figure out if the following can be broken out of the vector structs.
var c12 = new Vector2D(a1X - (a2X * 2) + a3X, a1Y - (a2Y * 2) + a3Y);
var c11 = new Vector2D(2 * (a2X - a1X), 2 * (a2Y - a1Y));
// c10 is a1X and a1Y
var c22 = new Vector2D(b1X - (b2X * 2) + b3X, b1Y - (b2Y * 2) + b3Y);
var c21 = new Vector2D(2 * (b2X - b1X), 2 * (b2Y - b1Y));
// c20 is b1X and b1Y
var a = (c12.I * c11.J) - (c11.I * c12.J);
var b = (c22.I * c11.J) - (c11.I * c22.J);
var c = (c21.I * c11.J) - (c11.I * c21.J);
var d = (c11.I * (a1Y - b1Y)) - (c11.J * (b1X - a1X));
var e = (-c22.I * c12.J) - (c12.I * c22.J);
var f = (c21.I * c12.J) - (c12.I * c21.J);
var g = (c12.I * (a1Y - b1Y)) - (c12.J * (b1X - a1X));
IList <double> roots;
if ((a * d) - (g * g) == 0)
{
var v0 = (a * c) - (2 * f * g);
var v1 = (a * b) - (f * f) - (2 * e * g);
var v2 = -2 * e * f;
var v3 = -e * e;
roots = CubicRootsTests.CubicRoots(
/* t^3 */ -v3,
/* t^2 */ -v2,
/* t^1 */ -v1,
/* C */ -v0,
epsilon);
}
else
{
var v0 = (a * d) - (g * g);
var v1 = (a * c) - (2 * f * g);
var v2 = (a * b) - (f * f) - (2 * e * g);
var v3 = -2 * e * f;
var v4 = -e * e;
roots = QuarticRootsTests.QuarticRoots(
/* t^4 */ -v4,
/* t^3 */ -v3,
/* t^2 */ -v2,
/* t^1 */ -v1,
/* C */ -v0,
epsilon);
}
//roots.Reverse();
foreach (var s in roots)
{
var point = new Point2D(
(c22.I * s * s) + (c21.I * s) + b1X,
(c22.J * s * s) + (c21.J * s) + b1Y);
if (s >= 0 && s <= 1)
{
var v0 = a1X - point.X;
var v1 = c11.I;
var v2 = c12.I;
var xRoots = QuadraticRootsTests.QuadraticRoots(
/* t^2 */ -v2,
/* t^1 */ -v1,
/* C */ -v0,
epsilon);
v0 = a1Y - point.Y;
v1 = c11.J;
v2 = c12.J;
var yRoots = QuadraticRootsTests.QuadraticRoots(
/* t^2 */ -v2,
/* t^1 */ -v1,
/* C */ -v0,
epsilon);
if (xRoots.Count > 0 && yRoots.Count > 0)
{
// Find the nearest matching x and y roots in the ranges 0 < x < 1; 0 < y < 1.
foreach (var xRoot in xRoots)
{
if (xRoot >= 0 && xRoot <= 1)
{
foreach (var yRoot in yRoots)
{
var t = xRoot - yRoot;
if ((t >= 0 ? t : -t) < 0.1)
{
result.AppendPoint(point);
goto checkRoots; // Break through two levels of foreach loops. Using goto for performance.
}
}
}
}
checkRoots :;
}
}
}
if (result.Items.Count > 0)
{
result.State = IntersectionStates.Intersection;
}
return(result);
}
public static Intersection QuadraticBezierSegmentQuadraticBezierSegmentIntersection0(
double a1X, double a1Y, double a2X, double a2Y, double a3X, double a3Y,
double b1X, double b1Y, double b2X, double b2Y, double b3X, double b3Y,
double epsilon = Epsilon)
{
// Initialize the intersection.
var result = new Intersection(IntersectionStates.NoIntersection);
var xCoeffA = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(a1X, a2X, a3X);
var yCoeffA = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(a1Y, a2Y, a3Y);
var xCoeffB = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(b1X, b2X, b3X);
var yCoeffB = QuadraticBezierBernsteinBasisTests.QuadraticBezierBernsteinBasis(b1Y, b2Y, b3Y);
var a = (xCoeffA.C * yCoeffA.B) - (xCoeffA.B * yCoeffA.C);
var b = (xCoeffB.C * yCoeffA.B) - (xCoeffA.B * yCoeffB.C);
var c = (xCoeffB.B * yCoeffA.B) - (xCoeffA.B * yCoeffB.B);
var d = (xCoeffA.B * (yCoeffA.A - yCoeffB.A)) - (yCoeffA.B * (xCoeffB.A - xCoeffA.A));
var e = (xCoeffB.C * yCoeffA.C) - (xCoeffA.C * yCoeffB.C);
var f = (xCoeffB.B * yCoeffA.C) - (xCoeffA.C * yCoeffB.B);
var g = (xCoeffA.C * (yCoeffA.A - yCoeffB.A)) - (yCoeffA.C * (yCoeffB.A - xCoeffA.A));
var roots = QuarticRootsTests.QuarticRoots(
/* t^4 */ e * e,
/* t^3 */ 2 * e * f,
/* t^2 */ (-a * b) + (f * f) + (2 * e * g),
/* t^1 */ (-a * c) + (2 * f * g),
/* C */ (-a * d) + (g * g),
epsilon);
foreach (var s in roots)
{
var point = new Point2D(
(xCoeffB.C * s * s) + (xCoeffB.B * s) + xCoeffB.A,
(yCoeffB.C * s * s) + (yCoeffB.B * s) + yCoeffB.A);
if (s >= 0 && s <= 1)
{
var xRoots = QuadraticRootsTests.QuadraticRoots(
/* t^2 */ -xCoeffA.C,
/* t^1 */ -xCoeffA.B,
/* C */ -xCoeffA.A + point.X,
epsilon);
var yRoots = QuadraticRootsTests.QuadraticRoots(
/* t^2 */ -yCoeffA.C,
/* t^1 */ -yCoeffA.B,
/* C */ -yCoeffA.A + point.Y,
epsilon);
if (xRoots.Count > 0 && yRoots.Count > 0)
{
// Find the nearest matching x and y roots in the ranges 0 < x < 1; 0 < y < 1.
foreach (var xRoot in xRoots)
{
if (xRoot >= 0 && xRoot <= 1)
{
foreach (var yRoot in yRoots)
{
var t = xRoot - yRoot;
if ((t >= 0 ? t : -t) < 0.06) // ToDo: Find the error and replace 0.06 with epsilon.
{
result.AppendPoint(point);
goto checkRoots; // Break through two levels of foreach loops. Using goto for performance.
}
}
}
}
checkRoots :;
}
}
}
if (result.Items.Count > 0)
{
result.State = IntersectionStates.Intersection;
}
return(result);
}
