private void vertex_PositionChanged(IShape sender, bool followUpMove)
{
if (!followUpMove)
{
if (closingPolygon)
{
closingPolygon = false;
checkBoundingBox();
base.OnChanged(false);
return;
}
if (tolerance.Equal(sender.RootX, points[0].RootX) &&
tolerance.Equal(sender.RootY, points[0].RootY))
{ // the first point was changed
closingPolygon = true;
points[points.Count - 1].moveTo(sender.RootX, sender.RootY, false, false);
}
else if (tolerance.Equal(sender.RootX, points[points.Count - 1].RootX) &&
tolerance.Equal(sender.RootY, points[points.Count - 1].RootY)
)
{ // the last point was changed
closingPolygon = true;
points[0].moveTo(sender.RootX, sender.RootY, false, false);
}
else
{
checkBoundingBox();
base.OnChanged(false);
}
}
//else
// closingPolygon = false;
}
/// <summary>
/// Returns a stereographically sphere representing us.
/// </summary>
public Sphere ToSphere()
{
// Equatorial sphere?
if (Pole == new Vector3D(0, 0, 0, 1))
{
return(new Sphere());
}
// A plane?
Vector3D poleR3 = Sterographic.S3toR3(Pole);
if (Tolerance.Equal(poleR3.Abs(), 1))
{
return(Sphere.Plane(poleR3));
}
// Get 4 points on the sphere.
Vector3D e1 = new Vector3D(1, 0, 0, 0);
Vector3D e2 = new Vector3D(0, 1, 0, 0);
Vector3D e3 = new Vector3D(0, 0, 1, 0);
Vector3D e4 = new Vector3D(0, 0, 0, 1);
System.Func <Vector3D, Vector3D> one = v =>
{
v = Euclidean3D.ProjectOntoPlane(Pole, new Vector3D(), v);
v.Normalize();
return(Sterographic.S3toR3(v));
};
return(Sphere.From4Points(one(e1), one(e2), one(e3), one(e4)));
}
/// <summary>
/// Comparator used for ordering point first by ascending X, then by ascending Y.
/// </summary>
/// <param name="other">The <see cref="Point"/> to compare.</param>
/// <returns>
/// 0 if the points are spatially equal or both empty; 1 if <paramref name="other"/> is empty or
/// if this point has a greater <see cref="X"/> value or equal X values and a greater <see cref="Y"/> value;
/// -1 if this point is empty or if <paramref name="other"/> has a greater <see cref="X"/> value or equal X values
/// and a greater <see cref="Y"/> value.
/// </returns>
/// <exception cref="ArgumentNullException">Thrown if <paramref name="other"/> is null.</exception>
public virtual Int32 CompareTo(Point other)
{
if (other == null)
{
throw new ArgumentNullException("other");
}
if (Equals(other)) // This handles the case where both are empty.
{
return(0);
}
else if (IsEmpty)
{
return(-1);
}
else if (other.IsEmpty)
{
return(1);
}
else if (Tolerance.Less(X, other.X) || Tolerance.Equal(X, other.X) && Tolerance.Less(Y, other.Y))
{
return(-1);
}
else if (Tolerance.Greater(X, other.X) || Tolerance.Equal(X, other.X) && Tolerance.Greater(Y, other.Y))
{
return(1);
}
throw new InvalidOperationException("Points cannot be compared.");
}
/// <summary>
/// Calculates the point of our simplex that is at a vertex.
/// </summary>
public static Vector3D VertexPointBall(int p, int q, int r)
{
Geometry vertexGeometry = Geometry2D.GetGeometry(q, r);
if (vertexGeometry == Geometry.Hyperbolic)
{
// Outside the ball, and not in a good way. Use the Klein version.
//throw new System.NotImplementedException();
return(new Vector3D(0, 0, -1));
}
// Get in UHS first.
Sphere cellFacet = Mirrors(p, q, r, moveToBall: false).First();
double rSquared = Math.Pow(cellFacet.Radius, 2);
double cSquared = Math.Pow(cellFacet.Center.Abs(), 2);
Vector3D uhs;
if (Tolerance.Equal(rSquared, cSquared)) // e.g. 363
{
uhs = new Vector3D();
}
else
{
double height = Math.Sqrt(rSquared - cSquared);
uhs = new Vector3D(0, 0, height);
}
return(H3Models.UHSToBall(uhs));
}
// XXX - Do we only want to compare the surface, or also the orientation?
// This just does the surface.
public override bool Equals(object obj)
{
Sphere s = (Sphere)obj;
if (IsPlane)
{
if (!s.IsPlane)
{
return(false);
}
Vector3D n1 = this.Normal;
n1.Normalize();
Vector3D n2 = s.Normal;
n2.Normalize();
return
((n1 == n2 || n1 == -n2) &&
Offset == s.Offset);
}
if (s.IsPlane)
{
return(false);
}
return
(Tolerance.Equal(Radius, s.Radius) &&
Center == s.Center /*&&
* Offset == s.Offset &&
* Invert == s.Invert*/);
}
private static H3.Cell.Edge CalcFiber(Vector3D boundaryPointBall)
{
bool parallel = false;
if (parallel)
{
Vector3D v2 = new Vector3D(0, 0, -1); // Really we can pick any boundary point.
return(new H3.Cell.Edge(boundaryPointBall, v2));
}
else
{
// If the point is above the real line, it will have been connected to z.
// If below the real line, it will have been connected to -i
// If on the real line, it's degenerate.
Vector3D pUHS = H3Models.BallToUHS(boundaryPointBall);
if (Tolerance.Equal(pUHS.Y, 0))
{
return(null);
}
Vector3D z = Vector3D.FromComplex(ZParam);
Vector3D minusi = new Vector3D(0, -1);
double dilation = double.NaN;
double translation = double.NaN;
if (pUHS.Y > 0)
{
dilation = pUHS.Y / z.Y;
z *= dilation;
translation = pUHS.X - z.X;
z.X += translation;
if (H3Models.UHSToBall(z) != boundaryPointBall)
{
throw new System.Exception();
}
minusi *= dilation;
minusi.X += translation;
return(new H3.Cell.Edge(H3Models.UHSToBall(minusi), boundaryPointBall));
//return null;
}
else
{
dilation = pUHS.Y / -1;
minusi *= dilation;
translation = pUHS.X - minusi.X;
minusi.X += translation;
if (H3Models.UHSToBall(minusi) != boundaryPointBall)
{
throw new System.Exception();
}
z *= dilation;
z.X += translation;
return(new H3.Cell.Edge(boundaryPointBall, H3Models.UHSToBall(z)));
//return null;
}
}
}
/// <summary>
/// Construct a circle from 3 points
/// </summary>
/// <returns>false if the construction failed (if we are a line).</returns>
public bool From3Points(Vector3D p1, Vector3D p2, Vector3D p3)
{
Reset();
// Check for any infinite points, in which case we are a line.
// I'm not sure these checks are smart, since our IsInfinite check is so inclusive,
// but Big Chop puzzle doesn't work if we don't do this.
// ZZZ - Still, I need to think on this more.
if (Infinity.IsInfinite(p1))
{
this.From2Points(p2, p3);
return(false);
}
else if (Infinity.IsInfinite(p2))
{
this.From2Points(p1, p3);
return(false);
}
else if (Infinity.IsInfinite(p3))
{
this.From2Points(p1, p2);
return(false);
}
/* Some links
* http://mathforum.org/library/drmath/view/54323.html
* http://delphiforfun.org/Programs/Math_Topics/circle_from_3_points.htm
* There is lots of info out there about solving via equations,
* but as with other code in this project, I wanted to use geometrical constructions. */
// Midpoints.
Vector3D m1 = (p1 + p2) / 2;
Vector3D m2 = (p1 + p3) / 2;
// Perpendicular bisectors.
Vector3D b1 = (p2 - p1) / 2;
Vector3D b2 = (p3 - p1) / 2;
b1.Normalize();
b2.Normalize();
b1.Rotate90();
b2.Rotate90();
Vector3D newCenter;
int found = Euclidean2D.IntersectionLineLine(m1, m1 + b1, m2, m2 + b2, out newCenter);
Center = newCenter;
if (0 == found)
{
// The points are collinear, so we are a line.
From2Points(p1, p2);
return(false);
}
Radius = (p1 - Center).Abs();
Debug.Assert(Tolerance.Equal(Radius, (p2 - Center).Abs()));
Debug.Assert(Tolerance.Equal(Radius, (p3 - Center).Abs()));
return(true);
}
private static void HopfFibration(Tiling tiling)
{
int segDivisions = 10;
int circleDivisions = 125;
Shapeways mesh = new Shapeways();
HashSet <Vector3D> done = new HashSet <Vector3D>();
foreach (Tile tile in tiling.Tiles)
{
foreach (Segment seg in tile.Boundary.Segments)
{
if (done.Contains(seg.Midpoint))
{
continue;
}
// Subdivide the segment, and project points to S2.
Vector3D[] points = seg.Subdivide(segDivisions).Select(v => Spherical2D.PlaneToSphere(v)).ToArray();
foreach (Vector3D point in points)
{
// Get the hopf circle and add to mesh.
// http://en.wikipedia.org/wiki/Hopf_fibration#Explicit_formulae
double a = point.X;
double b = point.Y;
double c = point.Z;
double factor = 1 / (Math.Sqrt(1 + c));
if (Tolerance.Equal(c, -1))
{
continue;
}
List <Vector3D> circlePoints = new List <Vector3D>();
double angleInc = 2 * Math.PI / circleDivisions;
double angle = 0;
for (int i = 0; i <= circleDivisions; i++)
{
double sinTheta = Math.Sin(angle);
double cosTheta = Math.Cos(angle);
circlePoints.Add(new Vector3D(
(1 + c) * cosTheta,
a * sinTheta - b * cosTheta,
a * cosTheta + b * sinTheta,
(1 + c) * sinTheta));
angle += angleInc;
}
bool shrink = false;
ProjectAndAddS3Points(mesh, circlePoints.ToArray(), shrink);
}
done.Add(seg.Midpoint);
}
}
STL.SaveMeshToSTL(mesh.Mesh, @"D:\p4\R3\sample\out1.stl");
}
public static int IntersectionLineCircle(Vector3D lineP1, Vector3D lineP2, Circle circle,
out Vector3D p1, out Vector3D p2)
{
p1 = new Vector3D();
p2 = new Vector3D();
// Distance from the circle center to the closest point on the line.
double d = DistancePointLine(circle.Center, lineP1, lineP2);
// No intersection points.
double r = circle.Radius;
if (d > r)
{
return(0);
}
// One intersection point.
p1 = ProjectOntoLine(circle.Center, lineP1, lineP2);
if (Tolerance.Equal(d, r))
{
return(1);
}
// Two intersection points.
// Special case when the line goes through the circle center,
// because we can see numerical issues otherwise.
//
// I had further issues where my default tolerance was too strict for this check.
// The line was close to going through the center and the second block was used,
// so I had to loosen the tolerance used by my comparison macros.
if (Tolerance.Zero(d))
{
Vector3D line = lineP2 - lineP1;
line.Normalize();
line *= r;
p1 = circle.Center + line;
p2 = circle.Center - line;
}
else
{
// To origin.
p1 -= circle.Center;
p1.Normalize();
p1 *= r;
p2 = p1;
double angle = Math.Acos(d / r);
p1.RotateXY(angle);
p2.RotateXY(-angle);
// Back out.
p1 += circle.Center;
p2 += circle.Center;
}
return(2);
}
/// <summary>
/// Checks if a point is anywhere on a segment.
/// </summary>
public static bool PointOnSegment(Vector3D s1, Vector3D s2, Vector3D point)
{
// Look for a degenerate triangle.
double d1 = (point - s1).MagSquared();
double d2 = (s2 - point).MagSquared();
double d3 = (s2 - s1).MagSquared();
return(Tolerance.Equal(d1 + d2, d3));
}
public bool IsPointOn(Vector3D test)
{
if (this.IsLine)
{
return(Tolerance.Zero(Euclidean2D.DistancePointLine(test, P1, P2)));
}
return(Tolerance.Equal((test - Center).Abs(), Radius));
}
private static bool PointOnLineSegment(Vector3D p, Segment seg)
{
// This will be so if the point and the segment ends represent
// the vertices of a degenerate triangle.
double d1 = (seg.P2 - seg.P1).Abs();
double d2 = (p - seg.P1).Abs();
double d3 = (seg.P2 - p).Abs();
return(Tolerance.Equal(d1, d2 + d3));
}
public bool IsPointOn(Vector3D test)
{
if (IsPlane)
{
double dist = Euclidean3D.DistancePointPlane(this.Normal, this.Offset, test);
return(Tolerance.Zero(dist));
}
return(Tolerance.Equal((test - Center).Abs(), Radius));
}
/// <summary>
/// Outputs edges to an stl file.
/// </summary>
public void Output()
{
System.Func <Vector3D, Vector3D> p2s = v => Spherical2D.PlaneToSphere(v);
System.Func <Vector3D, Vector3D> transform = v => H3Models.Ball.ApplyMobius(Mobius.Scale(3), v);
double min = double.MaxValue;
Cell cell = m_cells.First();
foreach (Vector3D v1 in cell.Tiles[0].Boundary.Vertices)
{
foreach (Vector3D v2 in cell.Tiles[1].Boundary.Vertices)
{
min = Math.Min(min, p2s(v1).Dist(p2s(v2)));
}
}
// XXX - code below so ugly to read!
Dictionary <TileVertex, TileVertex> vMap = new Dictionary <TileVertex, TileVertex>();
for (int tile_i = 0; tile_i < cell.Tiles.Length; tile_i++)
{
for (int tile_j = tile_i + 1; tile_j < cell.Tiles.Length; tile_j++)
{
for (int vertex_i = 0; vertex_i < cell.Tiles[tile_i].Boundary.Vertices.Length; vertex_i++)
{
for (int vertex_j = 0; vertex_j < cell.Tiles[tile_j].Boundary.Vertices.Length; vertex_j++)
{
Vector3D v1 = cell.Tiles[tile_i].Boundary.Vertices[vertex_i];
Vector3D v2 = cell.Tiles[tile_j].Boundary.Vertices[vertex_j];
if (Tolerance.Equal(p2s(v1).Dist(p2s(v2)), min))
{
vMap[new TileVertex(tile_i, vertex_i)] = new TileVertex(tile_j, vertex_j);
}
}
}
}
}
HashSet <H3.Cell.Edge> edges = new HashSet <H3.Cell.Edge>(new H3.Cell.EdgeEqualityComparer());
foreach (Cell c in m_cells)
{
foreach (KeyValuePair <TileVertex, TileVertex> kvp in vMap)
{
Vector3D v1 = transform(p2s(c.Tiles[kvp.Key.Item1].Boundary.Vertices[kvp.Key.Item2]));
Vector3D v2 = transform(p2s(c.Tiles[kvp.Value.Item1].Boundary.Vertices[kvp.Value.Item2]));
edges.Add(new H3.Cell.Edge(v1, v2));
}
}
//H3.m_settings.ThinEdges = true;
H3.SaveToFile("ultrainf", edges.ToArray(), finite: false);
}
private int PolyWithAngle(double angle)
{
for (int i = 3; i < m_max; i++)
{
if (Tolerance.Equal(angle, Polygon.InteriorAngle(i)))
{
return(i);
}
}
return(0);
}
/// <summary>
/// Calculates the two poles of a great circle defined by two points.
/// </summary>
public static void GreatCirclePole(Vector3D sphereCenter, Vector3D p1, Vector3D p2,
out Vector3D pole1, out Vector3D pole2)
{
double sphereRadius = p1.Dist(sphereCenter);
Debug.Assert(Tolerance.Equal(sphereRadius, p2.Dist(sphereCenter)));
Vector3D v1 = p1 - sphereCenter;
Vector3D v2 = p2 - sphereCenter;
pole1 = v1.Cross(v2) + sphereCenter;
pole2 = v2.Cross(v1) + sphereCenter;
}
private static bool PointOnArcSegment(Vector3D p, Segment seg)
{
double maxAngle = seg.Angle;
Vector3D v1 = seg.P1 - seg.Center;
Vector3D v2 = p - seg.Center;
Debug.Assert(Tolerance.Equal(v1.Abs(), v2.Abs()));
double angle = seg.Clockwise ?
Euclidean2D.AngleToClock(v1, v2) :
Euclidean2D.AngleToCounterClock(v1, v2);
return(Tolerance.LessThanOrEqual(angle, maxAngle));
}
public static Vector3D S3toR3(Vector3D p)
{
double w = p.W;
if (Tolerance.Equal(w, 1))
{
return(Vector3D.DneVector());
}
return(new Vector3D(
p.X / (1 - w),
p.Y / (1 - w),
p.Z / (1 - w)));
}
// Returns the geometry induced by a polygon with p points, q meeting at a vertex.
public static Geometry GetGeometry(double p, double q)
{
double test = 1.0 / p + 1.0 / q;
if (test > 0.5)
{
return(Geometry.Spherical);
}
else if (Tolerance.Equal(test, 0.5))
{
return(Geometry.Euclidean);
}
return(Geometry.Hyperbolic);
}
private static void AvoidNorthPole(ref Vector3D v, Vector3D direction)
{
if (!Tolerance.Equal(v.W, 1))
{
return;
}
Vector3D cutEnd = v - direction;
double abs = cutEnd.Abs();
abs -= 0.35;
cutEnd.Normalize();
cutEnd *= abs;
v = direction + cutEnd;
v.Normalize();
}
public static Geometry GetGeometry(int p, int q, int r)
{
double t1 = Math.Sin(PiOverNSafe(p)) * Math.Sin(PiOverNSafe(r));
double t2 = Math.Cos(PiOverNSafe(q));
if (Tolerance.Equal(t1, t2))
{
return(Geometry.Euclidean);
}
if (Tolerance.GreaterThan(t1, t2))
{
return(Geometry.Spherical);
}
return(Geometry.Hyperbolic);
}
public static Circle3D FromCenterAnd2Points(Vector3D cen, Vector3D p1, Vector3D p2)
{
Circle3D circle = new Circle3D();
circle.Center = cen;
circle.Radius = (p1 - cen).Abs();
if (!Tolerance.Equal(circle.Radius, (p2 - cen).Abs()))
{
throw new System.ArgumentException("Points are not on the same circle.");
}
Vector3D normal = (p2 - cen).Cross(p1 - cen);
normal.Normalize();
circle.Normal = normal;
return(circle);
}
/// <summary>
/// Finds the intersection (a circle) between us and another sphere.
/// Returns null if sphere centers are coincident or no intersection exists.
/// Does not currently work for planes.
/// </summary>
public Circle3D Intersection(Sphere s)
{
if (this.IsPlane || s.IsPlane)
{
throw new System.NotImplementedException();
}
double r = s.Radius;
double R = this.Radius;
Vector3D diff = this.Center - s.Center;
double d = diff.Abs();
if (Tolerance.Equal(d, r + R))
{
diff.Normalize();
return(new Circle3D()
{
Center = s.Center + diff * s.Radius,
Radius = 0
});
}
if (Tolerance.Zero(d) || d > r + R)
{
return(null);
}
// Sphere's inside spheres and not touching.
//if( d < Math.Abs( R - r ) )
// return null;
double x = (d * d + r * r - R * R) / (2 * d);
double y = Math.Sqrt(r * r - x * x);
Circle3D result = new Circle3D();
diff.Normalize();
result.Normal = diff;
result.Center = s.Center + diff * x;
result.Radius = y;
return(result);
}
/// <summary>
/// Checks whether this instance is spatially equal to <paramref name="p"/>.
/// </summary>
/// <param name="p">Point to compare to</param>
/// <returns>True if the points are either both empty or have the same coordinates, false otherwise.</returns>
public virtual Boolean Equals(Point p)
{
if (ReferenceEquals(p, null))
{
return(false);
}
if (IsEmpty && p.IsEmpty)
{
return(true);
}
if (IsEmpty || p.IsEmpty)
{
return(false);
}
return(Tolerance.Equal(p.X, X) && Tolerance.Equal(p.Y, Y));
}
// ZZZ - I wonder if we want to do normalization of lines before comparing.
public bool Equals(CircleNE c1, CircleNE c2)
{
bool radiusEqual =
Tolerance.Equal(c1.Radius, c2.Radius) ||
(Infinity.IsInfinite(c1.Radius) && Infinity.IsInfinite(c2.Radius));
if (c1.IsLine)
{
return(c1.P1 == c2.P1 &&
c1.P2 == c2.P2 &&
radiusEqual);
}
else
{
return
(c1.Center == c2.Center &&
c1.CenterNE == c2.CenterNE &&
radiusEqual);
}
}
public static Circle3D GeodesicFrom2Points(Vector3D a, Vector3D b)
{
if (a == b || a == -b)
{
throw new System.Exception("Geodesic not unique");
}
System.Func <Vector3D, Circle3D> lineFunc = p =>
{
Circle3D circ = new Circle3D();
circ.Radius = double.PositiveInfinity;
p.Normalize();
circ.Normal = p; // Hacky representation of a line.
return(circ);
};
if (a.IsOrigin || b.IsOrigin)
{
Vector3D p = a.IsOrigin ? b : a;
return(lineFunc(p));
}
double mag1 = a.Abs(), mag2 = b.Abs();
if (Tolerance.Equal(mag1, 1) && Tolerance.Equal(mag2, 1))
{
return(new Circle3D(a, b, a * -1));
}
// The antipode in S^3 of a or b will give us a 3rd point.
Vector3D antipode = Sterographic.S3toR3(Sterographic.R3toS3(a) * -1);
// If the antipode is also an an antipode in R3, the points are colinear
// (or they are on the equatorial 2-sphere, but that is checked above).
if (a == antipode * -1)
{
return(lineFunc(a));
}
return(new Circle3D(a, b, antipode));
}
private static H3.Cell.Edge[] Cull120Cell(H3.Cell.Edge[] edges)
{
Func <Vector3D, bool> passes = new Func <Vector3D, bool>(v =>
{
//return
// Math.Pow( v.Z, 2 ) < 0.13 &&
// Math.Pow( v.W, 2 ) < 0.13;
return(Tolerance.Equal(v.W, 0.0));
});
H3.Cell.Edge[] result = edges.Where(e =>
{
Vector3D start = Sterographic.R3toS3(e.Start);
Vector3D end = Sterographic.R3toS3(e.End);
return(passes(start) && passes(end));
}).ToArray();
// Now cull valence-2 edges.
//result = CullValence2Edges( result );
return(result);
}
public static Vector3D[] OneHopfCircle(Vector3D s2Point, bool anti = false)
{
int circleDivisions = 125;
// Get the hopf circle.
// http://en.wikipedia.org/wiki/Hopf_fibration#Explicit_formulae
double a = s2Point.X;
double b = s2Point.Y;
double c = s2Point.Z;
double factor = 1 / (Math.Sqrt(1 + c));
if (Tolerance.Equal(c, -1))
{
return new Vector3D[] {}
}
;
List <Vector3D> circlePoints = new List <Vector3D>();
double angleInc = 2 * Math.PI / circleDivisions;
double angle = 0;
for (int i = 0; i <= circleDivisions; i++)
{
double sinTheta = Math.Sin(angle);
double cosTheta = Math.Cos(angle);
Vector3D point = new Vector3D(
(1 + c) * cosTheta,
anti ? -a * sinTheta - b * cosTheta : a * sinTheta - b * cosTheta,
anti ? a * cosTheta - b * sinTheta : a * cosTheta + b * sinTheta,
(1 + c) * sinTheta);
point.Normalize();
circlePoints.Add(point);
angle += angleInc;
}
return(circlePoints.ToArray());
}
public bool Compare(Vector3D other, double threshold)
{
// NOTE: This is here because when the vector is infinite, it fails the tolerance checks below.
if ((X == other.X) && (Y == other.Y) && (Z == other.Z) && W == other.W)
{
return(true);
}
if (this.DNE && other.DNE)
{
return(true);
}
if (this.DNE || other.DNE)
{
return(false);
}
return(Tolerance.Equal(X, other.X, threshold) &&
Tolerance.Equal(Y, other.Y, threshold) &&
Tolerance.Equal(Z, other.Z, threshold) &&
Tolerance.Equal(W, other.W, threshold));
}
public static Segment Arc(Vector3D start, Vector3D mid, Vector3D end)
{
Segment newSeg = new Segment();
newSeg.Type = SegmentType.Arc;
newSeg.P1 = start;
newSeg.P2 = end;
Circle c = new Circle();
c.From3Points(start, mid, end);
newSeg.Center = c.Center;
// Obtain vectors from center point of circle (as if at the origin)
Vector3D startOrigin = start - c.Center;
Vector3D midOrigin = mid - c.Center;
Vector3D endOrigin = end - c.Center;
// Calculate the normal vector and angle to traverse.
// ZZZ - worry about failure of cross product here.
Vector3D normalVector = startOrigin.Cross(endOrigin);
newSeg.Clockwise = normalVector.Z < 0;
double angleToTraverse = startOrigin.AngleTo(endOrigin);
// The normal vector might need to be reversed and the angleToTraverse adjusted.
// This happens depending on the location of the midpoint relative to the start and end points.
double compareAngle = startOrigin.AngleTo(midOrigin) + midOrigin.AngleTo(endOrigin);
bool reverse = !Tolerance.Equal(angleToTraverse, compareAngle);
if (reverse)
{
newSeg.Clockwise = !newSeg.Clockwise;
}
return(newSeg);
}