/// <summary>
/// Reflect triangle in given plane
/// </summary>
public Triangle ReflectIn(Plane3d s)
{
return(new Triangle(_a.ReflectIn(s), _b.ReflectIn(s), _c.ReflectIn(s)));
}
/// <summary>
/// Reflect ray in given plane
/// </summary>
public Ray3d ReflectIn(Plane3d s)
{
return(new Ray3d(this.Point.ReflectIn(s), this.Direction.ReflectIn(s)));
}
/// <summary>
/// Reflect ellipse in given plane
/// </summary>
public Ellipse ReflectIn(Plane3d s)
{
return(new Ellipse(this.Center.ReflectIn(s), _v1.ReflectIn(s), _v2.ReflectIn(s)));
}
/// <summary>
/// Get intersection of line with triangle.
/// Returns 'null' (no intersection) or object of type 'Point3d' or 'Segment3d'.
/// </summary>
public object IntersectionWith(Line3d l)
{
// Relative tolerance ================================
if (!GeometRi3D.UseAbsoluteTolerance)
{
double tol = GeometRi3D.Tolerance;
GeometRi3D.Tolerance = tol * Max(AB, Max(BC, AC));
GeometRi3D.UseAbsoluteTolerance = true;
object result = this.IntersectionWith(l);
GeometRi3D.UseAbsoluteTolerance = false;
GeometRi3D.Tolerance = tol;
return(result);
}
//====================================================
Plane3d s = new Plane3d(this.A, this.Normal);
object obj = l.IntersectionWith(s);
if (obj == null)
{
return(null);
}
else
{
if (obj.GetType() == typeof(Line3d))
{
Segment3d sAB = new Segment3d(A, B);
Segment3d sBC = new Segment3d(B, C);
Segment3d sAC = new Segment3d(A, C);
// Line coincides with one side, return segment
if (sAB.BelongsTo(l))
{
return(sAB);
}
if (sBC.BelongsTo(l))
{
return(sBC);
}
if (sAC.BelongsTo(l))
{
return(sAC);
}
Point3d pAB = (Point3d)sAB.IntersectionWith(l);
Point3d pBC = (Point3d)sBC.IntersectionWith(l);
Point3d pAC = (Point3d)sAC.IntersectionWith(l);
bool bAB = (object.ReferenceEquals(null, pAB)) ? false : pAB.BelongsTo(sAB);
bool bAC = (object.ReferenceEquals(null, pAC)) ? false : pAC.BelongsTo(sAC);
bool bBC = (object.ReferenceEquals(null, pBC)) ? false : pBC.BelongsTo(sBC);
// Line crosses one corner, return point
if (bAB && bBC && pAB == pBC && !bAC)
{
return(pAB);
}
if (bAB && bAC && pAB == pAC && !bBC)
{
return(pAB);
}
if (bAC && bBC && pAC == pBC && !bAB)
{
return(pAC);
}
// Line crosses two sides, return segment
if (bAB && bBC && !bAC)
{
return(new Segment3d(pAB, pBC));
}
if (bAB && bAC && !bBC)
{
return(new Segment3d(pAB, pAC));
}
if (bAC && bBC && !bAB)
{
return(new Segment3d(pAC, pBC));
}
// Line crosses one corner and one side, return segment
if (pAB == pBC && bAC)
{
return(new Segment3d(pAB, pAC));
}
if (pAB == pAC && bBC)
{
return(new Segment3d(pAB, pBC));
}
if (pAC == pBC && bAB)
{
return(new Segment3d(pAB, pAC));
}
//else
return(null);
}
else
{
// result of intersection is point
Point3d p = (Point3d)obj;
if (p.BelongsTo(this))
{
return(p);
}
else
{
return(null);
}
}
}
}
/// <summary>
/// Intersection of ellipsoid with plane.
/// Returns 'null' (no intersection) or object of type 'Point3d' or 'Ellipse'.
/// </summary>
public object IntersectionWith(Plane3d plane)
{
// Solution 1:
// Peter Paul Klein
// On the Ellipsoid and Plane Intersection Equation
// Applied Mathematics, 2012, 3, 1634-1640 (DOI:10.4236/am.2012.311226)
// Solution 2:
// Sebahattin Bektas
// Intersection of an Ellipsoid and a Plane
// International Journal of Research in Engineering and Applied Sciences, VOLUME 6, ISSUE 6 (June, 2016)
Coord3d lc = new Coord3d(_point, _v1, _v2, "LC1");
plane.SetCoord(lc);
double Ax, Ay, Az, Ad;
double a, b, c;
if (Abs(plane.C) >= Abs(plane.A) && Abs(plane.C) >= Abs(plane.B))
{
a = this.A; b = this.B; c = this.C;
}
else
{
lc = new Coord3d(_point, _v2, _v3, "LC2");
plane.SetCoord(lc);
if (Abs(plane.C) >= Abs(plane.A) && Abs(plane.C) >= Abs(plane.B))
{
a = this.B; b = this.C; c = this.A;
}
else
{
lc = new Coord3d(_point, _v3, _v1, "LC3");
plane.SetCoord(lc);
a = this.C; b = this.A; c = this.B;
}
}
Ax = plane.A; Ay = plane.B; Az = plane.C; Ad = plane.D;
double tmp = (Az * Az * c * c);
double AA = 1.0 / (a * a) + Ax * Ax / tmp;
double BB = 2.0 * Ax * Ay / tmp;
double CC = 1.0 / (b * b) + Ay * Ay / tmp;
double DD = 2.0 * Ax * Ad / tmp;
double EE = 2.0 * Ay * Ad / tmp;
double FF = Ad * Ad / tmp - 1.0;
double det = 4.0 * AA * CC - BB * BB;
if (GeometRi3D.AlmostEqual(det, 0))
{
return(null);
}
double X0 = (BB * EE - 2 * CC * DD) / det;
double Y0 = (BB * DD - 2 * AA * EE) / det;
double Z0 = -(Ax * X0 + Ay * Y0 + Ad) / Az;
Point3d P0 = new Point3d(X0, Y0, Z0, lc);
if (P0.BelongsTo(this))
{
// the plane is tangent to ellipsoid
return(P0);
}
else if (P0.IsInside(this))
{
Vector3d q = P0.ToVector.ConvertTo(lc);
Matrix3d D1 = Matrix3d.DiagonalMatrix(1 / a, 1 / b, 1 / c);
Vector3d r = plane.Normal.ConvertTo(lc).OrthogonalVector.Normalized;
Vector3d s = plane.Normal.ConvertTo(lc).Cross(r).Normalized;
double omega = 0;
double qq, qr, qs, rr, ss, rs;
if (!GeometRi3D.AlmostEqual((D1 * r) * (D1 * s), 0))
{
rr = (D1 * r) * (D1 * r);
rs = (D1 * r) * (D1 * s);
ss = (D1 * s) * (D1 * s);
if (GeometRi3D.AlmostEqual(rr - ss, 0))
{
omega = PI / 4;
}
else
{
omega = 0.5 * Atan(2.0 * rs / (rr - ss));
}
Vector3d rprim = Cos(omega) * r + Sin(omega) * s;
Vector3d sprim = -Sin(omega) * r + Cos(omega) * s;
r = rprim;
s = sprim;
}
qq = (D1 * q) * (D1 * q);
qr = (D1 * q) * (D1 * r);
qs = (D1 * q) * (D1 * s);
rr = (D1 * r) * (D1 * r);
ss = (D1 * s) * (D1 * s);
double d = qq - qr * qr / rr - qs * qs / ss;
AA = Sqrt((1 - d) / rr);
BB = Sqrt((1 - d) / ss);
return(new Ellipse(P0, AA * r, BB * s));
}
else
{
return(null);
}
}
/// <summary>
/// Intersection of ellipse with plane.
/// Returns 'null' (no intersection) or object of type 'Ellipse', 'Point3d' or 'Segment3d'.
/// </summary>
public object IntersectionWith(Plane3d s)
{
if (this.Normal.IsParallelTo(s.Normal))
{
if (this.Center.BelongsTo(s))
{
// coplanar objects
return(this.Copy());
}
else
{
// parallel objects
return(null);
}
}
else
{
Line3d l = (Line3d)s.IntersectionWith(new Plane3d(this.Center, this.Normal));
Coord3d local_coord = new Coord3d(this.Center, this._v1, this._v2);
Point3d p = l.Point.ConvertTo(local_coord);
Vector3d v = l.Direction.ConvertTo(local_coord);
double a = this.A;
double b = this.B;
if (Abs(v.Y / v.X) > 100)
{
// line is almost vertical, rotate local coord
local_coord = new Coord3d(this.Center, this._v2, this._v1);
p = l.Point.ConvertTo(local_coord);
v = l.Direction.ConvertTo(local_coord);
a = this.B;
b = this.A;
}
// Find intersection of line and ellipse (2D)
// Solution from: http://www.ambrsoft.com/TrigoCalc/Circles2/Ellipse/EllipseLine.htm
// Line equation in form: y = mx + c
double m = v.Y / v.X;
double c = p.Y - m * p.X;
double amb = Math.Pow(a, 2) * Math.Pow(m, 2) + Math.Pow(b, 2);
double det = amb - Math.Pow(c, 2);
if (det < -GeometRi3D.Tolerance)
{
return(null);
}
else if (GeometRi3D.AlmostEqual(det, 0))
{
double x = -Math.Pow(a, 2) * m * c / amb;
double y = Math.Pow(b, 2) * c / amb;
return(new Point3d(x, y, 0, local_coord));
}
else
{
double x1 = (-Math.Pow(a, 2) * m * c + a * b * Sqrt(det)) / amb;
double x2 = (-Math.Pow(a, 2) * m * c - a * b * Sqrt(det)) / amb;
double y1 = (Math.Pow(b, 2) * c + a * b * m * Sqrt(det)) / amb;
double y2 = (Math.Pow(b, 2) * c - a * b * m * Sqrt(det)) / amb;
return(new Segment3d(new Point3d(x1, y1, 0, local_coord), new Point3d(x2, y2, 0, local_coord)));
}
}
}
/// <summary>
/// Reflect sphere in given plane
/// </summary>
public Sphere ReflectIn(Plane3d s)
{
return(new Sphere(this.Center.ReflectIn(s), this.R));
}
/// <summary>
/// Intersection of two circles.
/// Returns 'null' (no intersection) or object of type 'Circle3d', 'Point3d' or 'Segment3d'.
/// In 2D (coplanar circles) the segment will define two intersection points.
/// </summary>
public object IntersectionWith(Circle3d c)
{
// Relative tolerance ================================
if (!GeometRi3D.UseAbsoluteTolerance)
{
double tol = GeometRi3D.Tolerance;
GeometRi3D.Tolerance = tol * Max(this.R, c.R);
GeometRi3D.UseAbsoluteTolerance = true;
object result = this.IntersectionWith(c);
GeometRi3D.UseAbsoluteTolerance = false;
GeometRi3D.Tolerance = tol;
return(result);
}
//====================================================
if (this.Normal.IsParallelTo(c.Normal))
{
if (this.Center.BelongsTo(new Plane3d(c.Center, c.Normal)))
{
// Coplanar objects
// Search 2D intersection of two circles
// Equal circles
if (this.Center == c.Center && GeometRi3D.AlmostEqual(this.R, c.R))
{
return(this.Copy());
}
double d = this.Center.DistanceTo(c.Center);
// Separated circles
if (GeometRi3D.Greater(d, this.R + c.R))
{
return(null);
}
// One circle inside the other
if (d < Abs(this.R - c.R) - GeometRi3D.Tolerance)
{
return(null);
}
// Outer tangency
if (GeometRi3D.AlmostEqual(d, this.R + c.R))
{
Vector3d vec = new Vector3d(this.Center, c.Center);
return(this.Center.Translate(this.R * vec.Normalized));
}
// Inner tangency
if (Abs(Abs(this.R - c.R) - d) < GeometRi3D.Tolerance)
{
Vector3d vec = new Vector3d(this.Center, c.Center);
if (this.R > c.R)
{
return(this.Center.Translate(this.R * vec.Normalized));
}
else
{
return(this.Center.Translate(-this.R * vec.Normalized));
}
}
// intersecting circles
// Create local CS with origin in circle's center
Vector3d vec1 = new Vector3d(this.Center, c.Center);
Vector3d vec2 = vec1.Cross(this.Normal);
Coord3d local_cs = new Coord3d(this.Center, vec1, vec2);
double x = 0.5 * (d * d - c.R * c.R + this.R * this.R) / d;
double y = 0.5 * Sqrt((-d + c.R - this.R) * (-d - c.R + this.R) * (-d + c.R + this.R) * (d + c.R + this.R)) / d;
Point3d p1 = new Point3d(x, y, 0, local_cs);
Point3d p2 = new Point3d(x, -y, 0, local_cs);
return(new Segment3d(p1, p2));
}
else
{
// parallel objects
return(null);
}
}
else
{
// Check 3D intersection
Plane3d plane = new Plane3d(this.Center, this.Normal);
object obj = plane.IntersectionWith(c);
if (obj == null)
{
return(null);
}
else if (obj.GetType() == typeof(Point3d))
{
Point3d p = (Point3d)obj;
if (p.BelongsTo(c))
{
return(p);
}
else
{
return(null);
}
}
else
{
Segment3d s = (Segment3d)obj;
return(s.IntersectionWith(this));
}
}
}
/// <summary>
/// Get intersection of line with triangle.
/// Returns 'null' (no intersection) or object of type 'Point3d' or 'Segment3d'.
/// </summary>
public object IntersectionWith(Line3d l)
{
// Relative tolerance ================================
if (!GeometRi3D.UseAbsoluteTolerance)
{
double tol = GeometRi3D.Tolerance;
GeometRi3D.Tolerance = tol * Max(AB, Max(BC, AC));
GeometRi3D.UseAbsoluteTolerance = true;
object result = this.IntersectionWith(l);
GeometRi3D.UseAbsoluteTolerance = false;
GeometRi3D.Tolerance = tol;
return(result);
}
//====================================================
Plane3d s = new Plane3d(this.A, this.Normal);
object obj = l.IntersectionWith(s);
if (obj == null)
{
return(null);
}
else
{
if (obj.GetType() == typeof(Line3d))
{
// Coplanar line and triangle
// Check intersection in one corner
// or in corner and opposite side
if (_a.BelongsTo(l))
{
object obj2 = new Segment3d(_b, _c).IntersectionWith(l);
if (obj2 != null && obj2.GetType() == typeof(Point3d))
{
return(new Segment3d(_a, (Point3d)obj2));
}
else
{
return(A);
}
}
if (_b.BelongsTo(l))
{
object obj2 = new Segment3d(_a, _c).IntersectionWith(l);
if (obj2 != null && obj2.GetType() == typeof(Point3d))
{
return(new Segment3d(_b, (Point3d)obj2));
}
else
{
return(B);
}
}
if (_c.BelongsTo(l))
{
object obj2 = new Segment3d(_a, _b).IntersectionWith(l);
if (obj2 != null && obj2.GetType() == typeof(Point3d))
{
return(new Segment3d(_c, (Point3d)obj2));
}
else
{
return(C);
}
}
// Check intersection with two sides
object objAB = new Segment3d(_a, _b).IntersectionWith(l);
object objBC = new Segment3d(_b, _c).IntersectionWith(l);
if (objAB != null && objAB.GetType() == typeof(Point3d))
{
if (objBC != null && objBC.GetType() == typeof(Point3d))
{
return(new Segment3d((Point3d)objAB, (Point3d)objBC));
}
else
{
object objAC = new Segment3d(_a, _c).IntersectionWith(l);
if (objAC != null && objAC.GetType() == typeof(Point3d))
{
return(new Segment3d((Point3d)objAB, (Point3d)objAC));
}
else
{
return((Point3d)objAB);
}
}
}
if (objBC != null && objBC.GetType() == typeof(Point3d))
{
object objAC = new Segment3d(_a, _c).IntersectionWith(l);
if (objAC != null && objAC.GetType() == typeof(Point3d))
{
return(new Segment3d((Point3d)objBC, (Point3d)objAC));
}
else
{
return((Point3d)objBC);
}
}
object objAC2 = new Segment3d(_a, _c).IntersectionWith(l);
if (objAC2 != null && objAC2.GetType() == typeof(Point3d))
{
return((Point3d)objAC2);
}
else
{
return(null);
}
}
else
{
// result of intersection is point
Point3d p = (Point3d)obj;
if (p.BelongsTo(this))
{
return(p);
}
else
{
return(null);
}
}
}
}
/// <summary>
/// Orthogonal projection of the sphere to the plane
/// </summary>
public Circle3d ProjectionTo(Plane3d s)
{
Point3d p = this.Center.ProjectionTo(s);
return(new Circle3d(p, this.R, s.Normal));
}
/// <summary>
/// Reflect point in given plane
/// </summary>
public Point3d ReflectIn(Plane3d s)
{
Vector3d v = new Vector3d(this, this.ProjectionTo(s));
return(this.Translate(2 * v));
}
/// <summary>
/// Returns shortest distance from point to the plane
/// </summary>
public double DistanceTo(Plane3d s)
{
s.SetCoord(this.Coord);
return(Abs(X * s.A + Y * s.B + Z * s.C + s.D) / Sqrt(s.A * s.A + s.B * s.B + s.C * s.C));
}
/// <summary>
/// Reflect segment in given plane
/// </summary>
public virtual Segment3d ReflectIn(Plane3d s)
{
return(new Segment3d(P1.ReflectIn(s), P2.ReflectIn(s)));
}
/// <summary>
/// Reflect plane in given plane
/// </summary>
public Plane3d ReflectIn(Plane3d s)
{
return(new Plane3d(this.Point.ReflectIn(s), this.Normal.ReflectIn(s)));
}
/// <summary>
/// Get intersection of line with plane.
/// Returns 'null' (no intersection) or object of type 'Point3d' or 'Line3d'.
/// </summary>
public virtual object IntersectionWith(Plane3d s)
{
return(s.IntersectionWith(this));
}
/// <summary>
/// Orthogonal projection of the circle to plane
/// </summary>
public Ellipse ProjectionTo(Plane3d s)
{
return(this.ToEllipse.ProjectionTo(s));
}
/// <summary>
/// Reflect line in given plane
/// </summary>
public virtual Line3d ReflectIn(Plane3d s)
{
return(new Line3d(this.Point.ReflectIn(s), this.Direction.ReflectIn(s)));
}
/// <summary>
/// Reflect circle in given plane
/// </summary>
public Circle3d ReflectIn(Plane3d s)
{
return(new Circle3d(this.Center.ReflectIn(s), this.R, this.Normal.ReflectIn(s)));
}
/// <summary>
/// Returns shortest distance from point to the plane
/// </summary>
public double DistanceTo(Plane3d s)
{
s.SetCoord(this.Coord);
return(Abs(X * s.A + Y * s.B + Z * s.C + s.D) / Sqrt(Math.Pow(s.A, 2) + Math.Pow(s.B, 2) + Math.Pow(s.C, 2)));
}