/// <summary>
/// Get intersection of plane with sphere.
/// Returns 'null' (no intersection) or object of type 'Point3d' or 'Circle3d'.
/// </summary>
public object IntersectionWith(Plane3d s)
{
s.SetCoord(this.Center.Coord);
double d1 = s.A * this.X + s.B * this.Y + s.C * this.Z + s.D;
double d2 = Math.Pow(s.A, 2) + Math.Pow(s.B, 2) + Math.Pow(s.C, 2);
double d = Abs(d1) / Sqrt(d2);
if (d > this.R + GeometRi3D.Tolerance)
{
return(null);
}
else
{
double Xc = this.X - s.A * d1 / d2;
double Yc = this.Y - s.B * d1 / d2;
double Zc = this.Z - s.C * d1 / d2;
if (Abs(d - this.R) < GeometRi3D.Tolerance)
{
return(new Point3d(Xc, Yc, Zc, this.Center.Coord));
}
else
{
double R = Sqrt(Math.Pow(this.R, 2) - Math.Pow(d, 2));
return(new Circle3d(new Point3d(Xc, Yc, Zc, this.Center.Coord), R, s.Normal));
}
}
}
/// <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)
{
Vector3d r1 = this.Point.ToVector;
Vector3d s1 = this.Direction;
Vector3d n2 = s.Normal;
if (Abs(s1 * n2) < GeometRi3D.Tolerance)
{
// Line and plane are parallel
if (this.Point.BelongsTo(s))
{
// Line lies in the plane
return(this);
}
else
{
return(null);
}
}
else
{
// Intersection point
s.SetCoord(r1.Coord);
r1 = r1 - ((r1 * n2) + s.D) / (s1 * n2) * s1;
return(r1.ToPoint);
}
}
/// <summary>
/// Returns orthogonal projection of the point to the plane
/// </summary>
public Point3d ProjectionTo(Plane3d s)
{
Vector3d r0 = new Vector3d(this, _coord);
s.SetCoord(this.Coord);
Vector3d n = new Vector3d(s.A, s.B, s.C, _coord);
r0 = r0 - (r0 * n + s.D) / (n * n) * n;
return(r0.ToPoint);
}
/// <summary>
/// Finds the common intersection of three planes.
/// Return 'null' (no common intersection) or object of type 'Point3d', 'Line3d' or 'Plane3d'
/// </summary>
public object IntersectionWith(Plane3d s2, Plane3d s3)
{
// Set all planes to global CS
this.SetCoord(Coord3d.GlobalCS);
s2.SetCoord(Coord3d.GlobalCS);
s3.SetCoord(Coord3d.GlobalCS);
double det = new Matrix3d(new[] { A, B, C }, new[] { s2.A, s2.B, s2.C }, new [] { s3.A, s3.B, s3.C }).Det;
if (Abs(det) < GeometRi3D.Tolerance)
{
if (this.Normal.IsParallelTo(s2.Normal) && this.Normal.IsParallelTo(s3.Normal))
{
// Planes are coplanar
if (this.Point.BelongsTo(s2) && this.Point.BelongsTo(s3))
{
return(this);
}
else
{
return(null);
}
}
if (this.Normal.IsNotParallelTo(s2.Normal) && this.Normal.IsNotParallelTo(s3.Normal))
{
// Planes are not parallel
// Find the intersection (Me,s2) and (Me,s3) and check if it is the same line
Line3d l1 = (Line3d)this.IntersectionWith(s2);
Line3d l2 = (Line3d)this.IntersectionWith(s3);
if (l1 == l2)
{
return(l1);
}
else
{
return(null);
}
}
// Two planes are parallel, third plane is not
return(null);
}
else
{
double x = -new Matrix3d(new[] { D, B, C }, new[] { s2.D, s2.B, s2.C }, new[] { s3.D, s3.B, s3.C }).Det / det;
double y = -new Matrix3d(new[] { A, D, C }, new[] { s2.A, s2.D, s2.C }, new[] { s3.A, s3.D, s3.C }).Det / det;
double z = -new Matrix3d(new[] { A, B, D }, new[] { s2.A, s2.B, s2.D }, new[] { s3.A, s3.B, s3.D }).Det / det;
return(new Point3d(x, y, z));
}
}
/// <summary>
/// <para>Test if point is located in the epsilon neighborhood of the plane.</para>
/// <para>Epsilon neighborhood is defined by a GeometRi3D.Tolerance property.</para>
/// </summary>
public bool BelongsTo(Plane3d s)
{
s.SetCoord(this.Coord);
if (GeometRi3D.UseAbsoluteTolerance)
{
return(Abs(s.A * X + s.B * Y + s.C * Z + s.D) / Sqrt(s.A * s.A + s.B * s.B + s.C * s.C) < GeometRi3D.Tolerance);
}
else
{
double d = this.DistanceTo(this._coord.Origin);
if (d > 0.0)
{
return(Abs(s.A * X + s.B * Y + s.C * Z + s.D) / Sqrt(s.A * s.A + s.B * s.B + s.C * s.C) / d < GeometRi3D.Tolerance);
}
else
{
return(Abs(s.A * X + s.B * Y + s.C * Z + s.D) / Sqrt(s.A * s.A + s.B * s.B + s.C * s.C) < GeometRi3D.Tolerance);
}
}
}
/// <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.IsOnBoundary(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>
/// 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));
}