/// <summary> /// Orthogonal projection of ellipsoid to line. /// </summary> public Segment3d ProjectionTo(Line3d l) { //Stephen B. Pope "Algorithms for Ellipsoids" // https://tcg.mae.cornell.edu/pubs/Pope_FDA_08.pdf Coord3d lc = new Coord3d(_point, _v1, _v2); Point3d x0 = l.Point.ConvertTo(lc); Vector3d v = l.Direction.ConvertTo(lc); Matrix3d L_T = Matrix3d.DiagonalMatrix(this.A, this.B, this.C); Vector3d c = new Vector3d(0.0, 0.0, 0.0, lc); double s0 = v * (c - x0.ToVector) / (v * v); Vector3d w = L_T * v / (v * v); Point3d P1 = x0.Translate((s0 + w.Norm) * v); Point3d P2 = x0.Translate((s0 - w.Norm) * v); return(new Segment3d(P1, P2)); }
/// <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); } }