计算表面包含4个点的球体中心（C＃）

我正在使用名为MIConvexHull的3D Voronoi库，它为3D空间中的一系列点计算3D Voronoi图。 但是，它没有提供有关Voronoi图结构的高级信息; 报告的边是简单的坐标对列表，然后必须计算圆周。

现在，该库提供了一系列2D点的圆周计算的实现。 如您所见，显示了开始（橙色）和结束（绿色）的坐标对：

您可以直观地看到，如果您获取每个边缘中列出的顶点并创建一个圆形，使得该圆的圆周接触所有边缘，则中心就是边缘开始的位置。

我遇到的问题是我的点是3D，因此它不会是返回的圆心，而是球体的中心。 不幸的是，高级数学不是我的脑袋能够真正处理好的，所以我不知道如何解决这个问题。

如果给出3D空间中的4个点，我是否可以获得球体的中心，使得所有点都位于球体的表面上？

编辑：在3D中，将提供4个点，而不是3个。

Answer 1

这是一个Javascript实现：

http://www.convertalot.com/sphere_solver.html

还有一些数学解释：

http://steve.hollasch.net/cgindex/geometry/sphere4pts.html

通过将以下行列式设置为零来给出球体……的等式：

| x^2 + y^2 + z^2 xyz 1 | | x1^2 + y1^2 + z1^2 x1 y1 z1 1 | | x2^2 + y2^2 + z2^2 x2 y2 z2 1 | = 0. | x3^2 + y3^2 + z3^2 x3 y3 z3 1 | | x4^2 + y4^2 + z4^2 x4 y4 z4 1 | 

Answer 2

我将上面链接的Javascript实现转换为C＃。 这里是：

 /// 
 /// Given four points in 3D space, solves for a sphere such that all four points /// lie on the sphere's surface. /// 
 ///  /// Translated from Javascript on http://www.convertalot.com/sphere_solver.html, originally /// linked to by http://stackoverflow.com/questions/13600739/calculate-centre-of-sphere-whose-surface-contains-4-points-c. ///  public class CircumcentreSolver { private const float ZERO = 0; private double m_X0, m_Y0, m_Z0; private double m_Radius; private double[,] P = { { ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO } }; /// 
 /// The centre of the resulting sphere. /// 
 public double[] Centre { get { return new double[] { this.m_X0, this.m_Y0, this.m_Z0 }; } } /// 
 /// The radius of the resulting sphere. /// 
 public double Radius { get { return this.m_Radius; } } /// 
 /// Whether the result was a valid sphere. /// 
 public bool Valid { get { return this.m_Radius != 0; } } /// 
 /// Computes the centre of a sphere such that all four specified points in /// 3D space lie on the sphere's surface. /// 
 /// The first point (array of 3 doubles for X, Y, Z). /// The second point (array of 3 doubles for X, Y, Z). /// The third point (array of 3 doubles for X, Y, Z). /// The fourth point (array of 3 doubles for X, Y, Z). public CircumcentreSolver(double[] a, double[] b, double[] c, double[] d) { this.Compute(a, b, c, d); } /// 
 /// Evaluate the determinant. /// 
 private void Compute(double[] a, double[] b, double[] c, double[] d) { P[0, 0] = a[0]; P[0, 1] = a[1]; P[0, 2] = a[2]; P[1, 0] = b[0]; P[1, 1] = b[1]; P[1, 2] = b[2]; P[2, 0] = c[0]; P[2, 1] = c[1]; P[2, 2] = c[2]; P[3, 0] = d[0]; P[3, 1] = d[1]; P[3, 2] = d[2]; // Compute result sphere. this.Sphere(); } private void Sphere() { double r, m11, m12, m13, m14, m15; double[,] a = { { ZERO, ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO, ZERO } }; // Find minor 1, 1. for (int i = 0; i < 4; i++) { a[i, 0] = P[i, 0]; a[i, 1] = P[i, 1]; a[i, 2] = P[i, 2]; a[i, 3] = 1; } m11 = this.Determinant(a, 4); // Find minor 1, 2. for (int i = 0; i < 4; i++) { a[i, 0] = P[i, 0] * P[i, 0] + P[i, 1] * P[i, 1] + P[i, 2] * P[i, 2]; a[i, 1] = P[i, 1]; a[i, 2] = P[i, 2]; a[i, 3] = 1; } m12 = this.Determinant(a, 4); // Find minor 1, 3. for (int i = 0; i < 4; i++) { a[i, 0] = P[i, 0] * P[i, 0] + P[i, 1] * P[i, 1] + P[i, 2] * P[i, 2]; a[i, 1] = P[i, 0]; a[i, 2] = P[i, 2]; a[i, 3] = 1; } m13 = this.Determinant(a, 4); // Find minor 1, 4. for (int i = 0; i < 4; i++) { a[i, 0] = P[i, 0] * P[i, 0] + P[i, 1] * P[i, 1] + P[i, 2] * P[i, 2]; a[i, 1] = P[i, 0]; a[i, 2] = P[i, 1]; a[i, 3] = 1; } m14 = this.Determinant(a, 4); // Find minor 1, 5. for (int i = 0; i < 4; i++) { a[i, 0] = P[i, 0] * P[i, 0] + P[i, 1] * P[i, 1] + P[i, 2] * P[i, 2]; a[i, 1] = P[i, 0]; a[i, 2] = P[i, 1]; a[i, 3] = P[i, 2]; } m15 = this.Determinant(a, 4); // Calculate result. if (m11 == 0) { this.m_X0 = 0; this.m_Y0 = 0; this.m_Z0 = 0; this.m_Radius = 0; } else { this.m_X0 = 0.5 * m12 / m11; this.m_Y0 = -0.5 * m13 / m11; this.m_Z0 = 0.5 * m14 / m11; this.m_Radius = System.Math.Sqrt(this.m_X0 * this.m_X0 + this.m_Y0 * this.m_Y0 + this.m_Z0 * this.m_Z0 - m15 / m11); } } /// 
 /// Recursive definition of determinate using expansion by minors. /// 
 private double Determinant(double[,] a, double n) { int i, j, j1, j2; double d = 0; double[,] m = { { ZERO, ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO, ZERO }, { ZERO, ZERO, ZERO, ZERO } }; if (n == 2) { // Terminate recursion. d = a[0, 0] * a[1, 1] - a[1, 0] * a[0, 1]; } else { d = 0; for (j1 = 0; j1 < n; j1++) // Do each column. { for (i = 1; i < n; i++) // Create minor. { j2 = 0; for (j = 0; j < n; j++) { if (j == j1) continue; m[i - 1, j2] = a[i, j]; j2++; } } // Sum (+/-)cofactor * minor. d = d + System.Math.Pow(-1.0, j1) * a[0, j1] * this.Determinant(m, n - 1); } } return d; } }