public void EigenValueDecomposition2()
{
double[][] evals = { new double[] { 0.0, 1.0, 0.0, 0.0 }, new double[] { 1.0, 0.0, 2e-7, 0.0 }, new double[] { 0.0, -2e-7, 0.0, 1.0 }, new double[] { 0.0, 0.0, 1.0, 0.0 } };
GeneralMatrix A = new GeneralMatrix(evals);
EigenvalueDecomposition Eig = A.Eigen();
GeneralMatrix D = Eig.D;
GeneralMatrix V = Eig.GetV();
Assert.IsTrue(GeneralTests.Check(A.Multiply(V), V.Multiply(D)));
}
private static void CalculatePCA(List <Point3D> _points, out Point3D pivotO,
out Vector3D vecX, out Vector3D vecY, out Vector3D vecZ)
{
pivotO = new Point3D(0, 0, 0);
vecX = new Vector3D(0, 0, 0);
vecY = new Vector3D(0, 0, 0);
vecZ = new Vector3D(0, 0, 0);
if (_points == null || _points.Count < 1)
{
return;
}
Point3D pivot = GeometricTransforms.GetPivot(_points);
pivotO = new Point3D(pivot.X, pivot.Y, pivot.Z);
List <Vector3D> point_deviations = _points.Select(x => x - pivot).ToList();
int nrP = _points.Count;
#region COVARIANCE:Old
//// compute the covariance matrix
//double[] m = new double[3*nrP];
//for(int i = 0; i < nrP; i++)
//{
// m[i*3] = point_deviations[i].X;
// m[i*3 + 1] = point_deviations[i].Y;
// m[i*3 + 2] = point_deviations[i].Z;
//}
//MatrixNxN M = new MatrixNxN(nrP, 3, m);
//MatrixNxN MtxM = MatrixNxN.Squared(M);
//MtxM.Scale(1.0 / nrP);
#endregion
// compute the covariance matrix ...
// using 3rd party library DotNetMatrix
double[][] pd_as_array = new double[nrP][];
for (int i = 0; i < nrP; i++)
{
pd_as_array[i] = new double[] { point_deviations[i].X, point_deviations[i].Y, point_deviations[i].Z };
}
GeneralMatrix gm_M = new GeneralMatrix(pd_as_array);
GeneralMatrix gm_Mt = gm_M.Transpose();
GeneralMatrix gm_Msq = gm_Mt.Multiply(gm_M);
GeneralMatrix gm_Msqn = gm_Msq.Multiply(1.0 / nrP);
// extract the sorted Eigenvalues of the matrix...
// using 3rd party library DotNetMatrix
EigenvalueDecomposition decomp = gm_Msqn.Eigen();
GeneralMatrix gm_EVec = decomp.GetV();
double[] gm_EVal = decomp.RealEigenvalues;
// from smallest to largest eigenvalue
vecX = new Vector3D(gm_EVec.GetElement(0, 0), gm_EVec.GetElement(1, 0), gm_EVec.GetElement(2, 0));
vecY = new Vector3D(gm_EVec.GetElement(0, 1), gm_EVec.GetElement(1, 1), gm_EVec.GetElement(2, 1));
vecZ = new Vector3D(gm_EVec.GetElement(0, 2), gm_EVec.GetElement(1, 2), gm_EVec.GetElement(2, 2));
}
public void CalculateEigenvalueDecomposition()
{
double[] i = new double[3];
double[] j = new double[3];
double[] k = new double[3];
double[] u = new double[3];
double[] t_vol = new double[3];
double[] vol = new double[3];
vol[0] = 0;
vol[1] = 0;
vol[2] = 0;
double[] func_sum = new double[3];
func_sum[0] = 0;
func_sum[1] = 0;
func_sum[2] = 0;
double[] func_sum_inertia = new double[3];
func_sum_inertia[0] = 0;
func_sum_inertia[1] = 0;
func_sum_inertia[2] = 0;
double func_sum_xy = 0;
double func_sum_xz = 0;
double func_sum_yz = 0;
double surfaceArea = 0;
//loop through all the triangles
for (int count = 0; count < _connections.Length; count++)
{
//get the 3 points of the triangle
double[] p1 = new double[3];
double[] p2 = new double[3];
double[] p3 = new double[3];
//Fix so its relative to the centroid center of mass
p1[0] = (double)_pts[_connections[count][0]][0] - _centroid[0]; //fix x's first
p2[0] = (double)_pts[_connections[count][1]][0] - _centroid[0];
p3[0] = (double)_pts[_connections[count][2]][0] - _centroid[0];
p1[1] = (double)_pts[_connections[count][0]][1] - _centroid[1]; //fix y's
p2[1] = (double)_pts[_connections[count][1]][1] - _centroid[1];
p3[1] = (double)_pts[_connections[count][2]][1] - _centroid[1];
p1[2] = (double)_pts[_connections[count][0]][2] - _centroid[2]; //fix z's
p2[2] = (double)_pts[_connections[count][1]][2] - _centroid[2];
p3[2] = (double)_pts[_connections[count][2]][2] - _centroid[2];
//calculate the i, j, k vectors
i[0] = p2[0] - p1[0]; j[0] = p2[1] - p1[1]; k[0] = p2[2] - p1[2];
i[1] = p3[0] - p1[0]; j[1] = p3[1] - p1[1]; k[1] = p3[2] - p1[2];
i[2] = p3[0] - p2[0]; j[2] = p3[1] - p2[1]; k[2] = p3[2] - p2[2];
//cross product between two vectors, to determine normal vector
u[0] = j[0] * k[1] - k[0] * j[1];
u[1] = k[0] * i[1] - i[0] * k[1];
u[2] = i[0] * j[1] - j[0] * i[1];
//Normalize vector to 1
double norm = Math.Sqrt(u[0] * u[0] + u[1] * u[1] + u[2] * u[2]);
if (norm != 0.0)
{
u[0] = u[0] / norm;
u[1] = u[1] / norm;
u[2] = u[2] / norm;
}
else
{
u[0] = 0.0;
u[1] = 0.0;
u[2] = 0.0;
}
//This is reduced to ...
//area of a triangle...
double a = Math.Sqrt(i[1] * i[1] + j[1] * j[1] + k[1] * k[1]);
double b = Math.Sqrt(i[0] * i[0] + j[0] * j[0] + k[0] * k[0]);
double c = Math.Sqrt(i[2] * i[2] + j[2] * j[2] + k[2] * k[2]);
double s = 0.5 * (a + b + c);
double area = Math.Sqrt(Math.Abs(s * (s - a) * (s - b) * (s - c)));
//patches(count,1) = area
//
surfaceArea += area;
//volume elements ...
double zavg = (p1[2] + p2[2] + p3[2]) / 3.0;
double yavg = (p1[1] + p2[1] + p3[1]) / 3.0;
double xavg = (p1[0] + p2[0] + p3[0]) / 3.0;
//sum of function for centroid calculation
func_sum[0] += t_vol[0] * xavg;
func_sum[1] += t_vol[1] * yavg;
func_sum[2] += t_vol[2] * zavg;
//sum of function for inertia calculation
func_sum_inertia[0] += area * u[0] * xavg * xavg * xavg;
func_sum_inertia[1] += area * u[1] * yavg * yavg * yavg;
func_sum_inertia[2] += area * u[2] * zavg * zavg * zavg;
//sum of function for products of inertia calculation
func_sum_xz += area * u[0] * xavg * xavg * zavg;
func_sum_xy += area * u[1] * yavg * yavg * xavg;
func_sum_yz += area * u[2] * zavg * zavg * yavg;
}
func_sum_inertia[0] /= 3;
func_sum_inertia[1] /= 3;
func_sum_inertia[2] /= 3;
double Ixy = -1 * func_sum_xy / 2;
double Ixz = -1 * func_sum_xz / 2;
double Iyz = -1 * func_sum_yz / 2;
double Iyx = Ixy;
double Izx = Ixz;
double Izy = Iyz;
double Ixx = func_sum_inertia[1] + func_sum_inertia[2];
double Iyy = func_sum_inertia[0] + func_sum_inertia[2];
double Izz = func_sum_inertia[0] + func_sum_inertia[1];
GeneralMatrix i_CoM = new GeneralMatrix(3, 3);
i_CoM.Array[0][0] = Ixx;
i_CoM.Array[0][1] = Ixy;
i_CoM.Array[0][2] = Ixz;
i_CoM.Array[1][0] = Iyx;
i_CoM.Array[1][1] = Iyy;
i_CoM.Array[1][2] = Iyz;
i_CoM.Array[2][0] = Izx;
i_CoM.Array[2][1] = Izy;
i_CoM.Array[2][2] = Izz;
EigenvalueDecomposition eig = i_CoM.Eigen();
_eigenvalues = eig.D;
_eigenvectors = eig.GetV();
//make sure this is a right handed matrix
if (_eigenvectors.Determinant() < 0)
{
_eigenvectors = _eigenvectors.Multiply(-1);
}
}
