/// <summary>
/// Assume the following structure, return the determinant coeficients for v0, v1, v2, v3
/// v0 v1 v2 v3
/// x00 x01 x02 x03
/// x10 x11 x12 x13
/// x20 x21 x22 x23
/// </summary>
/// <param name="matrix"></param>
/// <returns></returns>
private static Unity.Mathematics.float4 determinantCoef(Unity.Mathematics.float4x3 matrix)
{
Unity.Mathematics.float4 bottomRow = matrix.c2;
float[] determinants = Determinant2X2(matrix.c0, matrix.c1);
return(new Unity.Mathematics.float4(
bottomRow.y * determinants[5] - bottomRow.z * determinants[4] + bottomRow.w * determinants[3],
-(bottomRow.x * determinants[5] - bottomRow.z * determinants[2] +
bottomRow.w * determinants[3]),
bottomRow.x * determinants[4] - bottomRow.y * determinants[2] + bottomRow.w * determinants[0],
-(bottomRow.x * determinants[3] - bottomRow.y * determinants[1] + bottomRow.z * determinants[0])
));
}
/// <summary>
/// Return the basis of a hyper plane orthagonal to a given vector
/// </summary>
/// <param name="v"></param>
/// <returns></returns>
public static Unity.Mathematics.float4x3 basisSystem(this Unity.Mathematics.float4 v)
{
Unity.Mathematics.math.normalize(v);
//use method described here: https://www.geometrictools.com/Documentation/OrthonormalSets.pdf
if (v.x == 0 && v.y == 0 && v.z == 0 && v.w == 0)
{
UnityEngine.Debug.LogError("Can't form basis from zero vector");
}
//the vector is the first basis vector for the 4-space, orthag to the hyperplane
Unity.Mathematics.math.normalize(v);
//establish a second basis vector
Unity.Mathematics.float4 basis0;
if (v.x != 0 || v.y != 0)
{
basis0 = new UnityEngine.Vector4(v.y, v.x, 0, 0);
}
else
{
basis0 = new UnityEngine.Vector4(0, 0, v.w, v.z);
}
Unity.Mathematics.math.normalize(basis0);
float[] determinants = Determinant2X2(v, basis0);
//index of largest determinant
int idx = 0;
for (int i = 0; i < 6; i++)
{
if (determinants[i] > determinants[idx])
{
idx = i;
}
}
if (determinants[idx] != 0)
{
UnityEngine.Debug.LogError("No non-zero determinant");
}
//choose bottom row of det matrix to generate next basis vector
Unity.Mathematics.float4 bottomRow;
if (idx == 0 || idx == 1 || idx == 3)
{
bottomRow = new Unity.Mathematics.float4(0, 0, 0, 1);
}
else if (idx == 2 || idx == 4)
{
bottomRow = new Unity.Mathematics.float4(0, 0, 1, 0);
}
else
{
bottomRow = new Unity.Mathematics.float4(0, 1, 0, 0);
}
Unity.Mathematics.float4 basis1 = determinantCoef(new Unity.Mathematics.float4x3(v, basis0, bottomRow));
Unity.Mathematics.math.normalize(basis1);
Unity.Mathematics.float4 basis2 = determinantCoef(new Unity.Mathematics.float4x3(v, basis0, basis1));
Unity.Mathematics.math.normalize(basis2);
//returns the basis that spans the hyperplane orthogonal to v
Unity.Mathematics.float4x3 basis = new Unity.Mathematics.float4x3(basis0, basis1, basis2);
//check that v is orthogonal.
// v.projectDownDimension(basis, ProjectionMethod.parallel, null, null, null);
// if (v.x != 0 || v.y != 0 || v.z != 0) UnityEngine.Debug.LogError("Basis is not orthogonal to v");
return(basis);
}
