public static Matrix4D operator *( Matrix4D m, double s )
{
Matrix4D result = new Matrix4D();
for( int i=0; i<4; i++ )
for( int j=0; j<4; j++ )
result[i, j] = m[i, j] * s;
return result;
}
public static Matrix4D operator *( Matrix4D m1, Matrix4D m2 )
{
Matrix4D result = new Matrix4D();
for( int i=0; i<4; i++ )
for( int j=0; j<4; j++ )
for( int k=0; k<4; k++ )
result[i, j] += m1[i, k] * m2[k, j];
return result;
}
/// <summary>
/// Handles updating our rotation matrices based on mouse dragging.
/// </summary>
public void MouseDragged( double dx, double dy,
bool xz_yz, bool xw_yw, bool xy_zw )
{
Matrix4D spinDelta = new Matrix4D();
// Sensitivity.
dx *= 0.012;
dy *= 0.012;
if( xz_yz )
{
spinDelta[0,2] += dx;
spinDelta[2,0] -= dx;
spinDelta[1,2] += dy;
spinDelta[2,1] -= dy;
}
if( xw_yw )
{
spinDelta[0,3] -= dx;
spinDelta[3,0] += dx;
spinDelta[1,3] -= dy;
spinDelta[3,1] += dy;
}
if( xy_zw )
{
spinDelta[0,1] += dx;
spinDelta[1,0] -= dx;
spinDelta[3,2] -= dy;
spinDelta[2,3] += dy;
}
ApplySpinDelta( spinDelta );
}
public RotationHandler4D( Matrix4D initialMatrix )
{
Current4dView = initialMatrix;
}
private void ApplySpinDelta( Matrix4D spinDelta )
{
Matrix4D delta = Matrix4D.Identity() + spinDelta;
delta = Matrix4D.GramSchmidt( delta ); // Orthonormalize
ViewMat4d = delta * ViewMat4d;
ViewMat4d = Matrix4D.GramSchmidt( ViewMat4d ); // Orthonormalize
}
public static Matrix4D operator +( Matrix4D m1, Matrix4D m2 )
{
Matrix4D result = new Matrix4D();
for( int i=0; i<4; i++ )
for( int j=0; j<4; j++ )
result[i, j] = m1[i, j] + m2[i, j];
return result;
}
public Matrix4D Clone()
{
Matrix4D result = new Matrix4D();
for( int i=0; i<4; i++ )
for( int j=0; j<4; j++ )
result.Data[i][j] = this.Data[i][j];
return result;
}
public static Matrix4D Identity()
{
Matrix4D result = new Matrix4D();
for( int i=0; i<4; i++ )
result[i,i] = 1;
return result;
}
public static Matrix4D Transpose( Matrix4D m )
{
Matrix4D result = new Matrix4D();
for( int i=0; i<4; i++ )
for( int j=0; j<4; j++ )
result[i, j] = m[j, i];
return result;
}
/// <summary>
/// Gram-Schmidt orthonormalize
/// </summary>
public static Matrix4D GramSchmidt( Matrix4D input,
Func<VectorND, VectorND, double> innerProduct, Func<VectorND, VectorND> normalize )
{
Matrix4D result = input;
for( int i=0; i<4; i++ )
{
for( int j=i+1; j<4; j++ )
{
VectorND iVec = result[i];
VectorND jVec = result[j];
iVec -= innerProduct( iVec, jVec ) * jVec;
result[i] = iVec;
}
result[i] = normalize( result[i] );
}
return result;
}
/// <summary>
/// Gram-Schmidt orthonormalize
/// </summary>
public static Matrix4D GramSchmidt( Matrix4D input )
{
Matrix4D result = input;
for( int i=0; i<4; i++ )
{
for( int j=0; j<i; j++ )
{
// result[j] is already unit length...
// result[i] -= (result[i] dot result[j])*result[j]
VectorND iVec = result[i];
VectorND jVec = result[j];
iVec -= ( iVec.Dot( jVec ) ) * jVec;
result[i] = iVec;
}
result[i].Normalize();
}
return result;
}
public static Matrix4D ReverseRows( Matrix4D m )
{
Matrix4D result = new Matrix4D();
for( int i=0; i<4; i++ )
result[i] = m[3-i];
return result;
}
public static Matrix4D GramSchmidt( Matrix4D input, Matrix4D innerProductValues )
{
Matrix4D result = input.Clone();
for( int i = 0; i < 4; i++ )
{
for( int j = 0; j < i; j++ )
{
VectorND iVec = result[i];
VectorND jVec = result[j];
double inner = innerProductValues[i].Dot( jVec );
iVec -= inner * jVec;
result[i] = iVec;
}
// Normalize. We don't use VectorND normalize because we might have timelike vectors.
double mag2 = innerProductValues[i].Dot( result[i] );
double abs = mag2 < 0 ? -Math.Sqrt( -mag2 ) : Math.Sqrt( mag2 );
result[i].Divide( abs );
}
return result;
}
/// <summary>
/// This calculates the 4 vertices of a general (but finite) Goursat Tetrahedron. Result is in the ball model.
///
/// The method comes from the dissertation "Hyperbolic polyhedra: volume and scissors congruence",
/// by Yana Zilberberg Mohanty, section 2.4, steps 1-5.
///
/// A,B,C are the three dihedral angles surrounding a vertex.
/// A_,B_,C_ are the three oppoite dihedral angles.
/// </summary>
public static Vector3D[] GoursatTetrahedron( double A, double B, double C, double A_, double B_, double C_ )
{
// Step 1: Construct Gram matrix with reversed rows/columns.
// NOTE: The sign of the diagonal in the paper was incorrect.
double pi = Math.PI;
double[,] gramMatrixData = new double[,]
{
{ -1, Math.Cos(pi/A_), Math.Cos(pi/B_), Math.Cos(pi/C) },
{ Math.Cos(pi/A_), -1, Math.Cos(pi/C_), Math.Cos(pi/B) },
{ Math.Cos(pi/B_), Math.Cos(pi/C_), -1, Math.Cos(pi/A) },
{ Math.Cos(pi/C), Math.Cos(pi/B), Math.Cos(pi/A), -1 },
};
Matrix4D gramMatrix = new Matrix4D( gramMatrixData );
gramMatrix *= -1;
Matrix4D identity = Matrix4D.Identity();
// Step 2: Gram-Schmidt.
Matrix4D W = GramSchmidt( identity, gramMatrix );
// Step 3: Divide 4th row by i (already effectively done in our Gram-Schmidt routine below), and reverse order of rows.
Matrix4D W_ = ReverseRows( W );
// Step 4
Matrix4D D = identity.Clone();
D[0, 0] = -1;
Matrix4D U = Matrix4D.Transpose( D * W_ );
// Step 5
Matrix4D U_ = ReverseRows( U );
for( int i=0; i<4; i++ )
MinkowskiNormalize( U_[i] );
// Now move from the hyperboloid model to the ball.
List<Vector3D> result = new List<Vector3D>();
for( int i=0; i<4; i++ )
result.Add( HyperboloidToBall( U_[i] ) );
return result.ToArray();
}