/// <summary>
/// Helper function which takes the blades listed in 'B' and returns them sorted by grade
/// in a double array. The relative order of blades of each grade left constant.
/// </summary>
/// <param name="B"></param>
/// <param name="spaceDim">dimension of space of the algebra; used to determine the length of the returned double array.</param>
/// <returns>'B' in an array of arrays, sorted by grade.</returns>
private RefGA.BasisBlade[][] SortByGrade(RefGA.BasisBlade[] B, int spaceDim)
{
RefGA.BasisBlade[][] L = new RefGA.BasisBlade[spaceDim + 1][];
{ // count, allocate number of elements of each grade
// count number of blades for each grade
int[] count = new int[spaceDim + 1];
for (int i = 0; i < B.Length; i++)
{
if (B[i].Grade() >= count.Length)
{
throw new Exception("G25.OM.SortByGrade(): unexpectedly high grade of basis blade: " + B[i].Grade());
}
else
{
count[B[i].Grade()]++;
}
}
// allocate
for (int g = 0; g <= spaceDim; g++)
{
L[g] = new RefGA.BasisBlade[count[g]];
}
}
// sort all blades:
int[] idx = new int[spaceDim + 1];
for (int i = 0; i < B.Length; i++)
{
L[B[i].Grade()][idx[B[i].Grade()]] = B[i];
idx[B[i].Grade()]++;
}
return(L);
}
/// <summary>
/// Little helper function for constructor.
/// </summary>
/// <returns>'basisBlades', but inside a newly allocated array of length 1.</returns>
protected static RefGA.BasisBlade[][] ToDoubleArray(G25.rsbbp.BasisBlade[] basisBlades)
{
RefGA.BasisBlade[][] B = new RefGA.BasisBlade[1][];
B[0] = new RefGA.BasisBlade[basisBlades.Length];
for (int i = 0; i < basisBlades.Length; i++)
{
B[0][i] = basisBlades[i].GetBasisBlade;
}
return(B);
}
/// <summary>
/// Returns the group index of basis blade 'B' or -1 when not represented by this MV.
///
/// Example: suppose you want to know the group and element index of B=e2^e3^no,
/// then (assuming it is represented), it is in m_basisBlades[GetGroupIdx(B)][GetElementIdx(B)].
/// </summary>
/// <param name="B">The basis blade whose bitmap is looked up.</param>
/// <returns>-1 if 'B' is cannot be represented by this MV, otherwise, the index of the group.</returns>
public int GetGroupIdx(RefGA.BasisBlade B)
{
if ((B.bitmap >= m_bitmapToGroup.Length) || (m_bitmapToGroup[B.bitmap] < 0))
{
return(-1);
}
else
{
return(m_bitmapToGroup[B.bitmap] >> 16);
}
}
/// <summary>
/// Returns the element index of basis blade 'B' or -1 when not represented by this MV.
///
/// Example: suppose you want to know the group and element index of B=e2^e3^no,
/// then (assuming it is represented), it is in BasisBlade(groupIdx, elemIdx).
/// </summary>
/// <param name="B">The basis blade whose bitmap is looked up.</param>
/// <returns>-1 if 'B' is cannot be represented by this MV, otherwise, the index of the element in the group.</returns>
public int GetElementIdx(RefGA.BasisBlade B)
{
if ((B.bitmap >= m_bitmapToGroup.Length) || (m_bitmapToGroup[B.bitmap] < 0))
{
return(-1);
}
else
{
return(m_bitmapToGroup[B.bitmap] & 0xFFFF);
}
}
/// <summary>
/// Constructor. Do not use directly. Use the constructors of G25.GMV and
/// G25.SMV instead.
///
/// If the multivector is specialized, basisBlades.Length must be 1.
/// </summary>
/// <param name="specialized">Whether this is a specialized MV or not.</param>
/// <param name="name">The name of the multivector, for example "mv" or "rotor".</param>
/// <param name="basisBlades">The basis blades, by group. Each entry in the array is a group of coordinates.
/// Specialized multivector have only one group.</param>
protected MV(bool specialized, String name, RefGA.BasisBlade[][] basisBlades)
{
if (specialized && (basisBlades.Length != 1))
{
throw new Exception("G25.MV(): specialized multivector classes must have one group");
}
m_specialized = specialized;
m_name = name;
m_basisBlades = new RefGA.BasisBlade[basisBlades.Length][];
for (int i = 0; i < basisBlades.Length; i++)
{
m_basisBlades[i] = (RefGA.BasisBlade[])basisBlades[i].Clone();
}
{ // init m_bitmapToGroup
// get maximum dimension used in any of the basis blades:
int maxDim = 0;
for (int i = 0; i < m_basisBlades.Length; i++)
{
for (int j = 0; j < m_basisBlades[i].Length; j++)
{
int d = RefGA.Bits.HighestOneBit(m_basisBlades[i][j].bitmap);
if (d > maxDim)
{
maxDim = d;
}
}
}
// allocate m_bitmapToGroup, set all to -1
m_bitmapToGroup = new int[1 << (maxDim + 1)];
for (int i = 0; i < m_bitmapToGroup.Length; i++)
{
m_bitmapToGroup[i] = -1;
}
// mark used position with group and element index
for (int i = 0; i < m_basisBlades.Length; i++)
{
for (int j = 0; j < m_basisBlades[i].Length; j++)
{
m_bitmapToGroup[m_basisBlades[i][j].bitmap] = (i << 16) | j;
}
}
}
}
/// <summary>
/// Returns highest grade basis blade in this MV.
/// </summary>
/// <returns>highest grade basis blade in this MV.</returns>
public int HighestGrade()
{
int hg = -1;
for (int g = 0; g < m_basisBlades.Length; g++)
{
for (int e = 0; e < m_basisBlades[g].Length; e++)
{
RefGA.BasisBlade srcB = m_basisBlades[g][e];
if (srcB.Grade() > hg)
{
hg = srcB.Grade();
}
}
}
return(hg);
}
/// <summary>
/// Returns lowest grade basis blade in this MV.
/// </summary>
/// <returns>lowest grade basis blade in this MV.</returns>
public int LowestGrade()
{
int lg = int.MaxValue;
for (int g = 0; g < m_basisBlades.Length; g++)
{
for (int e = 0; e < m_basisBlades[g].Length; e++)
{
RefGA.BasisBlade srcB = m_basisBlades[g][e];
if (srcB.Grade() < lg)
{
lg = srcB.Grade();
}
}
}
return(lg);
}
/// <summary>
/// Generates a map. For each coordinate in 'src', tells you where it is found in 'dst',
/// or that it is not present.
/// </summary>
/// <param name="src">Src multivector (can be SMV or GMV).</param>
/// <param name="dst">Dst multivector (can be SMV or GMV).</param>
/// <returns>The first entry of each pair is the group, the second entry is the entry.</returns>
public static Dictionary <Tuple <int, int>, Tuple <int, int> > GetCoordMap(MV src, MV dst)
{
Dictionary <Tuple <int, int>, Tuple <int, int> > D = new Dictionary <Tuple <int, int>, Tuple <int, int> >();
for (int g = 0; g < src.m_basisBlades.Length; g++)
{
for (int e = 0; e < src.m_basisBlades[g].Length; e++)
{
RefGA.BasisBlade srcB = src.m_basisBlades[g][e];
int groupIdx = dst.GetGroupIdx(srcB);
if (groupIdx >= 0)
{
int elementIdx = dst.GetElementIdx(srcB);
D[new Tuple <int, int>(g, e)] = new Tuple <int, int>(groupIdx, elementIdx);
}
}
}
return(D);
}
/// <summary>
/// This function takes the domain or range arrays (<c>RefGA.BasisBlade[][]</c>)
/// and returns the basis vectors which span this domain or range.
///
/// If <c>symbolicName</c> is not null, the vectors are given symbolic coordinates,
/// <c>symbolicName0</c>, <c>symbolicName1</c>, and so on.
/// </summary>
/// <param name="D">The double array (typically domain or range of this OM)</param>
/// <param name="symbolicName">Can be null. Optional symbolic coordinate base.</param>
/// <returns></returns>
protected static RefGA.BasisBlade[] GetVectors(RefGA.BasisBlade[][] D, string symbolicName)
{ // find out which vectors are in range, and create array of basis blades representing it as a vector type
// get union of all bitmaps
uint unionBitmap = 0;
for (int g = 0; g < D.Length; g++)
{
for (int c = 0; c < D[g].Length; c++)
{
unionBitmap |= D[g][c].bitmap;
}
}
// allocate memory for vectors
uint nbVectors = RefGA.Bits.BitCount(unionBitmap);
RefGA.BasisBlade[] vectors = new RefGA.BasisBlade[nbVectors];
// init vectors
uint v = 1;
int idx = 0;
while (v <= unionBitmap)
{
if ((v & unionBitmap) != 0)
{
if (symbolicName == null)
{
vectors[idx] = new RefGA.BasisBlade(v);
}
else
{
vectors[idx] = new RefGA.BasisBlade(v, 1.0, symbolicName + idx); // note the symbolic coordinate (for easy FindTightestMatch())
}
idx++;
}
v = v << 1;
}
return(vectors);
}
/// <summary>Shortcut to ScalarProduct() </summary>
public static BasisBlade scp(BasisBlade A, BasisBlade B, double[] m)
{
return ScalarProduct(A, B, m);
}
/// <summary>Shortcut to ScalarProduct() </summary>
public static ArrayList scp(BasisBlade A, BasisBlade B, Metric M)
{
return ScalarProduct(A, B, M);
}
/// <summary>
/// Returns the 'Hadamard product' of A and B.
/// If A and B are the same basis blades up to scale, then multiplies
/// the scalar parts of A and B with the unit basis blade.
/// Otherwise returns 0.
/// </summary>
/// <param name="A">input blade.</param>
/// <param name="B">input blade.</param>
/// <returns>'Hadamard' product of A and B.</returns>
public static BasisBlade HadamardProduct(BasisBlade A, BasisBlade B)
{
if (A.bitmap != B.bitmap) return BasisBlade.ZERO;
uint bitmap = 0;
B = new BasisBlade(B, bitmap); // get a copy of 'B', but as a scalar (0), so we can multiply the scalar part
return BasisBlade.gp(A, B);
}
/// <summary>Alternative way of calling B.GradeInvolution()</summary>
public static BasisBlade GradeInvolution(BasisBlade B)
{
return B.GradeInvolution();
}
/// <summary>
/// Shortcut to GeometricProduct()
/// </summary>
public static BasisBlade gp(BasisBlade a, BasisBlade b)
{
return GeometricProduct(a, b);
}
/// <summary>
/// Computes the inner product of two basis blades in Euclidean metric.
/// </summary>
/// <param name="A">input blade</param>
/// <param name="B">input blade</param>
/// <param name="m">an array of doubles giving the metric for each basis vector.</param>
/// <param name="type">inner product type to be used. Use one of the InnerProductType constants:
/// LEFT_CONTRACTION, RIGHT_CONTRACTION, HESTENES_INNER_PRODUCT, MODIFIED_HESTENES_INNER_PRODUCT or SCALAR_PRODUCT.</param>
/// <returns>Inner product of <paramref name="A"/> and <paramref name="B"/>.</returns>
public static BasisBlade InnerProduct(BasisBlade A, BasisBlade B, double[] m, InnerProductType type)
{
return InnerProductFilter(A.Grade(), B.Grade(), GeometricProduct(A, B, m), type);
}
/// <summary>
/// Shortcut to OuterProduct()
/// </summary>
public static BasisBlade op(BasisBlade a, BasisBlade b)
{
return OuterProduct(a, b);
}
/// <summary>
/// Transforms BasisBlade <paramref name="a"/> to the orthogonal basis of this metric.
/// </summary>
/// <param name="a">The BasisBlade (must be on the regular basis).</param>
/// <returns><paramref name="a"/> on the new basis.</returns>
public ArrayList ToEigenbasis(BasisBlade a)
{
return Transform(a, m_invEigMatrix);
}
/// <summary>Shortcut to InnerProduct() </summary>
public static ArrayList ip(BasisBlade A, BasisBlade B, Metric M, InnerProductType type)
{
return InnerProduct(A, B, M, type);
}
/// <summary>Alternative way of calling B.Negate()</summary>
public static BasisBlade Negate(BasisBlade B)
{
return B.Negate();
}
/// <summary>Shortcut to InnerProduct() </summary>
public static BasisBlade ip(BasisBlade A, BasisBlade B, double[] m, InnerProductType type)
{
return InnerProduct(A, B, m, type);
}
/// <summary>
/// Returns the 'Inverse Hadamard product' of A and B.
/// If A and B are the same basis blades up to scale, then divides
/// the scalar part of A by the scalar part of B with the unit basis blade.
/// Otherwise returns 0.
///
/// Do not call when scale of B is zero (returned result will be zero).
/// </summary>
/// <param name="A">input blade.</param>
/// <param name="B">input blade. Must not be zero.</param>
/// <returns>'Inverse Hadamard' product of A and B.</returns>
public static BasisBlade InverseHadamardProduct(BasisBlade A, BasisBlade B)
{
if (A.bitmap != B.bitmap) return BasisBlade.ZERO;
uint bitmap = 0;
if (B.IsSymbolic())
{
B = new BasisBlade(B, bitmap); // get a copy of 'B', but as a scalar (0), so we can multiply the scalar part
return BasisBlade.gp(A, new BasisBlade(0, 1.0, new Symbolic.UnaryScalarOp(Symbolic.UnaryScalarOp.INVERSE, new Multivector(B))));
}
else
{
double newScale = (B.scale == 0.0) ? 0.0 : A.scale / B.scale;
return new BasisBlade(A.bitmap, newScale, A.symScale);
}
}
/// <summary>
/// Computes the inner product of two basis blades in Euclidean metric.
/// </summary>
/// <param name="A">input blade</param>
/// <param name="B">input blade</param>
/// <param name="M">The Metric to be used.</param>
/// <param name="type">inner product type to be used. Use one of the InnerProductType constants:
/// LEFT_CONTRACTION, RIGHT_CONTRACTION, HESTENES_INNER_PRODUCT, MODIFIED_HESTENES_INNER_PRODUCT or SCALAR_PRODUCT.</param>
/// <returns>Inner product of <paramref name="A"/> and <paramref name="B"/>.</returns>
public static ArrayList InnerProduct(BasisBlade A, BasisBlade B, Metric M, InnerProductType type)
{
return InnerProductFilter(A.Grade(), B.Grade(), GeometricProduct(A, B, M), type);
}
/// <summary>
/// Version of CompareTo() that also compares the symbolic scalar part (the standard CompareTo() does not do so)
/// </summary>
/// <param name="B">The object to which 'this' is compared</param>
public int SymbolicCompareTo(BasisBlade B)
{
{ // first do regular comparison:
int c = CompareTo(B);
if (c != 0) return c;
}
{ // now try symbolic part:
System.Collections.IComparer SSAC = new Util.SimplifySymbolicArrayComparer();
int l1 = (symScale == null) ? 0 : symScale.Length;
int l2 = (B.symScale == null) ? 0 : B.symScale.Length;
for (int i = 0; ((i < l1) && (i < l2)); i++)
{
// compare both arrays of scalar values:
int c = SSAC.Compare(symScale[i], B.symScale[i]);
if (c != 0) return c; // if not equal, we're done
}
// end of (one of the) arrays reached: compare length of arrays to make comparison
return ((l1 < l2) ? -1 :((l1 > l2) ? 1 : 0));
}
}
/// <summary>
/// Little helper function for the constructor.
/// </summary>
/// <returns>'basisBlades', but inside a newly allocated array of length 1.</returns>
protected static RefGA.BasisBlade[][] ToDoubleArray(RefGA.BasisBlade[] basisBlades)
{
RefGA.BasisBlade[][] B = new RefGA.BasisBlade[1][];
B[0] = basisBlades;
return B;
}
/// <summary>
/// Transforms a basis blade to another basis which is represented by the columns of M.
/// </summary>
/// <param name="a">The basis blade to transform according to <paramref name="M"/>.</param>
/// <param name="M">The matrix to use to transform <paramref name="a"/>.</param>
/// <returns>a list of BasisBlade whose sum represents <paramref name="a"/> on the new basis.</returns>
public static ArrayList Transform(BasisBlade a, DotNetMatrix.GeneralMatrix M)
{
ArrayList A = new ArrayList();
A.Add(new BasisBlade(0, a.scale, a.symScale)); // start with just the scalar part;
int dim = M.RowDimension;
// for each 1 bit: convert to list of blades
int i = 0;
uint b = a.bitmap;
while (b != 0) {
if ((b & 1) != 0) {
// take column 'i' out of the matrix, wedge it to 'A'
ArrayList tmp = new ArrayList();
for (int j = 0; j < dim; j++) {
double m = M.GetElement(j, i);
if (m != 0.0) {
for (int k = 0; k < A.Count; k++)
{
BasisBlade o = BasisBlade.op((BasisBlade)A[k], new BasisBlade((uint)(1 << j), m));
if (o.scale != 0.0) tmp.Add(o);
}
}
}
A = tmp;
}
b >>= 1;
i++;
}
return A;
}
/// <summary>Computes the geometric product of two basis blades in arbitary non-Euclidean metric.</summary>
/// <param name="_A">Basisblade</param>
/// <param name="_B">Basisblade</param>
/// <param name="M">Metric to be used</param>
/// <returns>An ArrayList, because the result may not be a single BasisBlade anymore. For example,
/// <c>no ni = no.ni + no^ni = -1 + no^ni</c>. </returns>
public static ArrayList GeometricProduct(BasisBlade _A, BasisBlade _B, Metric M)
{
// convert arguments to 'eigenbasis'
ArrayList A = M.ToEigenbasis(_A);
ArrayList B = M.ToEigenbasis(_B);
ArrayList result = new ArrayList();
double[] EM = M.EigenMetric;
foreach (BasisBlade bA in A)
foreach (BasisBlade bB in B)
{
BasisBlade C = GeometricProduct(bA, bB, EM);
if (C.scale != 0.0) result.Add(C);
}
return M.ToMetricBasis(result);
}
/// <summary>
/// Transforms BasisBlade <paramref name="a"/> to the regular basis of this metric.
/// </summary>
/// <param name="a">The BasisBlade (must be on the orthogonal 'eigen' basis).</param>
/// <returns><paramref name="a"/> on the new basis.</returns>
public ArrayList ToMetricBasis(BasisBlade a)
{
return Transform(a, m_eig.GetV());
}
/// <summary>
/// Returns the 'Hadamard' product of A and B.
/// If A and B are the same basis blades up to scale, then multiplies
/// the scalar parts of A and B with the unit basis blade.
/// Otherwise returns 0.
/// </summary>
/// <param name="A">input blade.</param>
/// <param name="B">input blade.</param>
/// <returns>'Hadamard' product of A and B.</returns>
public static BasisBlade hp(BasisBlade A, BasisBlade B)
{
return HadamardProduct(A, B);
}
/// <summary>Computes the outer product of two basis blades.</summary>
/// <returns>Outer product of two basis blades <paramref name="a"/> and <paramref name="b"/></returns>
public static BasisBlade OuterProduct(BasisBlade a, BasisBlade b)
{
const bool outer = true;
return gp_op(a, b, outer);
}
/// <summary>
/// Returns the 'Inverse Hadamard product' of A and B.
/// If A and B are the same basis blades up to scale, then divides
/// the scalar part of A by the scalar part of B with the unit basis blade.
/// Otherwise returns 0.
/// </summary>
/// <param name="A">input blade.</param>
/// <param name="B">input blade.</param>
/// <returns>'Inverse Hadamard' product of A and B.</returns>
public static BasisBlade ihp(BasisBlade A, BasisBlade B)
{
return InverseHadamardProduct(A, B);
}
/// <summary>
/// Little helper function for constructor.
/// </summary>
/// <returns>'basisBlades', but inside a newly allocated array of length 1.</returns>
protected static RefGA.BasisBlade[][] ToDoubleArray(G25.rsbbp.BasisBlade[] basisBlades)
{
RefGA.BasisBlade[][] B = new RefGA.BasisBlade[1][];
B[0] = new RefGA.BasisBlade[basisBlades.Length];
for (int i = 0; i < basisBlades.Length; i++)
B[0][i] = basisBlades[i].GetBasisBlade;
return B;
}
/// <summary>Alternative way of calling B.Reverse()</summary>
public static BasisBlade Reverse(BasisBlade B)
{
return B.Reverse();
}
/// <summary>Computes the geometric product of two basis blades in diagonal non-Euclidean metric.</summary>
/// <param name="a">Basisblade</param>
/// <param name="b">Basisblade</param>
/// <param name="m">array of doubles giving the metric for each basis vector (i.e., the diagonal of the diagonal metric matrix)</param>
/// <returns>a new basis blade which is either the geometric product or the outer product of two basis blades <paramref name="a"/> and <paramref name="b"/>.</returns>
public static BasisBlade GeometricProduct(BasisBlade a, BasisBlade b, double[] m)
{
// compute the geometric product in Euclidean metric:
BasisBlade result = GeometricProduct(a, b);
double resultScale = result.scale;
// compute the meet (bitmap of annihilated vectors):
uint bitmap = a.bitmap & b.bitmap;
// change the scale according to the metric:
uint i = 0;
while (bitmap != 0)
{
if ((bitmap & 1) != 0) resultScale *= m[i];
i++;
bitmap = bitmap >> 1;
}
// return correctly scaled result
return new BasisBlade(result.bitmap, resultScale, result.symScale);
}
/// <summary>
/// Little helper function for the constructor.
/// </summary>
/// <returns>'basisBlades', but inside a newly allocated array of length 1.</returns>
protected static RefGA.BasisBlade[][] ToDoubleArray(RefGA.BasisBlade[] basisBlades)
{
RefGA.BasisBlade[][] B = new RefGA.BasisBlade[1][];
B[0] = basisBlades;
return(B);
}
/// <summary>Computes the geometric product of two basis blades.</summary>
/// <returns>geometric product of two basis blades <paramref name="a"/> and <paramref name="b"/></returns>
public static BasisBlade GeometricProduct(BasisBlade a, BasisBlade b)
{
const bool outer = false;
return gp_op(a, b, outer);
}
/// <summary>Shortcut to ScalarProduct() </summary>
public static BasisBlade scp(BasisBlade A, BasisBlade B)
{
return ScalarProduct(A, B);
}
/// <summary>
/// Computes the inner product of two basis blades in diagonal non-Euclidean metric.
/// </summary>
/// <param name="A">input blade</param>
/// <param name="B">input blade</param>
/// <param name="m">an array of doubles giving the metric for each basis vector</param>
/// <returns>Scalar product of <paramref name="A"/> and <paramref name="B"/>.</returns>
public static BasisBlade ScalarProduct(BasisBlade A, BasisBlade B, double[] m)
{
InnerProductType type = InnerProductType.SCALAR_PRODUCT;
return InnerProductFilter(A.Grade(), B.Grade(), GeometricProduct(A, B, m), type);
}
/// <summary>Shortcut to GeometricProduct()</summary>
public static BasisBlade gp(BasisBlade a, BasisBlade b, double[] m)
{
return GeometricProduct(a, b, m);
}
/// <summary>
/// Computes the inner product of two basis blades in arbitary non-Euclidean metric.
/// </summary>
/// <param name="A">input blade</param>
/// <param name="B">input blade</param>
/// <param name="M">The Metric to be used</param>
/// <returns>Scalar product of <paramref name="A"/> and <paramref name="B"/>.</returns>
public static ArrayList ScalarProduct(BasisBlade A, BasisBlade B, Metric M)
{
InnerProductType type = InnerProductType.SCALAR_PRODUCT;
return InnerProductFilter(A.Grade(), B.Grade(), GeometricProduct(A, B, M), type);
}
/// <summary>Alternative way of calling B.CliffordConjugate()</summary>
public static BasisBlade CliffordConjugate(BasisBlade B)
{
return B.CliffordConjugate();
}