public static void ComputePower()
{
var n = 4;
var uFrame = GaPoTNumFrame.CreateBasisFrame(n);
var cFrame = GaPoTNumFrame.CreateGramSchmidtFrame(n, n - 1);
var rotorsSequence =
GaPoTNumRotorsSequence.Create(
cFrame.GetRotorsToFrame(uFrame)
);
var v = "-4<1>, 8<2>, -3<3>, -1<4>".GaPoTNumParseVector();
var i = "-9<1>, 2<2>, -5<3>, 12<4>".GaPoTNumParseVector();
var m = v.Gp(i);
var vr = rotorsSequence.Rotate(v);
var ir = rotorsSequence.Rotate(i);
var mr = vr.Gp(ir);
Console.WriteLine($"m = {m.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine($"mr = {mr.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
}
//public static void DisplayGramSchmidtToClarkeMatrices()
//{
// for (var n = 2; n <= 12; n++)
// {
// var matrix1 = GaPoTNumFrame.CreateGramSchmidtFrame(n).GetMatrix(n);
// var matrix2 = GaPoTNumFrame.CreateClarkeFrame(n).GetMatrix(n);
// var matrix = Mfs.Dot[matrix2, Mfs.Transpose[matrix1]];
// Console.WriteLine($"m{n} = {matrix};");
// Console.WriteLine($"MatrixForm[FullSimplify[m{n}]]");
// Console.WriteLine($"MatrixForm[FullSimplify[Dot[m{n}, Transpose[m{n}]]]]");
// Console.WriteLine();
// }
//}
public static void DisplayGramSchmidtToClarkeRotors()
{
var n = 4;
var frame1 = GaPoTNumFrame.CreateGramSchmidtFrame(n).GetSwappedPairsFrame();
var frame2 = GaPoTNumFrame.CreateClarkeFrame(n);
var rotorsSequence = GaPoTNumRotorsSequence.Create(
frame1.GetRotorsToFrame(frame2)
);
var i = 1;
foreach (var rotor in rotorsSequence)
{
Console.WriteLine($"R_{i} = {rotor.TermsToLaTeX()}");
Console.WriteLine();
i++;
}
}
public static void DisplayFrame(GaPoTNumFrame sourceFrame, GaPoTNumFrame targetFrame)
{
var n = targetFrame.Count;
var matrixExpr = targetFrame.GetMatrix(n);
var orthoFrameEquation = targetFrame.ToLaTeXEquationsArray(
"c",
@"\mu"
);
Console.WriteLine(orthoFrameEquation);
Console.WriteLine();
Console.WriteLine("Rotation Matrix:");
Console.WriteLine();
Console.WriteLine($@"\[{matrixExpr.GetLaTeXArray()}\]");
Console.WriteLine();
var anglesTextList = sourceFrame
.GetAnglesToFrame(targetFrame)
.Select(a => a.GetLaTeXAngleInDegrees())
.ToArray();
Console.WriteLine("Angles between frames vectors:");
foreach (var angleText in anglesTextList)
{
Console.WriteLine($@"\[{angleText}\]");
Console.WriteLine();
}
var autoVectorProjection = GaPoTNumVector
.CreateAutoVector(n)
.GetProjectionOnFrame(targetFrame);
Console.WriteLine("Projection of auto-vector:");
Console.WriteLine($@"\[{autoVectorProjection.TermsToLaTeX()}\]");
Console.WriteLine();
var rotorsSequence =
GaPoTNumRotorsSequence.Create(
sourceFrame.GetRotorsToFrame(targetFrame)
);
//if (!rotorsSequence.ValidateRotation(sourceFrame, targetFrame))
// throw new InvalidOperationException("Error in rotation sequence");
//for (var i = 0; i < sourceFrame.Count - 1; i++)
//{
// var f1 = sourceFrame.GetSubFrame(0, i + 1);
// var f2 = targetFrame.GetSubFrame(0, i + 1);
// var rs = rotorsSequence.GetSubSequence(0, i + 1);
// if (!rs.ValidateRotation(f1, f2))
// throw new InvalidOperationException("Error in rotation sequence");
//}
Console.WriteLine("Rotors Sequence:");
Console.WriteLine();
for (var i = 0; i < rotorsSequence.Count; i++)
{
var rotorEquation = rotorsSequence[i].ToLaTeXEquationsArray(
$"R_{i + 1}",
@"\mu"
);
Console.WriteLine(rotorEquation);
Console.WriteLine();
}
if (n > 4)
{
return;
}
//var rotor21 = rotorsSequence[1].Gp(rotorsSequence[0]);
//var rotor21Equation = rotor21.ToLaTeXEquationsArray(
// $"R_{{21}}=R_2 R_1",
// @"\mu"
//);
//Console.WriteLine(rotor21Equation);
//Console.WriteLine();
//var rotor32 = rotorsSequence[2].Gp(rotorsSequence[1]);
//var rotor32Equation = rotor32.ToLaTeXEquationsArray(
// $"R_{{32}}=R_3 R_2",
// @"\mu"
//);
//Console.WriteLine(rotor32Equation);
//Console.WriteLine();
Console.WriteLine("Final Rotor $R = R_3 R_{21} = R_{32} R_1$:");
Console.WriteLine();
var finalRotor = rotorsSequence.GetFinalRotor();
var finalRotorEquation = finalRotor.ToLaTeXEquationsArray(
"R",
@"\mu"
);
Console.WriteLine(finalRotorEquation);
Console.WriteLine();
//Console.WriteLine($"m{n} = {matrixExpr};");
//Console.WriteLine($"MatrixForm[FullSimplify[m{n}]]");
//Console.WriteLine($"MatrixForm[FullSimplify[Dot[m{n}, Transpose[m{n}]]]]");
//Console.WriteLine($"Pseudo-Scalar = {frame.GetPseudoScalar()}");
//Console.WriteLine();
}
public static void Execute()
{
var n = 4;
var uFrame = GaPoTNumFrame.CreateBasisFrame(n);
var cFrame = GaPoTNumFrame.CreateGramSchmidtFrame(n, n - 1, out var eFrame);
var eFrameEquation = eFrame.ToLaTeXEquationsArray(
"e",
@"\mu"
);
Console.WriteLine("e-frame:");
Console.WriteLine(eFrameEquation);
Console.WriteLine();
var cFrameEquation =
cFrame.ToLaTeXEquationsArray(
"c",
@"\mu"
);
Console.WriteLine("c-frame:");
Console.WriteLine(cFrameEquation);
Console.WriteLine();
var cFrameMatrix =
cFrame.GetMatrix().GetLaTeXArray();
Console.WriteLine("c-frame matrix:");
Console.WriteLine(cFrameMatrix);
Console.WriteLine();
var rotorsSequence =
GaPoTNumRotorsSequence.Create(
cFrame.GetRotorsToFrame(uFrame)
);
var finalRotor =
rotorsSequence.GetFinalRotor();
var R1 = rotorsSequence[1].Gp(rotorsSequence[0]);
var R2 = rotorsSequence[2];
var autoVector = GaPoTNumVector.CreateAutoVector(n);
var rotatedAutoVector =
rotorsSequence.Rotate(autoVector);
var v1 = Math.Sqrt(2) * Math.Cos(0);
var v2 = Math.Sqrt(2) * Math.Cos(0 - 1 * 2 * Math.PI / 4);
var v3 = Math.Sqrt(2) * Math.Cos(0 - 2 * 2 * Math.PI / 4);
var v4 = -(v1 + v2 + v3);
var inputVector = new GaPoTNumVector()
.AddTerm(1, v1)
.AddTerm(2, v2)
.AddTerm(3, v3)
.AddTerm(4, v4);
Console.WriteLine("Final Rotor:");
Console.WriteLine($"{finalRotor.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine($"{finalRotor.TermsToText()}");
Console.WriteLine();
Console.WriteLine("R1:");
Console.WriteLine($"{R1.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
Console.WriteLine("R2:");
Console.WriteLine($"{R2.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
Console.WriteLine("R2 R1 - R:");
Console.WriteLine($"{(R2.Gp(R1) - finalRotor).TermsToText()}");
Console.WriteLine($"{(R1.Gp(R2) - finalRotor).TermsToText()}");
Console.WriteLine();
Console.WriteLine("Rotated Auto Vector:");
Console.WriteLine($"{rotatedAutoVector.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
Console.WriteLine("Input Vector:");
Console.WriteLine($"{inputVector.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
var rotatedVector =
rotorsSequence.Rotate(inputVector);
Console.WriteLine("Rotated Vector:");
Console.WriteLine($"{rotatedVector.TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
Console.WriteLine("Rotated Vector . Rotated Auto Vector:");
Console.WriteLine($"{rotatedVector.DotProduct(rotatedAutoVector):G}");
Console.WriteLine();
var eI = eFrame.GetPseudoScalar();
Console.WriteLine("Input Vector inside e-frame: ");
Console.WriteLine($"{inputVector.Op(eI).TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
Console.WriteLine("Rotated Vector inside e-frame: ");
Console.WriteLine($"{rotatedVector.Op(eI).TermsToLaTeX().GetLaTeXDisplayEquation()}");
Console.WriteLine();
Console.WriteLine("Input Vector squared norm: ");
Console.WriteLine($"{inputVector.Norm2()}");
Console.WriteLine();
Console.WriteLine("Rotated Vector squared norm: ");
Console.WriteLine($"{rotatedVector.Norm2()}");
Console.WriteLine();
}
public static void Execute()
{
var n = 3;
var uFrame = GaPoTNumFrame.CreateBasisFrame(n);
var cFrame = GaPoTNumFrame.CreateGramSchmidtFrame(n, n - 1, out var eFrame);
var rotorsSequence =
GaPoTNumRotorsSequence.Create(
uFrame.GetRotorsToFrame(cFrame)
).Reverse();
var finalRotor =
rotorsSequence.GetFinalRotor();
var autoVector = GaPoTNumVector.CreateAutoVector(n);
var rotatedAutoVector =
rotorsSequence.Rotate(autoVector);
//var v1 = Math.Sqrt(2);
//var v2 = Math.Sqrt(2) * Math.Cos(-Math.PI / 3);
//var v3 = Math.Sqrt(2) * Math.Cos(Math.PI / 3);
var v1 = 1.0d;
var v2 = Math.Cos(-2 * Math.PI / 3);
var v3 = Math.Cos(2 * Math.PI / 3);
var inputVector = new GaPoTNumVector()
.AddTerm(1, v1)
.AddTerm(2, v2)
.AddTerm(3, v3);
Console.WriteLine("Final Rotor:");
Console.WriteLine($"{finalRotor.TermsToLaTeX()}");
Console.WriteLine();
Console.WriteLine("Rotated Auto Vector:");
Console.WriteLine($"{rotatedAutoVector.TermsToLaTeX()}");
Console.WriteLine();
Console.WriteLine("Input Vector:");
Console.WriteLine($"{inputVector.TermsToLaTeX()}");
Console.WriteLine();
var rotatedVector =
rotorsSequence.Rotate(inputVector);
Console.WriteLine("Rotated Vector:");
Console.WriteLine($"{rotatedVector.TermsToLaTeX()}");
Console.WriteLine();
Console.WriteLine("Rotated Vector . Rotated Auto Vector:");
Console.WriteLine($"{rotatedVector.DotProduct(rotatedAutoVector):G}");
Console.WriteLine();
var eI = eFrame.GetPseudoScalar();
Console.WriteLine("Input Vector inside e-frame: ");
Console.WriteLine($"{inputVector.Op(eI).TermsToLaTeX()}");
Console.WriteLine();
Console.WriteLine("Rotated Vector inside e-frame: ");
Console.WriteLine($"{rotatedVector.Op(eI).TermsToLaTeX()}");
Console.WriteLine();
Console.WriteLine("Input Vector squared norm: ");
Console.WriteLine($"{inputVector.Norm2()}");
Console.WriteLine();
Console.WriteLine("Rotated Vector squared norm: ");
Console.WriteLine($"{rotatedVector.Norm2()}");
Console.WriteLine();
}
private static void DisplayFrames(GaPoTNumFrame sourceFrame, GaPoTNumFrame kirchhoffFrame, GaPoTNumFrame targetFrame)
{
var n = sourceFrame.Count;
var matrix = targetFrame.GetMatrix(n);
var kirchhoffFrameEquation = kirchhoffFrame.ToLaTeXEquationsArray(
"e",
@"\mu"
);
Console.WriteLine("Kirchhoff Frame:");
Console.WriteLine(kirchhoffFrameEquation);
Console.WriteLine();
var projectedFrameMatrix = sourceFrame
.GetProjectionOnFrame(kirchhoffFrame)
.GetMatrix(n)
.GetLaTeXArray();
Console.WriteLine("Projected Frame Matrix:");
Console.WriteLine(projectedFrameMatrix);
Console.WriteLine();
var orthoFrameEquation = targetFrame.ToLaTeXEquationsArray(
"c",
@"\mu"
);
Console.WriteLine("Orthonormal Frame:");
Console.WriteLine(orthoFrameEquation);
Console.WriteLine();
Console.WriteLine("Rotation Matrix:");
Console.WriteLine($"{matrix.GetLaTeXArray()}");
Console.WriteLine();
var anglesTextList = sourceFrame
.GetAnglesToFrame(targetFrame)
.Select(a => "$" + a.GetLaTeXAngleInDegrees() + "$")
.ToArray();
Console.WriteLine("Angles between frames vectors:");
foreach (var angleText in anglesTextList)
{
Console.WriteLine($"{angleText}");
Console.WriteLine();
}
var rotorsSequence =
GaPoTNumRotorsSequence.Create(
sourceFrame.GetRotorsToFrame(targetFrame)
);
//if (!rotorsSequence.ValidateRotation(sourceFrame, targetFrame))
// throw new InvalidOperationException("Error in rotation sequence");
//for (var i = 0; i < sourceFrame.Count - 1; i++)
//{
// var f1 = sourceFrame.GetSubFrame(0, i + 1);
// var f2 = targetFrame.GetSubFrame(0, i + 1);
// var rs = rotorsSequence.GetSubSequence(0, i + 1);
// if (!rs.ValidateRotation(f1, f2))
// throw new InvalidOperationException("Error in rotation sequence");
//}
Console.WriteLine("Rotors Sequence:");
Console.WriteLine();
for (var i = 0; i < rotorsSequence.Count; i++)
{
var rotorEquation = rotorsSequence[i].ToLaTeXEquationsArray(
$"R_{i + 1}",
@"\mu"
);
Console.WriteLine(rotorEquation);
Console.WriteLine();
}
//var rotor12Equation = rotorsSequence[1].Gp(rotorsSequence[0]).ToLaTeXEquationsArray(
// $"R_{{12}}",
// @"\mu"
//);
//Console.WriteLine(rotor12Equation);
//Console.WriteLine();
//var rotor23Equation = rotorsSequence[2].Gp(rotorsSequence[1]).ToLaTeXEquationsArray(
// $"R_{{23}}",
// @"\mu"
//);
//Console.WriteLine(rotor23Equation);
//Console.WriteLine();
Console.WriteLine("Final Rotor:");
Console.WriteLine();
var finalRotorEquation = rotorsSequence.GetFinalRotor().ToLaTeXEquationsArray(
"R",
@"\mu"
);
Console.WriteLine(finalRotorEquation);
Console.WriteLine();
}
public static GaPoTNumRotorsSequence CreateRotorsSequence(GaPoTNumFrame sourceFrame, GaPoTNumFrame targetFrame)
{
return(GaPoTNumRotorsSequence.Create(
sourceFrame.GetRotorsToFrame(targetFrame)
));
}
public static void Execute()
{
var refVectorIndex = 0;
for (var n = 3; n <= 8; n++)
{
var uFrame =
GaPoTNumFrame.CreateBasisFrame(n);
var uPseudoScalar =
GaPoTNumMultivector
.CreateZero()
.SetTerm((1 << n) - 1, 1.0d);
var eFrame =
GaPoTNumFrame.CreateKirchhoffFrame(n, refVectorIndex);
var pFrame =
uFrame.GetProjectionOnFrame(eFrame);
var fbdFrame = GaPoTNumFrame.CreateFbdFrame(n);
var rotorsSequence = GaPoTNumRotorsSequence.CreateFromFrames(
n,
pFrame,
fbdFrame
);
//var pFrame1 = pFrame
// .GetSubFrame(0, n - 1)
// .PrependVector(GaPoTNumVector.CreateAutoVector(n))
// .GetOrthogonalFrame(true);
//var fbdFrame1 = fbdFrame
// .GetSubFrame(0, n - 1)
// .PrependVector(GaPoTNumVector.CreateAutoVector(n))
// .GetOrthogonalFrame(true);
//var rs = GaPoTNumRotorsSequence.Create(
// pFrame1.GetRotorsToFrame(fbdFrame1)
//);
var pFrame2 = rotorsSequence.Rotate(pFrame);
Console.Write(@"\section{Dimensions: " + n + "}");
Console.WriteLine();
Console.Write(@"\subsection{FBD Frame:}");
Console.WriteLine();
DisplayFrame(fbdFrame);
Console.Write(@"\subsection{Projected Frame:}");
Console.WriteLine();
DisplayFrame(pFrame);
Console.Write(@"\subsection{Rotated Projected Frame:}");
Console.WriteLine();
DisplayFrame(pFrame2);
Console.WriteLine("Rotors Sequence:");
for (var i = 0; i < rotorsSequence.Count; i++)
{
var rotorEquation =
rotorsSequence[i].ToLaTeXEquationsArray(
$"R_{{{i + 1}}}",
@"\mu"
);
Console.WriteLine(rotorEquation);
Console.WriteLine();
}
}
}
public static void Execute(int n)
{
var uFrame =
GaPoTNumFrame.CreateBasisFrame(n);
var cFrame =
GaPoTNumFrame.CreateGramSchmidtFrame(n, 0);
var rs1 =
GaPoTNumRotorsSequence.CreateFromOrthonormalFrames(cFrame, uFrame);
var rotationMatrix = (Matrix)Matrix.Build.DenseOfArray(
rs1.GetFinalMatrix(n).ToComplexArray()
);
var cFrameMatrix = (Matrix)Matrix.Build.DenseOfArray(
cFrame.GetMatrix().ToComplexArray()
);
//Make sure the rotation matrix is correct
var m = rotationMatrix.Multiply(cFrameMatrix);
var finalRotor = rs1.GetFinalRotor();
Console.WriteLine("Rotation matrix:");
Console.WriteLine(rotationMatrix);
Console.WriteLine();
Console.WriteLine("Final rotor:");
Console.WriteLine(finalRotor.ToLaTeXEquationsArray("R", @"\mu"));
Console.WriteLine();
Console.WriteLine("Rotated cFrame using matrix:");
Console.WriteLine("MatrixForm[" + m + "]");
Console.WriteLine();
rotationMatrix.EigenDecomposition(out var eigenValuesExpr, out var eigenVectorsExpr);
Console.WriteLine("Eigen values:");
foreach (var value in eigenValuesExpr)
{
Console.WriteLine(value);
}
Console.WriteLine();
Console.WriteLine("Eigen vectors:");
foreach (var vector in eigenVectorsExpr)
{
Console.WriteLine(vector);
}
Console.WriteLine();
var eigenRotorsArray = new GaPoTNumMultivector[n];
for (var i = 0; i < n; i++)
{
eigenRotorsArray[i] = ComplexEigenPairToBivector(
eigenValuesExpr[i],
eigenVectorsExpr[i]
);
}
//Make sure the blades of these rotors are eigen blades of the rotation matrix
for (var i = 0; i < n; i++)
{
var(angleExpr, blade) =
eigenRotorsArray[i].GetSimpleRotorAngleBlade();
var bladeDiff = (rs1.Rotate(blade) - blade).Round(7);
Console.WriteLine("Angle:");
Console.WriteLine(angleExpr.ToString("G").GetLaTeXDisplayEquation());
Console.WriteLine();
Console.WriteLine("Blade:");
Console.WriteLine(blade.ToLaTeXEquationsArray("b_1", @"\mu"));
Console.WriteLine();
Console.WriteLine("Rotated Blade - Blade:");
Console.WriteLine(bladeDiff.ToLaTeXEquationsArray("b_1", @"\mu"));
Console.WriteLine();
}
var diff =
(eigenRotorsArray[0].Gp(eigenRotorsArray[1]) -
eigenRotorsArray[1].Gp(eigenRotorsArray[0]))
.Round(7);
Console.WriteLine("Difference:");
Console.WriteLine(diff.TermsToLaTeX().GetLaTeXDisplayEquation());
Console.WriteLine();
var rs = GaPoTNumRotorsSequence.Create(
eigenRotorsArray[0],
eigenRotorsArray[1]
);
//var rs = GaPoTNumRotorsSequence.Create(
// eigenRotorsArray[0]
//);
var uFrame1 = rs.Rotate(cFrame);
Console.WriteLine("Rotated cFrame:");
for (var i = 0; i < n; i++)
{
Console.WriteLine(uFrame1[i].Round(7).TermsToLaTeX().GetLaTeXDisplayEquation());
Console.WriteLine();
}
}
