Maths - 2D Multivector Functions

Maths - 2D multivector Functions

Dual

This multiplies by e₁₂ , for general information about n-dimentional dual see this page.

A_r	A_r^* = e₁₂ A_r
1	e₁₂
e₁	e₁₂e₁= -e₂
e₂	e₁₂e₂= e₁
e₁₂	e₁₂e₁₂= -1

Reverse

This reverses the order of the indexes, for general information about n-dimentional reverse see this page.

A_r	A_r†
1	1
e₁	e₁
e₂	e₂
e₁₂	e₂₁ = -e₁₂

Conjugate

The conjugate of A_r is denoted A_r^~ where: A_r^~*A_r = I = psudoscalar, for general information about n-dimentional conjugate see this page.

If A_r represents a transformation then A_r^~ reverses the transformation

A_r	A_r^~=(A_r†)^*	A_r^~*A_r
1	(1)^*= e₁₂	e₁₂
e₁	(e₁)^*= -e₂	e₁₂
e₂	(e₂)^*= e₁	e₁₂
e₁₂	(-e₁₂)^*= 1	e₁₂

The last column confirms that A_r^~*A_r = I = psudoscalar.

Inverse

try multipy by conjugate:

(e + ex + ey + exy)( e + ex + ey - exy)

	a.e	a.ex	a.ey	-a.exy
b.e	e	ex	ey	-exy
b.ex	ex	e	exy	-ey
b.ey	ey	-exy	e	+ex
b.exy	exy	-ey	ex	e

e = 4e

ex = 4ex

ey = 0

exy = -exy+exy-exy+exy =0

	a.e	a.ex	a.ey	a.exy
b.e	e	ex	ey	exy
b.ex	ex	e	exy	ey
b.ey	ey	-exy	e	-ex
b.exy	exy	-ey	ex	-e

public final void invert(Matrix4d m1) {
m00 = m12*m23*m31 - m13*m22*m31 + m13*m21*m32    - m11*m23*m32 - m12*m21*m33 + m11*m22*m33;
m01 = m03*m22*m31 - m02*m23*m31 -    m03*m21*m32 + m01*m23*m32 + m02*m21*m33 - m01*m22*m33;
m02 = m02*m13*m31 - m03*m12*m31    + m03*m11*m32 - m01*m13*m32 - m02*m11*m33 + m01*m12*m33;
m03 = m03*m12*m21 -    m02*m13*m21 - m03*m11*m22 + m01*m13*m22 + m02*m11*m23 - m01*m12*m23;
m10 = m13*m22*m30    - m12*m23*m30 - m13*m20*m32 + m10*m23*m32 + m12*m20*m33 - m10*m22*m33;
m11 =    m02*m23*m30 - m03*m22*m30 + m03*m20*m32 - m00*m23*m32 - m02*m20*m33 + m00*m22*m33;
m12 = m03*m12*m30 - m02*m13*m30 - m03*m10*m32 + m00*m13*m32 + m02*m10*m33 -    m00*m12*m33;
m13 = m02*m13*m20 - m03*m12*m20 + m03*m10*m22 - m00*m13*m22 - m02*m10*m23    + m00*m12*m23;
m20 = m11*m23*m30 - m13*m21*m30 + m13*m20*m31 - m10*m23*m31 -    m11*m20*m33 + m10*m21*m33;
m21 = m03*m21*m30 - m01*m23*m30 - m03*m20*m31 + m00*m23*m31    + m01*m20*m33 - m00*m21*m33;
m22 = m01*m13*m30 - m03*m11*m30 + m03*m10*m31 -    m00*m13*m31 - m01*m10*m33 + m00*m11*m33;
m23 = m03*m11*m20 - m01*m13*m20 - m03*m10*m21    + m00*m13*m21 + m01*m10*m23 - m00*m11*m23;
m30 = m12*m21*m30 - m11*m22*m30 -    m12*m20*m31 + m10*m22*m31 + m11*m20*m32 - m10*m21*m32;
m31 = m01*m22*m30 - m02*m21*m30    + m02*m20*m31 - m00*m22*m31 - m01*m20*m32 + m00*m21*m32;
m32 = m02*m11*m30 -    m01*m12*m30 - m02*m10*m31 + m00*m12*m31 + m01*m10*m32 - m00*m11*m32;
m33 = m01*m12*m20    - m02*m11*m20 + m02*m10*m21 - m00*m12*m21 - m01*m10*m22 + m00*m11*m22;
scale(1/m1.determinant());
}


public double determinant()
{ double value;
value = m03 * m12 * m21 * m30 - m02 * m13 * m21 * m30
        -m03 * m11 * m22 * m30+m01 * m13 * m22 * m30
        + m02 * m11 * m23    * m30-m01 * m12 * m23 * m30
        -m03 * m12 * m20 * m31+m02 * m13 * m20 * m31
        + m03    * m10 * m22 * m31-m00 * m13 * m22 * m31
        -m02 * m10 * m23 * m31+m00 * m12 * m23    * m31
        + m03 * m11 * m20 * m32-m01 * m13 * m20 * m32
        -m03 * m10 * m21 * m32+m00    * m13 * m21 * m32
        + m01 * m10 * m23 * m32-m00 * m11 * m23 * m32
        -m02 * m11 * m20    * m33+m01 * m12 * m20 * m33
        + m02 * m10 * m21 * m33-m00 * m12 * m21 * m33
        -m01    * m10 * m22 * m33+m00 * m11 * m22 * m33;
return value;
}

	a.e	a.ex	a.ey	a.exy
b.e	e	ex	ey	exy
b.ex	ex	e	exy	ey
b.ey	ey	-exy	e	-ex
b.exy	exy	-ey	ex	-e


public final void invert(Matrix4d m1) {
m00 = m1.m11;
m01 = -m1.m01;
m02 = -m1.m10;
m03 = m1.m00;

scale(1/m1.determinant());
} 



public double determinant() {
   return m00*m11 - m01*m10;
}

Exponential

Reverse

Involution

Determinant

EuclideanSpace

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