Maths - 2D multivector Functions

Dual

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

Ar
Ar* = e12 Ar
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.

Ar
Ar
1 1
e1 e1
e2 e2
e12 e21 = -e12

Conjugate

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

If Ar represents a transformation then Ar~ reverses the transformation

Ar
Ar~=(Ar†)* Ar~*Ar
1 (1)*= e12 e12
e1 (e1)*= -e2 e12
e2 (e2)*= e1 e12
e12 (-e12)*= 1 e12

The last column confirms that Ar~*Ar = 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


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see also:

 

Correspondence about this page

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Terminology and Notation

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