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

Generating Geometric Algebras

Vector multiplication (cross and dot product) can be very useful in physics but it also has its limitations and Geometric Algebra defines a new, more general, type of multiplication. This new type of multiplication generates new 'dimensions' so Geometric Algebra takes a vector algebra of dimension 'n' and generates an algebra of dimension n².

What are the properties of these new dimensions?
How much freedom do we have, in in choosing the basis vectors for example, to modify these properties?

This page discusses these questions .

grid image

Rules of Mathematical Operations

We get used to the rules of scalar algebra, for instance a+b=b+a, in many cases we do this without consiously thinking about it.

Are these rules true of all algebras?
Are these rules true of all operations in these algebras?

This page discusses these questions.

metadata block

see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

Mathmatica

Can you help?

Please send me any improvements to here. I would appreciate ideas to make the pages more useful including error correction, ideas for new pages, improvements to wording. It helps if you quote the full URL of the page.

Terminology and Notation

Specific to this page here:

program

I am working on a project which uses these principles, if you would like to help me with this you are welcome to join in, here:

http://sourceforge.net/projects/mjbworld/

This site may have errors. Don't use for critical systems.