Maths - Clifford Algebra - 3D Arithmetic

Maths - 3D Clifford Algebra - Inner Products

There are different definitions of the inner product as follows:

•	Dot product
•_H	Hestines inner product.	Like the dot product except that it is zero whenever one of its arguments is a scalar.
	left contraction inner product
	right contraction inner product

Dot product table:

a•b	b.e	b.e1	b.e2	b.e3	b.e12	b.e31	b.e23	b.e123
a.e	e	e1	e2	e3	e12	e31	e23	e123
a.e1	e1	e	0	0	e2	-e3	0	e23
a.e2	e2	0	e	0	-e1	0	e3	e31
a.e3	e3	0	0	e	0	e1	-e2	e12
a.e12	e12	-e2	e1	0	-e	0	0	-e3
a.e31	e31	e3	0	-e1	0	-e	0	-e2
a.e23	e23	0	-e3	e2	0	0	-e	-e1
a.e123	e123	e23	e31	e12	-e3	-e2	-e1	-e

Hestines inner product table:

a•_Hb	b.e	b.e1	b.e2	b.e3	b.e12	b.e31	b.e23	b.e123
a.e	0	0	0	0	0	0	0	0
a.e1	0	e	0	0	e2	-e3	0	e23
a.e2	0	0	e	0	-e1	0	e3	e31
a.e3	0	0	0	e	0	e1	-e2	e12
a.e12	0	-e2	e1	0	-e	0	0	-e3
a.e31	0	e3	0	-e1	0	-e	0	-e2
a.e23	0	0	-e3	e2	0	0	-e	-e1
a.e123	0	e23	e31	e12	-e3	-e2	-e1	-e

left contraction inner product table:

ab	b.e	b.e1	b.e2	b.e3	b.e12	b.e31	b.e23	b.e123
a.e	e	0	0	0	0	0	0	0
a.e1	e1	e	0	0	0	0	0	0
a.e2	e2	0	e	0	0	0	0	0
a.e3	e3	0	0	e	0	0	0	0
a.e12	e12	-e2	e1	0	-e	0	0	0
a.e31	e31	e3	0	-e1	0	-e	0	0
a.e23	e23	0	-e3	e2	0	0	-e	0
a.e123	e123	e23	e31	e12	-e3	-e2	-e1	-e

right contraction inner product table:

ab	b.e	b.e1	b.e2	b.e3	b.e12	b.e31	b.e23	b.e123
a.e	e	e1	e2	e3	e12	e31	e23	e123
a.e1	0	e	0	0	e2	-e3	0	e23
a.e2	0	0	e	0	-e1	0	e3	e31
a.e3	0	0	0	e	0	e1	-e2	e12
a.e12	0	0	0	0	-e	0	0	-e3
a.e31	0	0	0	0	0	-e	0	-e2
a.e23	0	0	0	0	0	0	-e	-e1
a.e123	0	0	0	0	0	0	0	-e

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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.

New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

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