Maths - Isometry Properties of 2D Multivectors

So far we have seen that, in 2 dimensions, we can calculate the result of combining two translations to give a total translation (see this page) and we can also combine two rotations to give a total rotations (see this page). the interesting thing is that the calculation to do both of these is the same:

  1. Take the exponent of both inputs.
  2. Multiply these exponents together.
  3. Take the loge of the result.

The way we distinguish between translations and rotations is that we use double numbers for one and complex numbers for the other:

 

2D Euclidean space
(2 space dimensions)

2D Minkowski space
(1 space and 1 time)
translations double numbers complex numbers
rotations complex numbers double numbers

Where:

So we can calculate translations and rotations separately for the different types of space.

What we now want to do is to combine translations and rotations into one operation and to do this in such a way that interactions between the translations are handled correctly. For example if we apply the following sequence:

  1. Translate from the origin 2 units to the right.
  2. Rotate 90° clockwise.

then we would expect to get a different result than applying the sequence in a different order:

  1. Rotate 90° clockwise.
  2. Translate from the origin 2 units to the right.

So how can we do this?

My best guess would be to apply the above sequence, but instead of just the even or odd parts we need to be able to take the exponent and inverse in functions of the whole multivector:

exp(a + b e1 + c e2 + d e12)

where:

I'm not sure how to calculate this? I think I will try applying an infinite sequence and then try breaking down the components.

exp(x) 1 + x1/1! + x2/2! + x3/3! ... + xr/(r)!

Then we need to check that it works for all types of sequence of translations and rotations.


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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.

flag flag flag flag flag flag 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.

 

Terminology and Notation

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