# Affine transformations & Affine spaces

Can Sage do affine transformation in affine spaces - adding a vector to a point?

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Yes, just represent your points with position vectors. Sage can add vectors to vectors and apply linear transformations to vectors.

For example:

sage: v = vector([1,2,3])
sage: M = matrix([[1,0,0],[0,2,0],[0,1,1]])
sage: M
[1 0 0]
[0 2 0]
[0 1 1]
sage: pt = vector([0,1,0])
sage: v + M*pt  # apply affine linear transformation to point pt
(1, 4, 4)
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Thanks Benjamin:

I am aware how it could be used in vector algebra. The source I use is Geometric Tools for Computer Graphics by Schneider & Eberly 2003.

There is a distinction unlike vector algebra between a vector and a point. *Add two vectors to get a third. *Multiply a vector by a scalar to get a vector. *Add a vector to a point to get another point. *Subtract two points to get vector. (edited) *etc.

In affine algebra, a point can be translated in a matrix multiplication where in vector algebra it cannot. I have to admit that vector algebra is used quite a lot in CG, for example, the affine matrix has a vector matrix embedded.

The elements of a point in affine space for example is [0, 1, 0, 1] for the vector [0, 1, 0] in vector space. The elements for a vector in affine space is [0, 1, 0, 0] for the vector [0, 1, 0] in vector space.

Please don't ask me to prove this. I do believe that sage developers can implement this as a subset to the module linear algebra. CG is a large and fast growing business.

I suppose the answer from Benjamin is NO. There is no affine algebra unless we define the vectors as points and the matrices as affine matrices....

Many Thanks

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It sounds to me like you are talking about realizing affine linear transformations of 3-dimensional space as linear transformations of 4 dimensional space followed by a projection. If you can describe the projection in terms of linear algebra, then you could easily implement affine linear transformations your sense extending the basic linear algebra framework in Sage.

( 2011-10-29 18:02:24 +0200 )edit

@benjaminfjones: This is in a lot of elementary Lie group or algebra books nowadays, and is sort of what you are saying. See http://en.wikipedia.org/wiki/Affine_transformation, for instance. It's actually a neat exercise to prove similar properties for these sets of matrices as for GL or SL or whatever. Anyway, I don't see why this couldn't be implemented pretty easily. I think we also have quaternions in Sage which also are used in a similar fashion for CG.

( 2011-10-29 21:16:15 +0200 )edit