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# Change of Basis

Suppose:

• We have two coordinate frames for a space, and ,
• Want to change from coordinates relative to to coordinates relative to .

Know
relative to .
Want
the coordinates of P relative to .
• Express each component of in terms of :

• The change of coordinates is given by the matrix

Factoring gives

• Consider:

How
do we get ?
• If is orthonormal:

• If is orthogonal:

• Otherwise, we have to solve a small system of linear equations, using .
• Change of basis from to Standard Cartesian Frame is trivial (since frame elements normally expressed with respect to Standard Cartesian Frame).

Example:

\

where is the standard coordinate frame.

• Matrices mapping from/to to/from :

• Check

Generalization
to 3D is straightforward ...
Example:
• Define two frames:

• All coordinates are specified relative to the standard frame in a Cartesian 3 space.
• The standard frame happens to be identical to .
• Note that both and are orthonormal.
• The matrix mapping to is given by

• Check this matrix:
Map each element of ,
See that we get that element relative to .
(e.g, )
• Question: What is the matrix mapping from to ?

Notes

• On the computer, frame elements usually specified in Standard Frame for space.

Eg, a frame is given by

relative to Standard Frame.

Question: What are coordinates of these basis elements relative to F?

• Frames are usually orthonormal.
• A point ``mapped'' by a change of basis does not change;

We have merely expressed its coordinates relative to a different frame.

Readings: Hearn and Baker, Section 5-5 (not as general as here, though). Red book, 5.9, White book, 5.8

Next: Compositions of Transformations Up: Affine Geometry and Previous: Geometric Transformations

CS488/688: Introduction to Interactive Computer Graphics
University of Waterloo
Computer Graphics Lab

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