**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:

- Express each component of in terms of :
**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).

- If is orthonormal:
- Example:
\

where is the standard coordinate frame.

- Matrices mapping from/to to/from :
- Check

- Matrices mapping from/to to/from :
**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 ?

- Define two frames:
**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.

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

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

University of Waterloo

Computer Graphics Lab