**Construct**- matrices for simple geometric transformations.
**Combine**- simple transformations into more complex ones.
- Assume that the range and domain frames are the Standard Frame.
- Will begin with 2D, generalize later.

**Translation:**- Specified by the vector
:
- A point will map to .
- A vector will remain unchanged under translation.
- Translation is
**NOT**a linear transformation.

If it were, then (ignoring the fact that point addition is not allowed)*T*(*P*+*Q*)=*T*(*P*)+*T*(*Q*) would hold,**BUT**: - Translation is linear on sums of vectors...

**Matrix representation of translation**-
- We can create a matrix representation of translation:
- will mean the above matrix.
- Note that vectors are unchanged by this matrix.
- Although more expensive to compute than the other
version of translation, we prefer this one:
- Uniform treatment of points and vectors
- Other transformations will also be in
matrix form.

We can compose transformations by matrix multiply. Thus, the composite operation less expensive if translation composed, too.

- We can create a matrix representation of translation:
**Scale about the origin:**-

Specified by factors .- Applies to points or vectors, is linear.
- A point will map to .
- A vector will map to .
- Matrix representation:

**Rotation:**- Counterclockwise about the origin, by angle .
- Applies to points or vectors, is linear.
- Matrix representation:

**Shear:**- Intermixes coordinates according to
:
- Applies to points or vectors, is linear.
- Matrix representation:
- Easiest to see if we set one of or to zero.

**Reflection:**- Through a line.
- Applies to points or vectors, is linear.
- Example: through
*x*-axis, matrix representation is - See book for other examples

**Note:**- Vectors map through all these transformations as
we want them to.

*
Readings: Hearn and Baker, 5-1, 5-2, and 5-4;
Red book, 5.2, 5.3; White book, 5.1, 5.2
*

University of Waterloo

Computer Graphics Lab