**The Truth:**- Can really only apply affine transforms to points.

Vectors can be transformed correctly iff they are defined by differences of points. **Transforming Normal Vectors:**-
- Normal vectors
**ARE NOT**defined by differences of points. - Tangent vectors
**ARE**defined by differences of points. - Normals are vectors perpendicular to all tangents at a point:
- Note that the natural representation of
is as a
*row vector*. - Suppose we have a transformation
*M*, a point , and a tangent at*P*. - Let be the ``linear part'' of
*M*, i.e. the upper submatrix. - Transform normals by inverse transpose of linear part of transformation: .
- If is O.N. (usual case for rigid body transforms), .

- Normal vectors
- Only worry if you have a non-uniform scale or a shear transformation.
**Transforming lines:**- Transform implicit form in a similar way.
**Transforming planes:**- Transform implicit form in a similar way.

*
Readings: Red Book, 5.8 (?);
White Book, 5.6.
*

CS488/688: Introduction to Interactive Computer Graphics

University of Waterloo

Computer Graphics Lab