CSci 480 Lecture Notes

(Fall) Week Five, Monday; (Summer) Week Two, Thursday: Geometrical Transformations

If you are not familiar with vector and matrix operations, please read chapter 5.1, "Mathematical Preliminaries."

Definition: a convex polygon is the intersection of a finite number of half-planes. A concave polygon is any polygon that is not convex. Any concave polygon is the union of a set of convex polygons.

2D Transformations

Translation: a rigid body transformation.

Scaling (uniform and differential): an affine transformation, but not a rigid body transformation.

Rotation: another rigid body transformation.

Homogeneous Coordinates and Matrix Representation of 2D Transformations

Homogeneous coordinates allow us to translate by multiplying by a translation matrix. They also allow us to apply perspective transformations on 3D objects later on when we get to 3D viewing.

Shear transformation: another affine transformation represented by a shear matrix.

Composition of 2D Transformations

We compose the various transformations into a single matrix and multiply every point in the object model by the matrix to accomplish the transformation.

Implication: students will need to create some functions to do matrix multiplication in both 2D and 3D using homogeneous coordinates.

This page established September 25, 1998; last updated September 17, 1998.