3D Computer Graphics. Mathem. Intro with OpenGL by Buss

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Angles between lines. The transformation is said to be orientation-preserving if it preserves the direction of angles, that is, if a counterclockwise direction of movement stays counterclockwise after being transformed by A. Rigid, orientation-preserving transformations are widely used. One application of these transformations is in animation: the position and orientation of a moving rigid body can be described by a time-varying transformation A(t). This transformation A(t) will be rigid and orientation-preserving provided the body does not deform or change size or shape.

The conventions adopted in this book are that points in space are represented by column vectors, and linear transformations with matrix representation M are computed as Mx. Thus, our matrices multiply on the left. Unfortunately, this convention is not universally followed, and it is also common in computer graphics applications to use row vectors for points and vectors and to use matrix representations that act on the right. That is, many workers in computer graphics use a row vector to represent a point: instead of using x, they use the row vector xT .

Homogeneous coordinates use vectors with four components to represent points in 3-space. The perspective division stage merely converts from homogeneous coordinates back into the usual three x-, y-, z-coordinates. The x- and y-coordinates determine the position of a vertex in the final graphics image. The z-coordinates measure the distance to the object, although they can represent a “pseudo-distance,” or “fake” distance, rather than a true distance. Homogeneous coordinates are described later in this chapter.

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