A 2D and 3D Transformation Library for Graphics - Computer Graphics: Principles and Practice - page 295

12.9 Other Approaches

We mentioned that there are other approaches to encapsulating the various trans-
formation operations needed in graphics, including basing them on coordinate
frames rather than linear transformations.
There's a particularly efficient way to create a restricted transformation library
in the case that we only want to describe rigid transformations of objects. This
eliminates all scaling—both uniform and nonuniform—and all nonaffine projec-
tive transformations. Thus, every transformation is simply a translation, rotation,
or combination of the two. There are two advantages of considering only such
transformations.

• There are no “degeneracies.” In the case of the PointsToPoints transfor-
mations we discussed earlier, there was a possibility of failure if the starting
points were not in general position. No such problem arises here.
• When we need to invert such a rigid transformation, no matrix inversion
procedure is needed, because the inverse of a rotation matrix A is its trans-
pose A T .

There are disadvantages as well.

• We can no longer easily use a PointsToPoints specification for a transfor-
mation. The pairwise distances of the starting points must exactly match
those of the target points, because a rigid transformation preserves dis-
tances between points. It's impractical to try to specify target points with
this property, even when the source points are ( 0, 0 ) , ( 1, 0 ) , and ( 0, 1 ) ,for
instance.
• We cannot make larger or smaller instances of objects in a scene using this
design. (A typical solution is to provide a method for reading objects from
afile with a scale factor so that you create a large sphere by reading a
standard sphere with a scale factor of 6.0, for instance.)

G3D, a package we'll use in Chapter 32 for the implementation of two ren-
derers, uses the rigid-motion approach. It contains a class CFrame (for “coordinate
frame”); the standard instance of this is the standard coordinate frame based at the
origin. The model for the scene in Figure 12.2 involves quite a lot of code, most
of which describes material properties, etc. The essence of the geometric part of
the modeling 3 is given in Listing 12.1, from which we've removed all modeling
of light sources and materials. Figure 12.2: A simple scene.

Listing 12.1: Modeling a simple scene.

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void World::loadWorld1() {
modeling of lights omitted
// A sphere, slightly to right, shiny and red.
addSphere(Point3(1.00f, 1.0f, -3.0f), 1.0f, material specification );
// LEFT sphere
addTransparentSphere(Point3(-0.95f, 0.7f, -3.0f), 0.7f, material specifications );

// And a ground plane...
addSquare(4.0, Point3(0.0f, -0.2f, -2.0f),

3. The camera is specified elsewhere in the program.