This is interpreted in the following way: Initially, the three objects (cube, cylinder, sphere) are coincident with their symmetry structures maximally aligned. This is the fiber-group copy called the alignment kernel above. The higher affine group moves the cylinder-sphere pair in relation to the cube. The lower affine group moves the sphere in relation to the cylinder.

The above discussion has been illustrating a class of groups we call telescope groups, which were proposed by us in Leyton (2001). To get an intuitive sense of a telescope group, think of an ordinary telescope. In an ordinary telescope, you have a set of rings that are initially maximally coincident. Then you pull them successively out of alignment with respect to each other. A telescope group is a group structured like this.

In fact, it is part of a still larger class of groups we call unfolding groups, which which were also proposed by us in Leyton (2001). Unfolding groups are the most important class of algebraic structures introduced in that book. The basic idea is that any complex structure such as a design in CAD is unfolded from a maximally collapsed form which we call the alignment kernel. Two main properties characterize unfolding groups:

Selection. The control group acts selectively on only part of its fiber.

Misalignment. The control group is symmetry-breaking by misalignment.

Now, in order to establish a group theory of CAD, our procedure was this: We spent several years working through every single operation in each of several of the main CAD, solid modeling, assembly, and animation programs, including several releases of AutoCAD, ProEngineer, 3D Studio Max and Viz, Architectural Desktop, Mechanical Desktop, etc., as well as all the major manuals on each of the programs - approximately 15,000 pages of text.

Each individual situation was characterized by a group, and a new class of groups was invented for any situation that could not be formalized in terms of a previously created class of groups. Proceeding in this manner, we eventually found that three classes of groups could handle any newly created situation. They were called:

0. Telescope groups.
0. Super-local unfoldings.
0. Sub-local unfoldings.

The above has looked so far (intuitively) at the structure of telescope groups. The book shows that serial-link manipulators are examples of telescope groups. Therefore:

That is, intuitively, modulation involves a set of initially coincident scales that successively slide against each other, out of alignment - like an opening telescope. This is illustrated in figure 16.