In this documentation we will be using some uncommon concepts, and some common terms will also be used with special meanings. Here we collect definitions of a few such terms and concepts for convenient reference.
Mathematicians have always used their "mind's eye" to visualize the abstract objects and processes that arise in all branches of mathematical research. By "mathematical visualization" I am referring not to that subjective and internal process, but rather to the modern use of computer imaging techniques that permit us to make these vague internal visualizations precise and to share them with others.
One initial goal for a mathematical theory of some new kind of object is usually a "classification theorem", roughly speaking, discovering the space of moduli. The next step is usually a detailed investigation of the moduli space to see how various properties of the object depend on the moduli, and to see what values of the moduli give rise to objects with some special properties. (For example, when the two semi-axes are equal, an ellipse becomes a circle, which has a larger symmetry group than a generic ellipse.)
If we can devise some good way of displaying an object graphically, depending on its moduli, then we can move along a curve in the moduli space, and draw frames consisting of the graphical representation of the object at various points along the curve. If we then play these frames back in rapid succession we get a movie showing how the object changes as we change the moduli along the curve, and this is what we call a morph. Clearly, this can be a powerful tool in investigating the moduli space.
Often, even when the moduli space is infinite dimensional, it will contain special curves that provide interesting morphs. For example, minimal surfaces come in one-parameter families (so-called associate families), all of whose elements are isometric, though not usually ambient isometric. Using this associated family parameter as a morphing parameter provides a particularly beautiful morph; one that could in principle be modeled in sheet metal.
Similarly, the moduli space for pseudo-spherical surfaces can be identified with the space of solutions of the Sine-Gordon equation. The latter contains certain n-parameter families (the pure n-soliton solutions) that correspond to particularly interesting surfaces. (In particular, the 1-solitons form the well-known Dini family that contains the pseudosphere.) It was a desire to be able to morph within the 2-soliton family that was the motivation for starting work on 3D-XplorMath.
To get away from surfaces, morphing is
the obvious method for displaying the bifurcations of solutions of
ODE. Indeed, the morphing process is such a powerful and revealing
tool, that whenever a new category of mathematical objects is
added to the 3D-XplorMath repertory, a lot of time goes into
thinking about and experimenting with creative ways to use morphs
that are particularly adapted to that category. For example, when
displaying conformal maps, we found that a morph between a given
map and the identity map is often a particularly good way to
reveal the structure of the map, so this is often the default
To create the default morphing animation of the current object, choose Morph from the Animate menu. (If you want a filmstrip animation rather than real-time, be sure that is checked in the Animate menu.) To create a custom morph, you should first familiarize yourself with how the nine parameters aa, bb,...,ii are used to determine the characteristics of the object (select About This Object... from the Action menu) and then decide the initial and final values of these parameters you want to morph between and the number of intermediate stages in the morph. To see how to set these before creating the morph, see the documentation for the Settings menu.
Morphing, defined above, is one of the most important processes, and in the design of 3D-XplorMath, we have put a lot of emphasis on making it easy to create good morphing animations.
Here are just a few other "processes" to illustrate better the scope of the term:
Given a plane curve, it is revealing to show an animation in which the osculating circles are drawn at a point that moves along the curve, the centers of curvature tracing out the evolute of the curve as the animation proceeds. In fact, there are many such classical processes that associate other curves with a given curve (pedals, involutes, strophoids, parallel curves, etc.) most of which become much easier to understand and illustrate with a computer.
For a space curve, an interesting process is the construction of a "tube" about the curve. This construction involves the choice of a framing of the normal bundle to the curve---usually the Frenet frame---and the tube serves to reveal the important (but usually invisible) framing. Another way to see the framing of the normal bundle provided by 3D-XplorMath, is to display it as a literally "moving frame (or repere mobile).
For a surface, important processes are
the construction of its focal sets and its parallel surfaces, and
for a polyhedron, two interesting processes are the constructions
of its stellations and its truncation (the latter is what converts
a regular icosahedron into a buckyball). It is revealing to morph
between the untruncated and truncated