There is a mathematical phenomenon, the homotopy, that lends itself particularly well to real-time interactive computer animation- Mathematically speaking, the notion of a homotopy spans a continuum of sophistication. At one end are the familiar, rigid Euclidean motions of translation, rotation and reflection; at the other are exotic metamorphoses of surfaces, such as sphere eversions, whose complexity resists holistic comprehension, and thus challenges computer graphics in a unique way. Simply put, one wishes to interact with the temporally extended homotopy as easily as with rigid objects.
Current hardware and graphics libraries deal well with objects in 3-space that do not change their shape during a rigid motion. Animating mild deformations that alter object shape without losing recognizable identity requires ingenuity and good technique, but is not intrinsically difficult. Non-linear interpolation between two given forms, such as morphing, is a familiar example of a less trivial homotopy that does generate animation problems. A topologist's (regular) homotopy, however, tends to be much more complicated than morphing. Turning a sphere inside out without tearing or excessively creasing its virtual fabric (everting it) is the paradigm example of such a homotopy. If a rendered teapot is the classical subject of computer graphics, sphere eversion is the ``teapot'' of visualizable geometry.
During an eversion the surface must be permitted to pass through itself. If either of the two constraints of continuity and regularity on a regular homotopy is relaxed, then eversion becomes trivial mathematically, though a graphical depiction may remain difficult. When both constraints are enforced, the problem has remained a challenge into this, the fourth decade since Smale proved the existence of an eversion. The collection of explicit examples has grown steadily over the years, and we discuss those that are most relevant to the present paper below.
In the early seventies, Nelson Max digitized Charles Pugh's wire mesh models of the stages in Bernard Morin's sphere eversion. Central to this eversion is an immersion of the sphere with symmetric but very complicated self-intersections. The homotopy simplifies this in stages until an embedded sphere is reached. There are two ways of proceeding that differ by an easily programmed symmetry. Reversing the one and following the other everts the sphere. With the technology of the time, Max could interact in real-time only with animated wire-frames of the homotopy, so that his film with fully rendered surfaces (see Sidebar B), had to be generated painstakingly frame-by-frame.
Mathematicians as well as computer animators require analytic expressions that parametrize homotopies. The former are obliged to mistrust purely qualitative depictions on logical grounds, while the latter find analytic representations far preferable to huge hand-generated data bases. Morin devised the first parametrizations of his eversion in the late seventies.
The power to manipulate a homotopy in real-time using a mouse did not appear until the eighties, when John Hughes used a Stardent graphics computer to realize an interactive parametrization of Morin's eversion. Like Max, he began with polyhedral models, but ones with very few vertices. Using techniques from Fourier analysis, he converted these first to power series in the frequency domain, and then mathematically manipulated the results so that their inverse transforms produced a fast and beautifully smooth eversion, a frame of which is shown in Figure 14.
More recently, François Apéry realized the Morin-Denner eversion as an illiView interactive animation, pictured in Figure 15. This polyhedral homotopy, influenced by a polyhedral Möbius band of Ulrich Brehm (who also inspired the trefoil knotbox in Figure 4), has the minimum number of vertices theoretically possible. It thus also solves an optimization problem. With the help of an illiView team, Apéry was also able to use an experimental smooth parametrization to accomplish the Morin-Apéry homotopically minimal sphere eversion.
A truly new sphere eversion based on an idea of William Thurston is the focus of the Geometry Center video Outside In, discussed in Sidebar B, and illustrated in Figure 16. From a mathematical viewpoint, the parametrization of this homotopy comes closest to Smale's original concept. The basic idea is that for any eversion there is another homotopy in an associated, higher dimensional manifold, which shadows it in an imperfect way. The equations for this doppelgänger are easy to find. Thurston solved the problem of producing an actual eversion from the higher dimensional ``shadow'' homotopy.