Computer Animation and the Geometry of Surfaces in 3- and 4-Space

Proceedings of the International Congress of Mathematicians Helsinki, 1978 Computer Animation and the Geometry of Surfaces in 3- and 4-Space Thomas F. Banchoff Geometers have always used any available media to help them illustrate their work with diagrams, pictures, and models. Modern computer graphics provides a new medium with great potential both for teaching and research. Older methods of representing curves and surfaces by drawings on a blackboard or models in wire or plaster are frequently found to be inadequate in many important geometric problems, specifically those which involve objects undergoing transformations or objects which exist properly in the fourth dimension or higher. A high-speed graphics computer makes it possible to approach and solve such problems by methods which were unavailable only a few years ago. Producing 30 different pictures per second, such a computer can display on a television tube a sequence of images which the viewer readily interprets as the projections of an object rotating in 3-dimensional space. By turning dials, a mathematician can investigate a curve or surface by having it rotate about different axes and stopping it at especially interesting positions. He or she can "fly inside" the object to focus on some local behavior or proceed to examine some specific singularity by deforming the object through a one-parameter family of curves or surfaces. Most of the classical objects of the calculus or differential geometry of curves and surfaces take on new meaning when they are reinvestigated using such methods. But these approaches also give insights into some areas where previous descriptive methods were very limited and entirely new aspects of geometry and topology become subjects for exploration. In this report, we describe five sets of films which give examples of the use of computer graphics techniques. Some are in finished form and have been used in

1006 Thomas F. Banchoff a variety of teaching situations. Others are in preliminary form, intended for mathematical research discussions. For the most part the films rely on direct and uncomplicated techniques. Most of the objects in the films are parametrically defined surfaces, given by three or four coordinate functions, each a function of two variables defined over a crosshatched domain which is usually either a rectangular or polar coordinate patch in the plane or a portion of the Riemann sphere. The images are then projected into a 3-dimensional subspace, orthogonally or centrally, and then projected again into the 2-dimensional plane of the television tube which is the output device. In some cases, for an extremely complicated object, it is possible to project two images which can be viewed with stereoscopic apparatus to give the effect of a single 3-dimensional image. Usually, however, a sense of 3-dimensionality is developed by having the image rotate slowly about an axis in the 3-dimensional space. For almost all viewers, this movement creates a spatial sensation which is interpreted readily as the shadows of a rotating transparent framework in ordinary 3-space. Subsequent deformations take place within the background context provided by this rotation. In particular as slices are made by planes parallel to a fixed direction, the curves of intersection on the rotating figure are perceived to be planar slices. Although slicing by a linear function, or more generally by some other function, does require some computational ability, the machine operates quickly enough that it is possible to view a sequence of slices in "real time", as if observing an object through a window as it rotated in the next room. The effect, however, is greater if in addition to the slice, the film displays as well the part of the surface lying below or above the slice the technique of "water-level slicing". Since this technique requires more time per picture and since it is especially well suited to representations using color, this technique is used primarily in the animation mode, where one picture is made at a time, and then the finished film is projected at 24 frames per second. For color, filters are used and each frame is exposed several times for the different portions of the picture. In addition to projection, rotation, and slicing it is possible to use linear interpolation between figures with the same parametrization. Again this is fast enough that the technique can be employed for real time manipulation of the figures for videotapes or for on-line research. More complicated programs require a recomputation of the data for every frame and are suited more for a filming mode. All of these techniques can be handled by a relatively small machine, in this case a META 4 A, B configuration with a Vector General scope, augmented by a parallel processor built at the Brown University Computing Laboratory. All films described here were produced in collaboration with Charles M. Strauss at Brown University. THE HYPERCUBE: PROJECTIONS AND SLICING treats the convex hull of the sixteen points (±1, ±1, ±1, ±1) in 4-space, first by orthogonal projection then by

Computer Animation 1007 central projection from 4-space to 3-space. In each case we rotate in the coordinate planes xy 9 yu 9 xw 9 yw 9 and zw ending at the original position. We then slice each figure by hyperplanes perpendicular to the vectors (1, 0, 0, 0) then (1,1, 0, 0) then (1,1,1,0) and finally (1,1,1,1), For a more thorough description of this film, see [4], COMPLEX FUNCTION GRAPHS treats graphs of complex functions w=f(z) considered as parametric surfaces (x 9 y 9 w, v) in 4-space, where z = x + iy and w = = u + iv. In each case orthographic projection into (x 9 y 9 u) is used to gett he graph of the real part of w (Figure 1), then rotation in the uv plane gives (x 9 y 9 v) 9 the graph of the imaginary part of w. Rotating the original graph in the xv plane leads to (y 9 w, v) the graph of the imaginary part of the inverse function of /, and finally projection to (x 9 u 9 v) gives the graph of the real part of the inverse function, FIGURES 1 AND 2 The first example is the squaring function w = z 2 with domain given by the lower half of the Riemann sphere and graph given by (x 9 y 9 x 2 y* 9 2xy) in 4-space. Projection of this locus into the (x 9 y 9 u) space gives a hyperbolic paraboloid (figure 1). Rotating in the uv plane gives the imaginary part of the squaring function, also a hyperbolic paraboloid. Rotating the original figure in the xv plane gives the imaginary part of the square root as the projection to (y 9 u 9 v) space. The graph of the inverse relation has a self-intersection curve along the positive w-axis and a singular point at the origin where the rank is 1. It represents a geometric realization of the Riemann surface of z=±]/w (Figure 2). In each projection the form of the parabolas x=constant is indicated. The special projection to the uv plane has a ramification point of order 2 which resolves into a hypocycloid with three cusps as the graph is rotated.

1008 Thomas F. Banchoff The second example is the exponential function w e z with the inverse relation z=log(w). The domain is 2TZ^X^47I 9 l^y-^l and the graph is (x 9 y 9 e x cos(y) 9 e x sin(y)) in 4-space. Projection to (x 9 y 9 u) gives the real part of the exponential (Figure 3). The projection (y 9 u 9 v) gives a right helicoid which represents the imaginary part of the Riemann surface for the logarithm (Figure 4), FIGURES 3 AND 4 The projection (x 9 u 9 v) gives a surface of revolution of a real exponential function as the real part of the logarithm (Figure 5). This example is also described in [5]. THE GAUSS MAP, A DYNAMIC APPROACH follows geometric ideas originated by Gauss in his paper defining total curvature of embedded surfaces. For the elliptic paraboloid, we show how the unit normals over a curve can be collected at a single point to form the boundary of the spherical image of the region bounded by the curve (Figure 6). ^The same procedure for the hyperbolic paraboloid produces FIGURES 5 AND 6

Computer Animation 1009 a spherical image with orientation reversed (Figure 7). For a parabolic cylinder, the spherical image degenerates to a single curve (Figure 8). We consider two examples which include elliptic as well as hyperbolic points and we examine in particular the singularities of the spherical image map. For almost all immersed surfaces, this mapping will have only folds and cusps as singularities and we indicate how two degenerate cases may be deformed to exhibit generic behavior at the cusps of this map. /r "V ^ tt:tv*' i y, ' FIGURES 7 AND 8 First we consider the monkey saddle, with an isolated point of zero Gaussian curvature, and perturb to get the graph of (x 9 y 9 x 3 3xy 2 +E(x 2 +y 2 )) (Figure 9). For =0, this surface has a Gauss mapping with a ramification point of order 2, and for fi^o, the image of the parabolic curve will have three cusps (Figure 10). Secondly we consider the biparabolic surface which is the graph of (x 9 y 9 (y x 2 )(y sx 2 )). For e^±l, the Gauss mapping of this surface will have exactly one cusp and the Gauss mapping will be stable (Figure 11). The case c=0 was first investigated by M. Menn [7] (Figure 12). In each case we show the spherical image of a circle x*+y*=r* as /* changes. We show the linear interpolation between the surface and its Gauss spherical image so that the singularities of the Gauss map are expressed as limits of singularities of homothetic images of parallel surfaces of the original surface. We then show the spherical image of a test curve centered on the curve r constant and indicate the behavior of the asymptotic vectors in a neighborhood of a cusp of the Gauss mapping.

1010 Thomas F. Banchoff Various characterizations of the singularities of the Gauss map in terms of lines of curvature, ridges, and double tangencies are included in the joint work of the author with T. Gaffney and C, McCrory [3]. THE VERONESE SURFACE is an embedding of the real projective plane which starts with the hemisphere x 2 +y 2 +z 2 =l 9 z^o and maps each point (x 9 y 9 z) to (x\ y 2 9 z 2 9 Ì2xy 9 Ì2yz 9 fäzx) in 6-space. The projection of this surface into 4-dimensional space given by (Y2xz, f2yz 9 (\\f2)(z 2 -x 2 \ f2xy) FIGURES 9 AND 10 f.- //, *\\ > FIGURES 11 AND 12

Computer Animation 1011 is again an embedding and we examine a family of projections of this surface into 3-dimensional subspaces (all of which must have local singularities) [1]. The projection into the first three coordinates gives a cross-cap with two pinch points (Whitney umbrella points). The linear interpolation of the lower hemisphere into the cross-cap is a regular homotopy right up to the last instant when opposite points on the equator are identified, forming a segment of double points (Figure 13). J W.< iii"* V ~""*- I -, - ^" r "'S'A J fr 4 - "' * 7, i FIGURES 13 AND 14 Rotating in the plane of the third and fourth coordinates gives a deformation from the cross-cap to Steiner's Roman surface (f2xz 9 Ì2yz 9 ]f2xy) (Figure 14) with tetrahedral symmetry. This projection has six pinch points which are the endpoints of three double point segments intersecting in a triple point. These examples are described further in [6]. The embedding in 4-space is tight (i.e. almost every height function when restricted to the surface has exactly one maximum and one minimum) and this property is shared by the images in 3-dimensional subspaces. These examples lead to the conjecture that any stable tight mapping of the real projective plane into 3-space must have either two pinch points or six pinch points. At the position in the rotation where the figure moves from cross-cap form to Roman surface form, the double point locus consists of two straight lines, and one of the orthogonal projections to a 2-plane is an equilateral triangle (Figures 15 and 16). In the film the cross-cap is sliced perpendicular to (1,0,0), then to (1, 1,0), then to (1,1,1). As the first two slices pass through the origin, they contain the line which is the image of the tangent plane at a pinch point and they intersect the surface in a pair of ellipses which are tangent to the line. After rotation in 4-space, Steiner's Roman surface is sliced in the same three directions, obtaining a cusp in the slice curve whenever the slicing plane passes a pinch point without containing the tangent line. The final slices have threefold symmetry,

1012 Thomas F. Banchoff :f\-r- FIGURES 15 AND 16 with a maximum, a curve with three cusps, then with three nodes, then with a triple point, then three nodes, ending at a doubly covered projective line with a Möbius band neighborhood containing three pinch points. THE TORUS is given as the surface of revolution ((2 + Y2 cos i/0 cos 0, (2 + y2 cos \j/) sin 0, j/2 sin ifr) and the slices in three different directions describe three different types of critical point behavior. Slicing perpendicular to (1, 0, 0) gives four non-degenerate critical points at different levels and the slice through the origin is a pair of congruent circles. Slicing perpendicular to (0, 0,1) gives two critical levels, each consisting of a circle of degenerate critical points and the slice through the origin is a pair of circles with the same center. A classical problem in differential geometry asks for a direction for which there are exactly three critical levels, and the film illustrates such a slice, perpendicular to (1, 0,1). In this case, the slice through the origin is again a pair of circles, this time intersecting in a pair of points [8]. THE FLAT TORUS is an embedding as a product of two circles in 4-space considered as the product of two planes, i.e. (coso, sino, cosp, sin<p). This torus is a surface on the 3-sphere of radius ]/2, as we may project stereographically from (0, 0, 0, j/2) and we obtain the torus in the previous paragraph (where sin^= /2cosc>/( /2-sinc))). Rotating the flat torus in the plane of the first and fourth coordinates produces a one-parameter family (cosa cos 0+sin a sinç>, sin0, cos<p, sin a cos 0+cos a sin<p)

Computer Animation 1013 which projects to a family of cyclides of Dupin, all conformally equivalent to the original torus. In particular when a=7t/2 and the point (0, 0, 0, j/2) lies on the torus, the result is a noncompact cyclide which separates all of 3-space into two congruent parts. The cyclides of Dupin and spheres are the only closed surfaces in 3-space which have the spherical two-piece property, so that any sphere separates them into at most two pieces. Their inverse stereographic projections are the only surfaces on the 3-sphere which are tight, so that every hyperplane separates them into at most two pieces [2]. Bibliography 1. T. Banchoff, Integral normal Euler classes jor surfaces in 4-space (to appear). 2. The spherical two piece property and tight surfaces in spheres, J. Differential Geometry 4 (1970), 193 205. 3. T. Banchoff, T. Gaffney and C. McCrory, Singularities of Gauss mappings (to appear). 4. T. Banchoff, and C. Strauss, Computer animated four dimensional geometry, Proc. Amer. Assoc. Advancement of Science, Washington, 1978. 5. T. Banchoff, Real time computer graphics techniques in geometry, The Influence of Computing on Mathematical Research and Education, Proc. Sympos. Appi. Math., vol. 20, Amer. Math. Soc, Providence, R. I., 1974, pp. 105 111. 6. D. Hilbert and S. Colin-Vossen, Geometry and the imagination, Chelsea, New York, 1952. 7. M. Menn, Generic geometry (mimeographed notes). 8. D. Struik, Lectures in classical differential geometry Addison-Wesley, Reading, Mass., 1961. BROWN UNIVERSITY PROVIDENCE, RHODE ISLAND 02912, U.S.A.