Mathematicians have discovered many properties of plane curves by relating them to the geometry of the catastrophe surface in three-dimensional space. That process of taking the life history of a changing object and stretching it out in a different spatial direction makes it possible for us to translate temporal phenomena in one space into static configurations in another space. By applying all of our visualization techniques to the static catastrophe surface, we can study the phenomena in new ways.
We could now go on to examine other plane curves by the same methods, investigating relationships between a curve and its focal curve to understand better how a planar object radiates waves. But since our primary object is to explore different dimensions, we will now take the experience we have gained in studying objects in two dimensions and apply it to phenomena in three and more dimensions.
What happens when our surrounding space is not two-dimensional but three-dimensional? When the simplest object, a point, radiates heat or sound or light, then the waves that emanate through the surrounding space are concentric spheres. The precise speed of the waves depends on the system's physical characteristics, but the geometric shape of the waves will invariably be described by these spherical wave fronts.
We could portray the entire history of the waves from a point by drawing a collection of concentric spheres, but the larger spheres will totally obscure the others. One remedy would be to use transparent spheres, or we could make use of the symmetry of the wave front by showing not the entire wave front but rather half of it, a lower hemisphere. The successive waves from a point will then be nested hemispherical shells, which we can see all at once. This representation clearly shows that the concentric circles in the plane of the equator display precisely the history of the waves emanating from a point in that plane.
The simplest one-dimensional object, a straight line, sends out wave fronts in space that are circular cylinders, all with the same axis. A cross section perpendicular to the axis will reproduce the same concentric circle pattern generated by a point in the plane. If we slice the cylinders by a plane containing the original line, we obtain the lower-dimensional case of a line radiating pairs of parallel lines.
Once again, the simplest closed curve is a circle. The waves that emanate from a circle in space are surfaces of revolution. At the beginning, such a wave front will be a torus, the surface formed by revolving a small circle centered at a point of the original circle and lying in a plane perpendicular to the plane of that circle. As the wave front moves outward from the circle, it begins to run into itself and develop singularities.
To prevent the larger parallel surfaces from obscuring what is happening inside, we can use slicing techniques to expose the family of parallel surfaces. If we slice by the plane of the original circle, we obtain a family of pairs of concentric circles identical to the wave fronts from a circle in the plane.
However, if we slice by a plane perpendicular to the plane of the circle and passing through the center of the circle, we obtain something quite new. Such a plane intersects the original circle in a pair of opposite points situated symmetrically with respect to a line through the center of the original circle. From each of these points there proceeds a family of concentric circles. As the wave front moves inward, the two circles do not intersect at first, but eventually the two outermost circles touch at one point and subsequently meet at a pair of points on the symmetry line.
The parallel surface at each stage is obtained by forming the surface of revolution of the pair of curves about the symmetry axis. At first this surface is a torus without singularities. When the circular waves first touch, their surface of revolution is what nineteenth-century geometers called a ``horn cyclide.'' When the circular waves intersect, then their surface of revolution becomes a ``spindle cyclide.'' It has two singular points, each resembling the singular point of a double cone.
We may also consider the parallel surfaces radiating from a surface in three-dimensional space. As before, if the distance to a smooth surface is small enough, the parallel surfaces will themselves be smooth. As the distance increases, the parallel surfaces may develop singularities. For example, the parallel surfaces to a sphere will collapse down to the sphere's center and reemerge, so there is just one focal point.
For an ellipsoid of revolution, formed by revolving an ellipse around one of its axes of symmetry, the parallel surfaces will themselves be surfaces formed by revolving the parallel curves of the ellipse around the same axis. The singularities of these parallel surfaces will be circular curves of cusps and ``spindle points'' and circles of double points. These singularities all lie either on the surface of revolution of the evolute curve of the ellipse or on a segment along the axis of revolution.
But what happens as we deform an ellipsoid of revolution to an ellipsoid with three unequal axes? This question was raised over 130 years ago by the British mathematician Arthur Cayley. By a great effort he was able to construct one picture to show the form of the focal surfaces for a single example. Today, using computer graphics, we can create entire families of parallel surfaces and their associated focal surfaces.
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