Geometers know that plane geometry is incomplete. The derivatives at are, and, indeed, we see that the curve is vertical near. Here is the same curve on the sphere, viewed to show zero and then. Making the interval somewhat larger loses details close to the origin. We illustrate this by plotting the function. In particular, at, the slopes of the different branches are exactly what they should be. On the other hand, points near infinity will appear close together, thus showing the asymptotic behavior that is the qualitative property of the curve far away from the origin.įaithfulness means that mapping the plane curve to the sphere does not alter the shape of the curve. This modified inverse stereographic projection is very sensitive close to the origin: two points near the origin in the plane will appear far apart on the sphere. To see through the sphere we use the Opacity option. The blue point zero is the image of the origin in the plane. Here is an animation where the view point goes once around the equator. The default view point shows, the image of the point at infinity. As an illustration, here are the graphs of a polar curve first in the plane (with asymptotes) and then on the sphere. Benefits of the Method Asymptotic BehaviorĪlthough the point at infinity cannot be reached, the mapping gives points so close to that it is as if we had reached it. The origin maps to the blue point on the sphere, and the point at infinity maps to the red point. We prefer a slightly modified version: we smoothly wrap the plane on the sphere using the inverse stereographic projection from the pole. The usual method is to map the plane graph to the sphere using the inverse stereographic projection. The remedy is to compactify the plane and represent graphs on the Riemann sphere. Furthermore, for most functions (e.g., polynomials of degree greater than four), graphing in a large window loses important details, while graphing in a small window loses the global features. This makes it difficult to understand the asymptotic behavior of complicated curves with various kinds of infinities. If the function to be plotted has a large domain or range, it is practically impossible to get a global view of the curve. Graphing a curve in the Cartesian plane can be done only in a restricted “window”. We give a procedure to plot parametric curves on the sphere whose advantages over classical graphs in the Cartesian plane are obvious whenever the graph involves infinite domains or infinite branches.
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