2 1. Various Ways of Representing Surfaces and Examples Figure 1.1. Three familiar surfaces. distance is equal to R, the equation for the cylinder is (1.2) x2 + y2 = R2. Another surface familiar from elementary geometry (and also from ice-cream parlours) is the cone, which is obtained by rotating a straight line around another line which intersects it. If the axis of rotation is again the z-axis and the initial line lies in the xz-plane, with the equation x = az, then the cone is given by the equation (1.3) x2 + y2 = a2z2. Exercise 1.1. If we construct a surface of revolution using parallel lines instead of intersecting lines (as we did with the cone), we obtain a cylinder. There is a third possibility the lines may be skew, that is, neither intersecting nor parallel. Describe the surface obtained in this case, and derive its equation. We feel immediately that the three objects expressed by equations (1.1), (1.2), and (1.3), which are shown in Figure 1.1, are very different in a variety of robust ways. For example, the sphere is bounded— in fact, compact—while the cylinder and cone are not (contrary to what the picture might suggest). The sphere and cylinder are smooth everywhere, while the cone has a special point, the intersection of the two lines in the construction, which is the origin in (1.3). These differences are qualitative, and would not be changed if we deformed each surface by a small amount—this reflects the fact that the three surfaces in question have different topologies. Such a deformation would, however, change the quantitative properties of a surface, which constitute its geometry. For example, stretching or

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.