24 1. Various Ways of Representing Surfaces and Examples The non-vanishing condition is suﬃcient, but not necessary, to have a smooth curve to see the latter, consider the curve x = t3, y = t3. The tangent vector vanishes when t = 0, but the curve itself is just the line x = y, which is as smooth as we could possibly ask for. In this case we could reparametrise the curve to obtain a parametric representation in which the tangent vector is everywhere non-vanishing. d. Diﬃculties with representation by embedding. Parametric representations of curves (and surfaces as well), along with repre- sentations as level sets of functions (the implicit representations we saw before) all embed the curve or surface into an ambient Euclidean space, which so far has usually been R3. Our subsequent dealings have sometimes relied on properties of this ambient space for exam- ple, the usual definition of the length of a curve relies on a broken line approach, in which the curve is approximated by a piecewise lin- ear ‘curve’, whose length we can compute using the usual notion of Euclidean distance. What happens, though, if our surface does not live in R3? We already touched upon this problem in Lecture 1(b), and now return to it in more depth, as R3 is not the proper setting for several of the surfaces we have seen so far. For example, RP 2 cannot be embedded in R3, so if we are to compute the length of curves in the elliptic plane, we must either embed it in R4 or some higher-dimensional space, or else come up with a new definition of length, an issue to which we shall return in Lecture 23. Our discussion of factor spaces in Lecture 1 was motivated by the example of the Klein bottle, which was defined as a factor space of the square, or rectangle, where the left and right edges are identified with direction reversed (as with the M¨ obius strip), but in addition, the top and bottom edges are identified (without reversing direction). We mentioned then that the Klein bottle cannot be embedded into R3, and that the closest one can come is to imagine rolling the square into a cylinder, then attaching the ends of the cylinder after passing one end through the wall of the cylinder into the interior.

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