22 1. Various Ways of Representing Surfaces and Examples Figure 1.16. Three curves in R3. Exercise 1.11. Find all geodesics on the round cylinder { (x, y, z) ∈ R3 | x2 + y2 = 1 } and the upper half of the round cone { (x, y, z) ∈ R3 | x2 + y2 − z2 = 0, z ≥ 0 }. c. Parametric representations of curves. We often write a curve in R2 as the solution of a particular equation the unit circle, for ex- ample, is the set of points satisfying x2 + y2 = 1. This implicit representation becomes more diﬃcult in higher dimensions in gen- eral, each equation we require the coordinates to satisfy will remove a degree of freedom (assuming independence) and hence a dimension, so to determine a curve in R3 we require not one, but two equations. Geometrically, we are obtaining a curve as the intersection of two sur- faces, each specified by one of the equations. For example, the unit circle lying in the xy-plane is the solution set of x2 + y2 = 1, z = 0. which is the intersection of this plane with a cylinder of unit radius. This is a simple example for which these equations and the visualisa- tion of the surfaces pose no real diﬃculty there are many examples which are more diﬃcult to deal with in this manner, but which can be easily written down using a parametric representation. That is, we define the curve in question as the set of all points given by (x, y, z) = (f1(t), f2(t), f3(t)),

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