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Chapter 4 Surfaces

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Chapter 4 Surfaces Chapter 4 Surfaces In this chapter we turn to surfaces in general. We discuss the following topics. Describing surfaces with equations and parametric descriptions. ² Some constructions of surfaces: surfaces of revolution and ruled surfaces. ² Tangent planes to surfaces. ² Surface area of surfaces. ² Curvature of surfaces. ² Figure 4.1: In this design for the 1972 Olympic Games in Munich, Germany, Frei Otto illustrates the role of surfaces in modern architecture. (Photo: Wim Huisman) 91 92 Surfaces 4.1 Describing general surfaces 4.1.1 After degree 2 surfaces, we could proceed to degree 3, but instead we turn to surfaces in general. We highlight some important aspects of these surfaces and ways to construct them. 4.1.2 Surfaces given as graphs of functions Suppose f : R2 R is a di®erentiable function, then the graph of f consisting of all points of the form!(x; y; f(x; y)) is a surface in 3{space with a smooth appearance due to the di®erentiability of f. Since we are not tied to degree 1 or 2 expressions for f, the resulting surfaces may have all sorts of fascinating shapes. The description as a graph can 1 1 0.5 0.5 0 0 2 2 -0.5 -0.5 -1 -1 0 -6 0 -2 -4 -2 0 -2 0 -2 2 2 Figure 4.2: Two `sinusoidal' graphs: the graphs of sin(xy) and sin(x). be seen as a special case of a parametric description: x = u y = v z = f(u; v); where u and v are the parameters. Just like in parametric descriptions of planes, two parameters are needed. This reflects the 2{dimensionality of a surface. Although we are used to thinking of graphs for which the points are of the form (x; y; f(x; y)), it is sometimes more convenient to think of y as a function of x and z, or of x as a function of y and z. For instance, a portion of the hyperboloid x2 + y2 z2 = 1 is described by the graph of x = 1 + z2 y2. So the function is a function of ¡y and z. ¡ So points of the graph are of the formp ( 1 + z2 y2; y; z). ¡ p 4.1.3 Surfaces given by an equation In principle, a single equation in x, y and z describes a surface in R3. Depending on the structure of the equation, the surface may look more or less smooth. Examples we have come across before include planes, like x + 2y 3z = 5, and quadrics, like the sphere x2 + y2 + z2 = 3. ¡ 4.1 Describing general surfaces 93 4.1.4 Surfaces given by a parametric description A surface is 2{dimensional. This 2{dimensionality is the reason why surfaces or parts of surfaces can be described using two parameters. So, if x, y and z all three depend on the two parameters u and v, then x, y and z will run through the points of a surface if u and v vary. We denote such a parametric description by a boldface symbol, usually x. Note that x is a `vector{valued function', so that x(u; v) is a 3d vector for every pair u; v. The coordinate functions are ordinary functions of two variables, whose values at (u; v) are denoted by x(u; v), y(u; v) and z(u; v), respectively. Whether the surface looks nice depends on the explicit expressions for x(u; v), y(u; v) and z(u; v). In general, if x(u; v), y(u; v) and z(u; v) depend di®erentiably on u and v, the surface usually does look nice. 4.1.5 Curves in space and their tangents Just like representing a curve in the plane by a parametric description of the form, x(t) = (x(t); y(t)), curves in 3{space can be described in the following way: x(t) = (x(t); y(t); z(t)); where t runs through (part of) the real numbers. For instance, x(t) = (cos t; sin t; 0) (with 0 t 2¼) is a circle in the x; y{plane, but x(t) = (cos t; sin t; t) is a kind of spiral in 3{space.· · By the way, both of these curves lie on the cylinder with equation x2 + y2 = 1, since cos2 t + sin2 t = 1. Just like in 2{space, a tangent vector to the curve at x(t0) is computed by di®erentiating the components of the curve, i.e., 0 0 0 0 x (t0) = (x (t0); y (t0); z (t0)) is the tangent vector (at least if not all three components are 0). The tangent line at x(t0) has parametric description 0 x(t0) + ¸x (t0): For example, in the case x(t) = (cos t; sin t; t), the tangent vector is x0(t) = ( sin t; cos t; 1). In particular, the tangent vector at (1; 0; 0) corresponding to t = 0, is the ¡vector (0; 1; 1). The tangent line is then described by (1; 0; 0) + ¸(0; 1; 1). The length of a (part of a) curve can be computed through an integral: t=b x0(t) dt Zt=a j j computes, under some mild conditions, the length of the part of the curve, where t ranges from a to b. In our example, the length of the curve if t varies from 0 to 4 is 4 ( sin t)2 + cos2 t + 12 dt = 4 p2 dt = 4p2. 0 ¡ 0 R pThe tangent vectors themselvRes can be used to compute the curvature of a curve, but we will not explore that topic in these notes. 94 Surfaces 4.2 Some constructions of surfaces 4.2.1 In this section we present a few ways of constructing surfaces: surfaces of revolution and ruled surfaces. Surfaces of revolution 4.2.2 Rotating a curve around a coordinate axis A surface of revolution arises by rotating a curve around a line. We have briefly discussed surfaces of revolution in Chapter 1, but only in a special case. Since the descriptions of surfaces of revolution in terms of equations or parametric descriptions can be quite complicated, we discuss this topic using an example. Suppose, we have a curve whose parametric description is as follows: x(t) = (t; t2; t3): Now suppose we rotate this curve around the x{axis and ask for a parametric description and an equation of the resulting surface of revolution. Constructing a parametric description. ² To construct a parametric description, take t to be ¯xed for the moment. The point (t; t2; t3) rotates along the circle in the plane x = t, whose radius is pt4 + t6 (the distance between (t; 0; 0) and (t; t2; t3)) and whose center is (t; 0; 0). A parametric description of this circle is (t; pt4 + t6 cos u; pt4 + t6 sin u). Varying t again, we ¯nd the parametric description of the surface of revolution: x(t; u) = (t; pt4 + t6 cos u; pt4 + t6 sin u); where 0 u 2¼ for instance. Here is maybe the most direct method: use a rotation · · Figure 4.3: Rotating the curve (t; t2; t3) around the x{axis. matrix. The matrix 1 0 0 0 0 cos u sin u 1 0 sin u ¡cos u @ A 4.2 Some constructions of surfaces 95 describes a rotation around the x{axis over an angle of u (radians, say). So we multiply each vector (t; t2; t3) with this matrix: 1 0 0 t t 0 0 cos u sin u 1 0 t2 1 = 0 t2 cos u t3 sin u 1 ; 0 sin u ¡cos u t3 t2 sin u +¡t3 cos u @ A @ A @ A and ¯nd the parametric description x(t; u) = (t; t2 cos u t3 sin u; t2 sin u + t3 cos u). ¡ Constructing an equation. ² There are several ways to ¯nd an equation. One is to start with the ¯rst parametric description obtained above and try to eliminate the parameters t and u. Since cos2 u+ sin2 u = 1 (for every u), the parameter u can be eliminated by taking the sum of the squares of the y{ and z{coordinates: y2 + z2 = (pt4 + t6 cos u)2 + (pt4 + t6 sin u)2 = t4 + t6: Since x = t, we ¯nally ¯nd y2 + z2 = x4 + x6. (Try eliminating t and u from the second parametric description yourself.) Here is another approach, without using the parametric description. Every point (t; t2; t3) will move along a circle in the plane x = t. This circle can be viewed as the intersection of the sphere with center (0; 0; 0) and radius x(t) , and the plane x = t. In terms of equations: j j x2 + y2 + z2 = t2 + t4 + t6; x = t: To get the equation of the surface of revolution, we have to eliminate the parameter t so that an equation in terms of x, y and z remains. This is easily done using the second equation: we replace t in the ¯rst equation by x, so we get x2 +y2 +z2 = x2 +x4 +x6. This simpli¯es to an equation of degree 6: x4 x6 + y2 + z2 = 0: ¡ ¡ In general, if you rotate the curve x(t) = (x(t); y(t); z(t)) around the x{axis, you have to eliminate t from x2 + y2 + z2 = x(t)2 + y(t)2 + z(t)2; x = x(t): Even for relatively simple curves, this may turn out to be a complicated matter. 4.2.3 The torus or `dough{nut' Take the circle (x 2)2 + z2 = 1, y = 0 in the x; z{plane and rotate this circle around the z{axis.
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