16 6 Parametric Surfaces and Their Areas Copyright Cengage Learning All rights reserved Parametric Surfaces and Their Areas Here we use vector functions to describe more general surfaces called parametric surfaces and compute their areas Then we take the general surface area formula and see how it applies to special surfaces 3 Parametric Surfaces 4 Parametric Surfaces In much the same way that we describe a space curve by a vector function r t of a single parameter t we can describe a surface by a vector function r u v of two parameters u and v We suppose that r u v x u v i y u v j z u v k is a vector valued function defined on a region D in the uv plane 5 Parametric Surfaces So x y and z the component functions of r are functions of the two variables u and v with domain D The set of all points x y z in x x u v such that y y u v z z u v and u v varies throughout D is called a parametric surface S and Equations 2 are called parametric equations of S 6 Parametric Surfaces Each choice of u and v gives a point on S by making all choices we get all of S In other words the surface is traced out by the tip of the position vector r u v as u v moves throughout the region D See Figure 1 A parametric surface Figure 1 7 Example 1 Identify and sketch the surface with vector equation r u v 2 cos u i v j 2 sin u k Solution The parametric equations for this surface are x 2 cos u y v z 2 sin u 8 Example 1 Solution cont d So for any point x y z on the surface we have x2 z2 4 cos2u 4 sin2u 4 This means that vertical cross sections parallel to the xz plane that is with y constant are all circles with radius 2 9 Example 1 Solution cont d Since y v and no restriction is placed on v the surface is a circular cylinder with radius 2 whose axis is the y axis see Figure 2 Figure 2 10 Parametric Surfaces If a parametric surface S is given by a vector function r u v then there are two useful families of curves that lie on S one family with u constant and the other with v constant These families correspond to vertical and horizontal lines in the uv plane 11 Parametric Surfaces If we keep u constant by putting u u0 then r u0 v becomes a vector function of the single parameter v and defines a curve C1 lying on S See Figure 4 Figure 4 12 Parametric Surfaces Similarly if we keep v constant by putting v v0 we get a curve C2 given by r u v0 that lies on S We call these curves grid curves In Example 1 for instance the grid curves obtained by letting u be constant are horizontal lines whereas the grid curves with v constant are circles In fact when a computer graphs a parametric surface it usually depicts the surface by plotting these grid curves as we see in the next example 13 Example 2 Use a computer algebra system to graph the surface r u v 2 sin v cos u 2 sin v sin u u cos v Which grid curves have u constant Which have v constant 14 Example 2 Solution We graph the portion of the surface with parameter domain 0 u 4 0 v 2 in Figure 5 Figure 5 15 Example 2 Solution cont d It has the appearance of a spiral tube To identify the grid curves we write the corresponding parametric equations x 2 sin v cos u y 2 sin v sin u z u cos v If v is constant then sin v and cos v are constant so the parametric equations resemble those of the helix Thus the grid curves with v constant are the spiral curves in Figure 5 16 Example 2 Solution cont d We deduce that the grid curves with u constant must be curves that look like circles in the figure Further evidence for this assertion is that if u is kept constant u u0 then the equation z u0 cos v shows that the z values vary from u0 1 to u0 1 17 Example 4 Find a parametric representation of the sphere x2 y2 z2 a2 Solution The sphere has a simple representation a in spherical coordinates so let s choose the angles and in spherical coordinates as the parameters 18 Example 4 Solution cont d Then putting a in the equations for conversion from spherical to rectangular coordinates we obtain x a sin cos y a sin sin z a cos as the parametric equations of the sphere The corresponding vector equation is r a sin cos i a sin sin j a cos k 19 Example 4 Solution cont d We have 0 and 0 2 so the parameter domain is the rectangle D 0 0 2 The grid curves with constant are the circles of constant latitude including the equator 20 Example 4 Solution cont d The grid curves with constant are the meridians semi circles which connect the north and south poles see Figure 7 Figure 7 21 Parametric Surfaces Note We saw in Example 4 that the grid curves for a sphere are curves of constant latitude and longitude For a general parametric surface we are really making a map and the grid curves are similar to lines of latitude and longitude Describing a point on a parametric surface like the one in Figure 5 by giving specific values of u and v is like giving the latitude and longitude of a point 22 Surfaces of Revolution 23 Surfaces of Revolution Surfaces of revolution can be represented parametrically and thus graphed using a computer For instance let s consider the surface S obtained by rotating the curve y f x a x b about the x axis where f x 0 Let be the angle of rotation as shown in Figure 10 Figure 10 24 Surfaces of Revolution If x y z is a point on S then x x y f x cos z f x sin Therefore we take x and as parameters and regard Equations 3 as parametric equations of S The parameter domain is given by a x b 0 2 25 Example 8 Find parametric equations for the surface generated by rotating the curve y sin x 0 x 2 about the x axis Use these equations to graph the surface of revolution Solution From Equations 3 the parametric equations are x x y sin x cos z sin x sin and the parameter domain is 0 x 2 0 2 26 Example 8 Solution cont d Using a computer to plot these equations and rotate the image we obtain the graph in Figure 11 Figure 11 27 Tangent Planes 28 Tangent Planes We now find the tangent plane to …
View Full Document
Unlocking...