22M:28Spring 07J. SimonPractice withParameterizing Curves and SurfacesPart 2: SurfacesTo calculate (in fact, to define) the surface area of a 2-dimensional surface in 3-space, and to calculate (in fact, to define) an integral whose domain is some surface, we need to parameterize the surface. Since the surface is a 2-dimensional thing, we expect to need 2 parameters; since the surface lives in 3-space, we expect to have 3 coordinate functions.So a parameterization of a surface in 3-space is a function from some subset of R^2 into R^3.We can denote such a function as X(s,t) = [ x(s,t) , y(s,t) , z(s,t) ] .Example 1.Parameters (s,t)Coordinate functions x(s,t) = s+t^2 y(s,t) = cos(s) z(s,t) = sin(t)Let's consider two different domains: Example 1a. s=0..1, t=0..1 a square in (s,t) space. Example 1b. s=-1..1 , t=-sqrt(1-s^2) .. sqrt(1-s^2) a circular disk in (s,t) space.> with(plots):Warning, the name changecoords has been redefined> X:= (s,t) -> [s+t^2, cos(s), sin(t)];X := s, t( ) ® s + t2, cos s( ), sin t( )[ ]> plot3d(X(s,t), s=0..1, t=0..1);The surface S may be easier to see if we make the picture prettier.> > plot3d(X(s,t), s=0..1, t=0..1, axes=boxed, lightmodel=light1,labels=[x,y,z], color=pink, orientation=[115,73]);0.60.72y00.81.50.20.4x10.90.6z0.50.801 Now let's use the same function X(s,t), but apply it to a different domain in (s,t) space.> plot3d(X(s,t), s=-1..1, t=-sqrt(1-s^2)..sqrt(1-s^2));It will be much easier to visualize the surface if you use the computer to do the plot, then play with the picture by rotating or zooming. But we can make a static picture a bit more informative.> plot3d(X(s,t), s=-1..1, t=-sqrt(1-s^2)..sqrt(1-s^2), color=yellow, labels=[x,y,z], axes=boxed, lightmodel=light1, orientation=[42,39]);-1-0.8-0.50.60.70-0.4x0.8y0.50.9101z0.40.8 ########################Examples with more familiar surfacesExample 2. S = a sphere, or part of a sphereHere are some parameterizations of a standard sphere: radius = 3, center = [0,0,0], or parts of that sphere.Solution 2a. Use "spherical coordinates" x = rho * sin(phi) * cos(theta) y = rho * sin(phi) * sin(theta) z = rho * cos(phi) For a sphere of fixed radius 3 , we just use rho = 3. The two parameters are the angles phi and theta.> plot3d([3*sin(phi)*cos(theta), 3*sin(phi)*sin(theta), 3*cos(phi)], phi=0..Pi, theta=0..2*Pi, scaling=constrained);To parameterize the upper hemisphere, we just restrict phi to run from 0 to Pi/2.> plot3d([3*sin(phi)*cos(theta), 3*sin(phi)*sin(theta), 3*cos(phi)], phi=0..Pi/2, theta=0..2*Pi, scaling=constrained, orientation=[51,104]);To parameterize the right hemisphere (i.e. x coordinates positive), use phi=0..Pi, but have theta to run between -pi/2 and pi/2 .> plot3d([3*sin(phi)*cos(theta), 3*sin(phi)*sin(theta), 3*cos(phi)], phi=0..Pi, theta=-Pi/2..Pi/2, scaling=constrained, orientation=[51,104], orientation=[-116,70], axes=boxed, labels=[x,y,z]);321y0-1-2-332.5x21.510.50-3-2-10z123 **Exercise: Parameterize the portion of the sphere of radius 5 centered at the origin in R^3 that lies in the first octant.**Exercise: Parameterize the portion of the sphere of radius 5 centered at the origin in R^3 that lies in the octant where y is negative, x is negative, and z is positive.What about the portion of the sphere of radius 3 that lies above some z-value (think of a polar ice cap).We want to restrict, e.g., z > 2. In spherical coordinates, z = rho * cos(phi) = [in our case] 3 cos(phi). So we want to restrict the angle phi to make 3 cos(phi) > 2. That says cos(phi) > 2/3, so we need to have phi run from 0 to arccos(2/3).> plot3d([3*sin(phi)*cos(theta), 3*sin(phi)*sin(theta), 3*cos(phi)], phi=0..arccos(2/3), theta=0..2*Pi, scaling=constrained, orientation=[104,96], axes=boxed, labels=[x,y,z]);-2-10z32.82.62.42.221210x-12-2 #####Example 3.#####Parameterize a cylinder, or part of a cylinder.For example, parameterize a circular cylinder (i.e. cross-section is a circle) in R^3 of radius 3 and height 5, whoseaxis is the y-axis. [I have to say "a" rather than "the" cylinder because I have not specified where the base is located. Let's say the base is supposed to lie in the plane y=1.]We need two parameters: To describe a cylinder we might expect to use cylindrical coordinates, something like (r,theta,z). But not all three coordinates are independent here; and the "z" coordinate that is supposed to correspond to "height" is supposed to run along the y-axis here, not the z-axis. This can get confusing. So we will use (s,t) instead of suggestive letters like "r" or "theta".s is the polar angle, running from 0 to 2*Pit is the height, running [we are instructed] from 1 to 1+5=6.X(s,t) = [x,y,z], wherex(s,t) = 3 cos(s)y(s,t) = tz(s,t) = 3 sin(s)> plot3d([3*cos(s), t, 3*sin(s)], s=0..2*Pi, t=1..6); Let's make the picture better, and include the axes so we can see we have the right cylinder.> plot3d([3*cos(s), t, 3*sin(s)], s=0..2*Pi, t=1..6, scaling=constrained, axes=normal, labels=[x,y,z], view=[-6..6, -3..12, -6..6]);-6-6-4-4-2-2-2000x2224z4646y861012 **Exercise: Parameterize a cylinder of radius 2 and length 4 whose axis is the x-axis.**Exercise: Parameterize a cylinder whose cross-section is an ellipse with radii 2 and 3, and whose axis isthe x-axis.**Exercise (for anyone who is finding these too easy) Parameterize a cylinder whose cross-section is a circle of radius 1, whose length is 5, and whose axis is the line y=x in the xy-plane. Here is a picture, just toprove it can be done. > >-2-2-1-2y0002z416224x68