Parameterizing Surfaces \302\251 Mike May, S.J. 2006 - maymk@slu.edu restart: with(plots):

With any new technique, we start by redoing old problems in a new way. When we looked at parametric curves, the first examples we saw simply used a parametric description to plot graphs of functions that we could already graph in functional format. For parametric surfaces we follow the same approach. The first parametric surfaces to consider are the graphs of functions. They are parameterized so naturally that we usually don't even notice the parameters we use, i.e., the original variables. Thus, if NiMvJSJ6RyomJSJ4RyIiIi0lJHNpbkc2IyUieUdGJw==, we can either tell Maple to plot the graph of NiMqJiUieEciIiItJSRzaW5HNiMlInlHRiU=, a function of x and y, or to plot the surface NiM3JSUieEclInlHKiZGJCIiIi0lJHNpbkc2I0YlRic= parameterized by x and y . The two commands produce the same graph. func := x*sin(y); plot3d(func,x=-1..1,y=-Pi..Pi,axes=BOXED); plot3d([x,y,func],x=-1..1,y=-Pi..Pi, axes=BOXED); One advantage of the parametric method is that we can easily change which variable is the function and which variables are the inputs. It thus becomes easy to plot the graph of y=sin(x)*sin(z), where y is a function of x and z func := sin(x)*sin(z); plot3d([x, func, z],x=-2*Pi..2*Pi,z=-2*Pi..2*Pi, axes=BOXED); Similarly, we may have a surface defined in spherical coordinates. We can either think of it as the graph with rho a function of phi and theta, or as a parameterized surface with rho, phi and theta all functions of the parameters phi and theta. plot3d(2,theta=0..Pi, phi = 0..Pi/2, axes=BOXED, coords=spherical); plot3d([2,theta,phi],theta=0..Pi, phi = 0..Pi/2, axes=BOXED, coords=spherical); Notice that in spherical coordinates, Maple expects the coordinates to come in the order [rho, theta, phi]. We can also use spherical coordinates to plot less simple figures, ones that are not easily expressible in Cartesian coordinates. plot3d([sin(2*phi)*cos(2*theta), theta, phi],theta=0..2*Pi, phi=0..Pi, axes=BOXED, coords=spherical); As with Cartesian coordinates, we can use parametric graphing if we want to make either phi or theta a function of the other two variables. plot3d([rho, theta, arcsin(1/rho)],theta=0..2*Pi, rho=1..3, axes=BOXED, coords=spherical);

A second place where we routinely use parameterized surfaces is when we are converting a surface from a coordinate system that is natural for a surface to a coordinate system chosen for some other reason. (We may, for example, want to consider a sphere of fixed radius centered about the origin, which is easy in spherical coordinates, in Cartesian coordinates, because we need to do something like integrating center of mass for a strange density function that has been expressed in Cartesian coordinates.) Thus to switch from spherical to Cartesian coordinates we use the formulas: x=rho*sin(theta)*sin(phi). y=rho*cos(theta)*sin(phi). z=rho*cos(phi). We are interested in the cases when rho is a function of theta and phi, so the surface is parameterized that way. Consider the two surfaces plotted in spherical coordinates above when written as parameterized surfaces in Cartesian coordinates. rad:=2; plot3d([rad*sin(theta)*sin(phi), rad*cos(theta)*sin(phi), rad*cos(phi)], theta=0..2*Pi, phi=0..Pi,axes=BOXED); rad:=sin(2*phi)*cos(2*theta); plot3d([rad*sin(theta)*sin(phi), rad*cos(theta)*sin(phi), rad*cos(phi)], theta=0..2*Pi, phi=0..Pi,axes=BOXED); A particularly useful variation of this occurs when a surface natural to one coordinate system undergoes a transformation natural to another coordinate system. For example, we might want to take a sphere at the origin, and move it to another location, or stretch it into an ellipsoid. (Problems 17-20 are of this kind.) Consider the case of a sphere of radius 2 stretched into an ellipsoid by stretching the x, y, and z axes by 2, 3, and 5, then shifted to be centered at (1, -1, 2). rad:=2; plot3d([1+2*rad*sin(t)*sin(p), -1+3*rad*cos(t)*sin(p), 2+5*rad*cos(p)], t=0..2*Pi, p=0..Pi,axes=BOXED, scaling=CONSTRAINED);

When we initially looked at tangent plane to a surface z=f(x,y) at p0=(x0,y0,z0), we started by looking at slices of the graph made by holding first x and then y as constants. This gave us an x-slope, m, and a y-slope, n. This also gives an x-tangent vector (1,0,m) and a y-tangent vector (0,1,n). We can parameterize the tangent plane as: T(s,t)=(x0,y0,z0)+s*(1,0,m)+t*(0,1,n). f := x^2+y^2; x0 := 3: y0 := 2: z0 := eval(f,{x=x0,y=y0}); del := 2: m := eval(diff(f,x),{x=x0,y=y0}); n := eval(diff(f,y),{x=x0,y=y0}); surface := plot3d(f, x=x0-del..x0+del, y=y0-del..y0+del, color=red): tangent := plot3d([x0+s,y0+t,z0+s*m+t*n], s=-del..del, t=-del..del, color=blue): display3d({surface, tangent}, axes=boxed);

The place where parameterization is most important in this class is not when we have a hard function to understand, but when we have a non-standard domain. The typical double integral 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 has us trying to find the volume under a surface where the domain is defined by the limits of integration. For each value of x0 from a to b, we want y0 to take on all values from g(x0) to h(x0). It is easy to graph z=f(x,y) in general. The challenge is to only graph the function over the region of integration. We can do this with a parameterization of the domain letting x0(s) = s*b+(1-s)*a, and y0(s,t) = t*h(x0(s))+(1-t)*g(x0(s)). With this parameterization, s=0 implies x=a and s=1 implies x=b. Similarly, t=0 implies y=g(x) and t=1 implies y=h(x). Consider the example from the section on iterated integrals, 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. We first plot the surface, then add in the domain in the x-y plane. a :=1; b :=5; g := x; h := 2*x; f := (x,y) -> sin(x); x0 := s -> s*b + (1-s)*a; y0 := (s,t) -> t*eval(g,x=x0(s)) + (1-t)*eval(h,x=x0(s)); plot3d([x0(s), y0(s,t), f(x0(s),y0(s,t))], s=0..1, t=0..1, axes=boxed); plot3d([[x0(s), y0(s,t), f(x0(s),y0(s,t))],[x0(s), y0(s,t), 0]], s=0..1, t=0..1, axes=boxed, color=[red,blue]); We can do the same thing for double integrals in polar coordinates, graphing in cylindrical coordinates. Consider the integral 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, where 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. Note that for cylindrical coordinates, Maple expects the coordinates in the order [r, theta, z]. a :=Pi/6; b :=5*Pi/6; g := csc(theta); h := 2; f := (r,theta) -> r*sin(theta); theta0 := s -> s*b + (1-s)*a; r0 := (s,t) -> t*eval(g,theta=theta0(s)) + (1-t)*eval(h,theta=theta0(s)); plot3d([r0(s,t), theta0(s), f(r0(s,t),theta0(s))], s=0..1, t=0..1, axes=boxed, coords=cylindrical, scaling=constrained); plot3d([[r0(s,t), theta0(s), f(r0(s,t),theta0(s))], [r0(s,t), theta0(s), 0]], s=0..1, t=0..1, axes=boxed, color=[red,blue], coords=cylindrical, scaling=constrained); In the plots, notice that the grid is made up of curves where the parameters are constant.

1) Plot the following surfaces in Cartesian coordinates, by making them parametric surfaces: (a) A sphere of radius 3, centered at the point (2, 4, 5); (b) A cylinder of radius 2 with central axis through the point (1,3,0); (c) The cone z=r/2 shifted to have its vertex at the point (1,-2, 3); 2) Plot the surface z=2*sin(x)*sin(y) + x + y along with the plane tangent at (Pi, 2, 2+Pi) 3) The double integrals below measure the volume beneath the surface defined by the limits of integration. Map the surface we are integrating over and describe the region in words. (a) 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; (b) 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; (c) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2MFErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSVmb3JtR1EncHJlZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGSi8lKG1pbnNpemVHUSFGJy8lKG1heHNpemVHRk8tRiM2Iy1JI21uR0YkNiRRIjBGJ0YyLUYjNiMtRlU2JFEiM0YnRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRmluLUYsNidGLi1GIzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR0Y8L0YzUSdpdGFsaWNGJy1GIzYjLUZVNiRRIjJGJ0YyRmduLUYvNjBRKiZ1bWludXMwO0YnRjIvRjZRJmZhbHNlRicvRjlGZHAvRjtGZHAvRj5GZHAvRkBGZHAvRkJGZHAvRkRGZHAvRkZRJmluZml4RicvRklRMG1lZGl1bW1hdGhzcGFjZUYnL0ZMRl5xL0ZOUSIxRicvRlFRKWluZmluaXR5RidGWkZjby1GIzYjRmNvRmduRmpuLUZVNiRGYXFGMi1GLzYwUSJ+RidGMkZjcEZlcEZmcEZncEZocEZpcEZqcC9GRkZPRkhGS0ZgcUZicS1GLzYwUTAmRGlmZmVyZW50aWFsRDtGJ0YyRjVGOC9GO0Y3Rj0vRkBGN0ZBRkNGW3IvRklGTy9GTEZPRk1GUC1GZG82JVEieEYnRmdvRmlvRmhxRlxyRmNv; (d) 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; (e) 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.

Needless to say, the parametric option on plotting is most useful on figures which were defined parametrically (i.e., when the surface is given to us by a parametric description). This is the set-up many of the exercises of the section on parameterized surfaces. Consider the following parameterized surface. plot3d([r*sin(t), r*cos(t),r],r=0..5, t=0..2*Pi, axes=BOXED); The surface is easily seen to be a cone. Once we see the graph, we easily notice that the function is easily describable in cylindrical coordinates as z=r. In contrast the figure plotted below does not seem to have an easy description as a function in any standard coordinate system. plot3d([cos(s)*sin(2*t), cos(s)*cos(3*t),sin(2*s)],s=0..2*Pi, t=0..2*Pi, axes=BOXED, grid=[30,30]);

4) Use Maple to graph the parametric surfaces described below. Add a description of each surface in words. (a) 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiNUYnRjktRiM2Ji1GLDYlUSFGJ0YvRjItRjY2MFEkc2luRidGOS9GPFEmdW5zZXRGJy9GP0Zfby9GQUZfby9GQ0Zfby9GRUZfby9GR0Zfby9GSUZfby9GS0Zqbi9GTkZqbi9GUUZqbi9GU0Zqbi9GVkZqbi1GNjYwUTAmQXBwbHlGdW5jdGlvbjtGJ0Y5Rl5vRmBvRmFvRmJvRmNvRmRvRmVvRkovRk5RJDBlbUYnL0ZRRl9wRmlvRmpvLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEidEYnRi9GMkY5Rmhu, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiN0YnRjk=, 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. (b) 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, 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiNUYnRjktRiw2JVEidEYnRi9GMg==, 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. 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We are of course, interested in using parameterization to describe surfaces that can easily be parameterized, but are hard to describe as graphs of functions. (This is often because they are not graphs of functions.) Another class of examples is surfaces of revolution. Since a surface of revolution has a cylindrical symmetry, it is easiest to describe it in terms of cylindrical coordinates. When we use cylindrical coordinates, Maple expects [r, theta, z]. Consider the curve y = f(x), with x in [a, b], revolved around the x axis. The surface is easily parameterized by x and theta, the angle of revolution. Consider the following example. f := x + 5*sin(x): plot(f,x=0..3, axes=BOXED); plot3d([f,t,x], t=0..2*Pi, x=0..3, axes=BOXED, coords=cylindrical); In the picture above we turned x into the axis of rotation, which we call z in cylindrical coordinates. Then y becomes the distance from the axis of rotation or r. With the same curve we can rotate around the y axis. Then x becomes r and y becomes z. f := x + 5*sin(x): plot3d([x,t,f], t=0..2*Pi, x=0..3, axes=BOXED, coords=cylindrical); The same construction works when the original curve is a parameterized curve rather than the graph of a function. Consider the case when the curve is an ellipse moved away from the origin. plot([3*sin(t),5+2*cos(t), t=0..2*Pi],x=-4..4, y=0..8, scaling=CONSTRAINED, axes=BOXED); plot3d([5+2*cos(t),s,3*sin(t)], t=0..2*Pi, s=0..2*Pi, axes=BOXED, coords=cylindrical);

Once we understand surfaces of revolution, it is an easy step to look at surfaces that are not constructed by revolution, but whose radial cross sections are easy to describe. Consider the surface with the cross section along angle theta being a circle of radius NiMsJiIiIyIiIi0lJHNpbkc2IyomIiIoRiUlJnRoZXRhR0YlRiU= centered a distance of 8 from the central axis. (Think of the surface as a doughnut pinched 7 times.) The surface is easiest to describe parametrically in cylindrical coordinates. For a given theta we want a vertical circle of radius NiMsJiIiIyIiIi0lJHNpbkc2IyomIiIoRiUlJnRoZXRhR0YlRiU=. The center of the circle should be on the x-y plane, 8 units from the origin. We plot both radius of the circle as a function of theta and the resulting surface. r1 := 2 + sin(7*t); plot(r1,t=0..2*Pi,y=0..3, axes=BOXED); z1 := r1*cos(s); r2 := 8+r1*sin(s); plot3d([r2*sin(t),r2*cos(t),z1],s=0..2*Pi, t=0..2*Pi, axes=BOXED); We can get a better picture of the surface by increasing the number of grid lines. plot3d([r2*sin(t),r2*cos(t),z1],s=0..2*Pi, t=0..2*Pi, grid=[25, 50], axes=BOXED);

5) Using parametric equations, graph the surface described as the vase formed by rotating the curve 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRkovJShtaW5zaXplR0YuLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRjItRiw2JFEiMkYnRi8=, around the z axis. 6) The surface S is a deformed torus described as follows. The radial cross section is an ellipse of height 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 and of width 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, centered 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 from the z-axis. Parameterize S and graph it. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn