Cylinders and Quadric Surfaces
Math 211  Prof. Leise

Email completed labs to tleise@amherst.edu by noon next Wednesday.

In Math 211, we will apply calculus methods to both curves and surfaces. Being able to visualize them will be extremely helpful, for instance, when setting up bounds on double and triple integrals or solving optimization problems. Today's lab will explore some important types of surfaces that will frequently appear in later chapters. See Section 12.6 to find out more about these surfaces.

Recall the three rules of thumb about working in Mathematica:

1) Commands and functions like Sin[x] and Sqrt[16] begin with capital letters.  Also, constants like Pi are capitalized.

2) Use square brackets with commands and functions, and use parentheses for grouping.  For example, Cos[(x+1)/(x-1)] or (1+Exp[x])^2.

3) Press shift-enter to evaluate a set of commands (grouped by brackets at far right into “cells”).

Cylinders

The textbook’s definition of a cylinder is a surface that consists of all lines that are parallel to a given line and pass through a given plane curve. Another way to think about it: a cylinder is a surface that is generated by moving a straight line along a curve while holding the line parallel to a fixed line. This process creates a “stack” of copies of the curve.

In the plot below for the parabolic cylinder y==x^2, observe that Mathematica forms the surface by combining parallel vertical lines perpendicular to a stack of parabolas:

ContourPlot3D[y==(x^2),{x,-3,3},{y,0,9},{z,0,5},AxesLabel->{x,y,z},BoxRatios->Automatic]

An example of a circular cylinder is given by x^2+y^2==1 (other examples are x^2+z^2==4 and y^2+z^2==9). These surfaces are composed of parallel lines aligned to a stack of circles.

ContourPlot3D[(x^2)+(y^2)==1,{x,-1,1},{y,-1,1},{z,0,2},AxesLabel->{x,y,z},BoxRatios->Automatic]

Any curve can be used to generate a cylinder. Note the general pattern in how we’re creating these cylinders is to define an equation in a coordinate plane (xy-plane, xz-plane, or yz-plane), so that one variable is missing from these equations.

ContourPlot3D[z==Sin[x],{x,0,2π},{y,-2,2},{z,-1,1},AxesLabel->{x,y,z},BoxRatios->Automatic]

Exercise 1.  Plot a cylinder different from any of those given above. You may use any curve you like that is distinct from the previous examples.

Quadric surfaces

Quadric surfaces are the counterparts in 3 dimensions of conic sections.

Recall that conic sections result from the intersection of a cone and a plane and include ellipses ((x^2)/(a^2))+ ((y^2)/(b^2))=1, parabolas y=((x^2)/a)+c, and hyperbolas ((x^2)/(a^2))- ((y^2)/(b^2))=1.  (More generally we can replace x and y in these equations with any pairs among x, y, and z.)

The most general equation describing quadric surfaces is

		A (x^2) + B (y^2) + C (z^2) + D xy + E yz + F xz + G x + H y + J z + K = 0

where  A, B, C, D, E, F, G, H, J, and K are constants.  By translation and rotation it can be brought into one of these standard forms:

((x^2)/(a^2)) + ((y^2)/(b^2))+((z^2)/(c^2)) = 1  (ellipsoid)
((x^2)/(a^2)) + ((y^2)/(b^2))-((z^2)/(c^2)) = 1 (hyperboloid of one sheet) or ((x^2)/(a^2)) - ((y^2)/(b^2))-((z^2)/(c^2)) = 1 (hyperboloid of two sheets)
 ((x^2)/(a^2))+  ((y^2)/(b^2))=((z^2)/(c^2))     (cone) 
 ((x^2)/(a^2)) + ((y^2)/(b^2))=(z/c)  (elliptic paraboloid)
 ((x^2)/(a^2)) - ((y^2)/(b^2))=(z/c)  (hyperbolic paraboloid)

We can swap around the roles of x, y, and z to get other forms of the same surfaces.

Exercise 2.  Graph the ellipsoid  ((x^2)/16)+ ((y^2)/9)+ ((z^2)/4) = 1 with  -4 < x < 4,  -3 < y < 3,  and -2 < z < 2. Include BoxRatios->Automatic in your Plot3D command as shown in the examples above, so the axes are scaled equally (otherwise it will look like a sphere instead of an ellipsoid).

Exercise 3.  Observe that the contour curves for the ellipsoid above all look like ellipses. We can use ContourPlot to verify that plugging in a constant value for z (or for x or y) yields an ellipse. For instance, you can plug in z=0 or z=1 (but what happens if you plug in z=2 or z=3?).

ContourPlot[(x^2)/16+(y^2)/9+1^2/4==1,{x,-4,4},{y,-3,3},AxesLabel->{x,y},AspectRatio->Automatic,Axes->True]

Exercise 4.  Graph the hyperboloid of one sheet given by  ((x^2)/16)+(( (y^2))/(9 ))- ((z^2)/4) = 1. Choose axis ranges that yield a good view of the surface.

Exercise 5.  Contour curves for the hyperboloid look like hyperbolas and ellipses. Use ContourPlot to verify that plugging in a constant value for z yields an ellipse, while plugging in a constant value for x or y yields a hyperbola.

Exercise 6.  Graph the hyperboloid of two sheets given by  ((x^2)/16)-(( (y^2))/9)- ((z^2)/4) = 1. Choose axis ranges that yield a good view of the surface. Again contour curves look like hyperbolas and ellipses for this type of surface.

Exercise 7.  Graph the cone given by  ((x^2)/16)+ (((y^2) )/9)= ((z^2)/4).

Exercise 8.  What curves are the contours for this cone? Use ContourPlot to create graphs of the different types of contour curves occurring for the cone.

Exercise 9.  Graph the elliptic paraboloid given by  (x^2)+(( (y^2))/4) = z. Choose axis ranges that yield a good view of the surface. What are the contour curves for this type of surface?

Example 10.  Graph the hyperbolic paraboloid given by  (x^2)-(y^2) = z. Choose axis ranges that yield a good view of the surface. What are the contour curves for this type of surface?

Example 11.  2.  Use ContourPlot3D to graph the following surfaces, make sure you label the axes and use BoxRatios->Automatic. Adjust the ranges for x, y, and z so that you can recognize the surfaces.  Classify each surface as an ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, cone, elliptic paraboloid, or hyperbolic paraboloid:

a)  4 (x^2) + 9 (y^2)= (z^2)

b)  144 z = 16(x^2) -  9(y^2)

c)  9(x^2) -  4(y^2) + 4(z^2) = 36

d)  4z =4 (x^2) +  (y^2)

e)   9(x^2) + 2 (y^2) + 4 (z^2) = 36

f)   (x^2) - 9 (y^2) - 9 (z^2) =9