CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE 41

12.6 and Quadric Surfaces

Definition. Given a surface, the traces of the surface are the intersection of surface with planes that are parallel to one of the coordinate . In other words, they are the curves obtained by setting one of the variables equal to a constant. They are also called cross sections. Definition. A is a surface that consists of all the lines that are parallel to a given line and through a given . Equivalently, it can be described as having constant along a given axis. Example 1. For each of the following surfaces, describe some of the traces and the resulting surface. (a) x2 + y2 =1 z2 (b) y2 + =3 4 (c) z = y2

Solution: (a) The traces with z = 0, z = 1, etc., are all circles with radius 1. This is a circular cylinder with a vertical axis.

(b) The traces with x = 0 is an with z-radius p12 and y-radius p3. So are the other traces. The cylinder axis is along the x-axis. (c) The trace with x = 0 is a . So is the trace with x = 1, etc. This is a parabolic cylinder with axis along the x-axis.

Definition. A quadric surface is the set of points (x, y, z) that satisfy a second degree equation

Ax2 + By2 + Cz2 + Dxy + Eyz + Fxz+ Gx + Hy + Iz + J =0

(A, B, . . . , I are constants) Luckily, we won’t study quadric surfaces with the equation given in this general form. Through rotation and translation, the same shapes will satisfy one of the following standard forms:

Ax2 + By2 + Cz2 + J =0, or Ax2 + By2 + Iz =0.

Definition. A surface that has all traces that are is called an .A surface that has traces along two axes given by is called a .A surface that has traces given along one axis is called a . Example 2. Describe the traces, in three di↵erent directions, for the following shape:

x2 z2 + y2 + =1 9 16 What kind of surface is the shape? CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE 42

Solution: If x = 0, then the equation becomes z2 y2 + =1 16 which is an ellipse in the yz-plane. If y = 0 then the equation becomes x2 z2 + =1 9 16 which is an ellipse in the xz-plane. If z = 0 then the equation becomes x2 + y2 =1 9 which is an ellipse in the xy-plane. The surface is an ellipsoid. Example 3. In the table below “var” stands for any variable: x, y or z. Also “#” stands for a positive number. Fill in the table with what kind of shape the equation makes: circle, ellipse, , or parabola.

Equation shape

var2 + var2 =# var2 var2 + =# # # var2 = var2 # ± var = var2 # ± ± Solution: var2 + var2 =# circle var2 var2 + = # ellipse (if #s on bottom aren’t equal) # # var2 = var2 # hyperbola ± var = var2 # parabola ± ± In at least three of the cases it should be pretty easy to verify that the answers shown above are correct. For instance, for var2+var2 = #, just imagine something like x2+y2 = 1, and then recall that that’s a circle. The one that is probably not very familiar is var2 =var2 #. Let’s make it more familiar: y2 = x2 + 1. Probably you don’t know what that looks± like, but you can graph it by solving for y and getting y = x2 + 1. Let’s look first at y = x2 + 1. For large values of x this should be y x±, which gives us a slant asymptote of y = x.On p p the other side if x is large and negative,⇡ we have y x, and so y = x is also a slant asymptote. In the middle, for instance when x = 0,⇡ we get y = 1. So, the graph should start at the point (0, 1) and then curve towards y = x: ± CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE 43

The graph of y = x2 + 1 gives us a mirror image on the bottom p

In a similar fashion, let’s look quickly at y2 = x2 1. The functions y = x2 1 should have slant asymptotes at y = x.Butx = 0 is not defined. What about x = 1? This p gives y = 0. So, this function “starts”± at the point (1, 0), and then curves up and down towards y = x. The function y = x2 1 is a mirror image, and these two functions together give± us the left and right sides of the following p CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE 44

These have exactly the same shape as the more familiar y =1/x,buthave been rotated 45 either clockwise or counter-clockwise. Example 4. The table on the next page shows most (all?) of the basic shapes that quadratic surfaces have. Can you figure out what some of the traces look like in each case? We show on the following page the Matlab code used to generate these graphs. Solution: (a) Each trace is an ellipse. (b) The z = 0 trace is a circle. The x = 0 and y = 0 traces are hyperbolas. (They are hyperbolas, and not parabolas, because their shape approaches a slant asymptote that is a straight line.) (c) The z = 0 trace is a point. The z = 1 trace is a circle. The x = 0 trace is a pair of straight lines, like z = y.Thex = 1 trace is a hyperbola. (It is a hyperbola since it approaches a straight line.± If you can’t picture this, and think that maybe it’s a parabola, that’s fine, it’s not easy to see the di↵erence.) The y-traces are just like the x-traces. (d) The z = 0 trace is empty, there is no point there. The z = 2 trace some sort of a closed curve, so either an ellipse or a circle. In this particular picture, it’s an ellipse, but it’s hard to tell just by looking at it. The x = 0 and y = 0 traces are hyperbolas. (e) The z = 1, z = 5, etc., traces are all ellipses. The x = 0 and y = 0 traces are parabolas. (f) The z = 0, z = 5, etc., traces are all hyperbolas. The x = 0 and y = 0 traces are parabolas that are perpendicular to each other, and open in opposite directions. This surface is almost always called a “saddle”.

This is where we ended on Monday, January 28 CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE 45

(b) One Sheet Hyperboloid

(a) Ellipsoid

(d) Two sheet hyperboloid (c)

(e) Elliptic Paraboloid (f) Hyperbolic Paraboloid CHAPTER 12. VECTORS AND THE GEOMETRY OF SPACE 46

%% plot hyperboloid of one sheet %% plot ellipsoid t= linspace(-10,10,25); % parameterizes z- value [x,y,z]=ellipsoid(0,0,0,6,3,3,30); [x,y,z]=cylinder( sqrt(0.003*t .^2+0.05)); surf(x,y,z); surf(x,y,z); axis off axis equal axis equal view(-38,13); grid off grid off axis off %% hyperboloid of two sheets x = linspace(-7,7,25); y = linspace(-4,4,25); %% plot cone [X,Y]=meshgrid(x,y); t =0:pi/30:2*pi; z = sqrt(X .^2/3 + Y .^2+1); [x,y,z]=cylinder([1 0 1]); surf(X,Y,z); hold on surf(x,y,z); z=-sqrt(X .^2/3+Y .^2+1); axis off surf(X,Y,z); hold off axis equal axis equal grid off zlim([-4 4]) view(-32,11) view(-38,11) grid off axis off %% hyperbolic paraboloid %% plot elliptic parabaloid x = linspace(-7,7,25); x = linspace(-5,5,25); y = linspace(-4,4,25); y = linspace(-2.5,2.5,25); [X,Y]=meshgrid(x,y); [X,Y]=meshgrid(x,y); z = X .^2/4 - Y .^2; z = X .^2/4+Y .^2; surf(X,Y,z); surf(X,Y,z) axis equal axis equal zlim([-4 4]) zlim([0 5]) view(-38,11) grid off grid off axis off axis off