Math 200 Week 3 - Friday Quadric Surfaces

Main Questions for Today Math 200 Main Questions for Today What are some of the main quadric surfaces? How do we distinguish between the various quadric surfaces? What is a trace? Given an equation in x, y, and z, how do we use traces to determine what the surface corresponding to the equation looks like?

Quadric Surfaces Surfaces that result from equations of the form Math 200 Quadric Surfaces Surfaces that result from equations of the form Examples:

Math 200 Traces To figure out what these look like, we’ll start by looking at traces. A trace of a surface is the intersection of the surface with a given plane This will be a curve, a point, or nothing Putting traces together, we’ll deduce what the whole surface looks like Often, traces on planes like x=0,1,2,3,…, y=0,1,2,3…, and z=0,1,2,3… will be enough

Example 1 Let’s start with z = x2 + y2 Math 200 Example 1 Let’s start with z = x2 + y2 Let’s look at the traces on the planes z = 0, z = 1, z = 2,… z = 0: x2 + y2 = 0 the only solution is the point (0,0) z = 1: x2 + y2 = 1 unit circle z = 2: x2 + y2 = 2 circle with radius sqrt(2)

Now, let’s look at traces on the planes x=0,1,-1 x=0: z = y2 Math 200 Now, let’s look at traces on the planes x=0,1,-1 x=0: z = y2 This is a parabola on the yz-plane x=1: z = 1 + y2 This is a parabola shifted up on the yz-plane x=-1: z = 1 + y2

Now, let’s look at traces on the planes y=0,1,-1 y=0: z = x2 Math 200 Now, let’s look at traces on the planes y=0,1,-1 y=0: z = x2 This is a parabola on the xz-plane y=1: z = x2+1 This is a parabola shifted up on the xz-plane y=-1: z = x2+1

Alright, now to put it all together… Math 200 Alright, now to put it all together… First, we’ll draw our traces for z=0,1,2 Then, let’s add in the ones for x=0,y=0 The shape is coming together… Here are the rest of the traces

Example 2 Let’s repeat the same process for z2 = x2 + y2 Math 200 Example 2 Let’s repeat the same process for z2 = x2 + y2 Draw traces by setting x, y, and z equal to various constant values (e.g. -1,1,0,1,1) First draw those traces in 2D Then combine them into a 3D sketch With a few traces in each “direction” you should be able to deduce the shape…

x = -1,1 both give the same trace: z2 = 1+y2 (hyperbola) Math 200 Traces z = constant z = 0: 0 = x2+y2 (only a point (0,0)) z = -1, 1 both give the same trace: 1 = x2+y2 (unit circle) x = constant x=0: z2 = y2, which is the same as |z| = |y| x = -1,1 both give the same trace: z2 = 1+y2 (hyperbola) y = constant y=0: z2 = x2, which is the same as |z| = |x| y = -1,1 both give the same trace: z2 = x2 + 1 (hyperbola)

We’ll start with the first few traces and see what we see Math 200 We’ll start with the first few traces and see what we see Already we can see that it’s going to be a double cone With too many traces drawn at once it can be tricky to visualize, but here’s what they look like on the surface

Clarifying a little bit Math 200 Clarifying a little bit We found the trace for y=1 in the last example to be the hyperbola z2=x2+1 In 2D, it looks like this This is really on the plane y=1, so isolating that curve in 3D looks like this

Math 200 Example 3 Looking at z2=x2+y2+1, we can tell one thing right away about the possible z-values… x2+y2+1 ≥ 1 which means z2≥1 …which means z≤-1 and z≥1 …which means there’s an empty space between -1 and 1 in the z-direction Draw some traces for (valid) constant values of z Draw traces for x=0 and y=0 See if that’s enough…

When z=const. we get circles. Math 200 With a few z traces and the x=0 and y=0 traces, we get a good sense of the shape When z=const. we get circles. When x=0 or y=0 we get hyperbolas We call this shape a hyperboloid of two sheets

Example 4 Hyperboloid of one sheet Math 200 Example 4 Hyperboloid of one sheet In the last example (z2=x2+y2+1) we notice that we couldn’t get z-values between -1 and 1 How is z2=x2+y2-1different? Writing it like this might help: z2+1=x2+y2 In this case, x2+y2≥1, so inside the unit circle/cylinder is empty. The traces are still circles and hyperbolas

If z = constant, we get circles (k2+1)=x2+y2 Math 200 If z = constant, we get circles (k2+1)=x2+y2 If x or y are constant, we get hyperbolas z2+1=k2+y2 z2+1=x2+k2 In combination, we get a hyperboloid of one sheet

Lastly…the hyperbolic paraboloid - AKA The Saddle Math 200 Lastly…the hyperbolic paraboloid - AKA The Saddle z = y2 - x2 Hyperbolas for z = constant (except zero) z = 0: |x| = |y| z = 1: y2 = x2+1 z = -1: x2 = y2+1 Parabolas in opposite directions for x=const. and y=const. x=0: z = y2 y=0: z = -x2

Math 200