1. Math 200
Week 3 - Friday
Quadric Surfaces

2. 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?

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

4. 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

5. 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)

6. 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

7. 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

8. 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

9. 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…

10. 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)

11. 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

12. 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

13. 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…

14. 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

15. 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

16. 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

17. 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

18. Math 200