QUADRIC SURFACES MATH 200 MAIN QUESTIONS FOR TODAY What are some - - PowerPoint PPT Presentation

QUADRIC SURFACES

MATH 200 WEEK 3 - FRIDAY

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?

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

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)

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 ▸ This is a parabola shifted

up on the yz-plane

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 ▸ This is a parabola shifted

up on the xz-plane

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

• nes for x=0,y=0

▸ The shape is coming

together…

▸ Here are the rest of the

traces

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…

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)

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

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…

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

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

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

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