Tuesday, January 31 ∗∗ Worksheet 5 - Visualizing quadric surfaces.
1. Elliptic paraboloid. Equation
z=Ax2+By2
where Aand Bhave the same sign.
(a) What happens if either Aor Bis 0? What if they both are? Should any of these
objects be called “elliptic” paraboloids?
(b) What would happen if the sliders included negative values for Aand B?
2. Hyperbolic paraboloid. Equation
z=Ax2+By2
where Aand Bhave different signs.
(a) What does the horizontal cross section given by z=0 look like? Check on the first
picture, and also look at the equation when z=0. Is this still a hyperbola?
(b) How would z=y2−x2look different from z=x2−y2?
3. Ellipsoid. Equation
x
A2
+y
B2
+z
C2
=1.
(a) What needs to happen for an ellipsoid to be a sphere?
(b) The sliders don’t actually go all the way to 0. Make the values as small as you can
and zoom in to verify this; you’ll find you have a very small sphere. (Its radius is
0.1, as it happens.) Why shouldn’t the sliders go all the way to 0?
4. Double cone. Equation
z2=Ax2+By2.
(a) Why aren’t any of the vertical or horizontal cross sections parabolas?
(b) Explain what happens when either A=0 or B=0. Why don’t you get a cone?
(c) Similarly, what are the cross sections given by x=0 or y=0? Are these hyperbolas?
5. Hyperboloid of one sheet. Equation
x
A2
+y
B2
−z
C2
=1.
(a) Once again, the sliders don’t go all the way to 0. Why not? Make all of them as small
as possible and zoom in to see the resulting hyperboloid.
(b) Look at the equation. What should happen when x=Aor x=−A? Check this in
the first picture; recall that A=1 there.
