Exercises

01

Translating away linear terms of a quadric

Show how you can eliminate the linear part $Gx+Hy+Iz$ from the general equation of a quadric by completing squares.

02

Describe surfaces

Describe the surface accurately using as few words as possible:

• (a) $y=5$
• (b) $y^2+z^2=9$
• (c) $x^2-y=0$
• (d) $4x^2+y^2=4$
• (e) $z-\sin (y)=0$

03

Describe surfaces

Describe the surface accurately using as few words as possible:

• (a) $z=2$
• (b) $x^2+y^2=25$
• (c) $y^2+z=6$
• (d) $y^2-z^2=16$
• (e) $z-e^y=0$

04

Surfaces: rectangular to spherical coordinates

Find an equation in spherical coordinates of the form $r=f(\theta ,\phi )$ for the following surfaces given in rectangular coordinates:

• (a) $x^2+y^2-3z^2=0$
• (b) $y=2$
• (c) $x^2+y^2=4y$
• (d) $x^2+y^2=36$

05

Hyperboloid with circular traces

Suppose a hyperboloid is such that every intersection with a horizontal plane is a circle. Give a simplified form of the general equation for such a surface.