Parametrized surface examples

Example 1

Parametrize the half-cone z = ∘ -2----2- x + y.

Solution: For a fixed z, the cross section is a circle with radius z. So, if z = u, the parameterization of that circle is x = u cos v, y = u sin v, for 0 ≤ v ≤ 2π.

The parameterization of whole surface is

(x,y,z) = Φ(u,v) = (u cos v,u sin v,u)
for 0 ≤ v ≤ 2π, 0 ≤ u ≤∞.

Of course, there’s nothing sacred about u and v. Could also use

(x,y,z) = Φ(r,θ) = (r cos θ,r sin θ,r).

Example 2

What happens if fix the radius of the circle to x = 3 cos θ, y = 3 sin θ?

Solution. The parameterization becomes

(x,y,z) = Φ(u,θ) = (3 cos θ, 3 sin θ,u)
This is a right circular cylinder of radius 3.