Parametrized surface examples
Example 1
Parametrize the half-cone
z =
.
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
2
.
The parameterization of whole surface is
| (x, y, z) = |
Of course, there's nothing sacred about u and v. Could also use
| (x, y, z) = |
Example 2
What happens if fix the radius of the circle to
x = 3 cos
,
y = 3 sin
?
Solution. The parameterization becomes
| (x, y, z) = |
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