Parametrized surface examples

Example 1

Parametrize the half-cone z = $ \sqrt{{x^2+y^2}}$.

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$ \le$v$ \le$2$ \pi$.

The parameterization of whole surface is

(x, y, z) = $\displaystyle \Phi$(u, v) = (u cos v, u sin v, u)    

for 0$ \le$v$ \le$2$ \pi$, 0$ \le$u$ \le$$ \infty$.

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

(x, y, z) = $\displaystyle \Phi$(r,$\displaystyle \theta$) = (r cos$\displaystyle \theta$, r sin$\displaystyle \theta$, r).    

Example 2

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

Solution. The parameterization becomes

(x, y, z) = $\displaystyle \Phi$(u,$\displaystyle \theta$) = (3 cos$\displaystyle \theta$, 3 sin$\displaystyle \theta$, u)    

This is a right circular cylinder of radius 3.

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Duane Nykamp
nykamp@math.umn.edu
2007-12-05
03696 hits since
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