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Surface Parameterization

A surface in 3-Space can be parameterized by two variables (or coordinates) $u$ and $v$ such that

$\displaystyle x$ $\textstyle =$ $\displaystyle x(u,v)$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle y(u,v)$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle z(u, v).$ (3)

If a surface is parameterized as above, then the tangent Vectors
{\bf T}_u = {\partial x\over \partial u}\hat {\bf x}
+ {\pa...
...tial u}\hat {\bf y} + {\partial z\over \partial u}\hat {\bf z}
\end{displaymath} (4)

{\bf T}_v = {\partial x\over \partial v}\hat {\bf x}
+ {\pa...
...tial v}\hat {\bf y} + {\partial z\over \partial v}\hat {\bf z}
\end{displaymath} (5)

are useful in computing the Surface Area and Surface Integral.

See also Smooth Surface, Surface Area, Surface Integral

© 1996-9 Eric W. Weisstein