is a vector-valued function defined on a domain D in the uv-plane. The set of all points (x,y,z) such that
as (u,v) vary over D is called a parametric surface S, and the equations 1 are called parametric equations of S. Curves on S corresponding to constant values of u and v are called grid curves.
If a smooth parametric surface S is given by the equation
and S is covered just once as (u,v) range over D, then the surface area of S is
Note that, in the event that x and y provide a parameterization,
which we've encounted before.
-
Surfaces of revolution can be easily parameterized: For example, if we have a
curve z=f(y), and we rotate this about the y-axis, then one choice of
parametric equations would be
So y and
would be the parameters. -
Tangent planes are simply determined: since the tangent vector in the case of a
parameterized space curve is
, we might assume that the tangent
plane is given by the plane through the point 
with slopes
in the u and v directions given by 
and
. Since 
is perpendicular to the pair, this is a normal vector for the plane, which has
parametric equations given by
Imagine the tip of a pencil in three-space, drawing out a surface (rather than
just a curve). That's what we're dealing with in this section. An example
surface is the 
band, which is a (fundamentally)
three-dimensional object created out of a rectangle. If the original rectangle
represents the parameter space, then the vector 
would draw out
the band.
In a sense this is a ``change of variables'' problem: for example, a sphere is
best parameterized by spherical coordinates 
and . Surfaces
overlaying rectangular regions are best parameterized by our old friends x
and y. Finding the best parameterization can be a challenge!
Once we have a parameterization, we can easily compute the usual kinds of animals: surface areas, tangent planes, etc.