2.1  Parameterized Surfaces

A parameterized surface is the 2 variable analog of a parameterized curve. Specifically we're dealing with the map from an open subset U R2 to R3.

Example 1:   A plane is easliy generated parametrically. All that is needed is two linearly independent vectors defining the plane. Multiplying either one of them by a variable would create a parametric line in R3. By summing the two with two different variables basically extrudes one line along the other, generating a plane. For example, the plane defined by 2x + 2y + z = 0 can be uniquely defined by the vectors a = (-1,0,2), b = (0,-1,2). Using two variables, u and v, the parameterized plane is given by the equation s(u,v) = (u a + v b ) = { (-u,0,2u) + (0,-v,2v)} = (-u,-v,2u + 2v). Note the similarities between this and a vector space, it will be used extensively later in the course.

 

Example 2:   The Sphere is easily parameterized. One parameterization is given by s(u,v) = (sin(u)cos(v),sin(u)sin(v),cos(u)). If we fix v, notice that the resulting parameterized curve traces out a circle with angle v with respect to the x-axis. If we specify a range of 0 < u < p and 0 < v < 2p, the resulting graph will trace a shpere by revolving a half-circle around the z-axis. This is shown in Fig. 1. Notice that since we are mapping from an open set to R3, a curve of points is not included, specifically when u = 0, this half circle is not included. Similiarly, neither pole is included in this parameterization. However, this is not the only parameterization of a sphere, others may include the missing points.

 

 

 

 

 

 
Fig. 1

 

Example 3:   The helicoid is parameterized by (u,v) (v cos(u), v sin(u) , u). The graph generated by this function is shown in Fig. 2. Notice the similarites between this surface and the helix, described earlier.

 

 

 

 

 

 

 

 

 
Fig.2

 

The definition of a parameterized surface is as follows:

A parameterized surface S is a map from an open set U R2 R3


Note: The reason we specify an open set as the map is to allow differentiability everywhere. (A closed set is not differentiable on the edges).
Note: There are many different parameterizations of a given surface. Any single parameterization is usually not enough to define a surface since the open set mapping rarely covers the entire surface.

 

 

Example 4:   A Torus. One parameterization is given by:
(u,v) (a cos(u) + b cos(u)cos(v), a sin(u) + b sin(u)cos(v), c sin(v)), where a is the center to center radius b is the x/y radius of the outer elipse and c is the eleptical height of the elipse. This, like the shpere can be thought of as a parameterized curve revolved around the z-axis. The graph shown in fig. 3 uses the values a = 3,b = 1, and c = 2.

 

 

 

 
Fig. 3