The 3-D parametric surface graph is the most complex of the 3-D surface and the most difficult with which to work. The key to understanding the 3-D parametric surface graph is in understanding that the surfaces are described parametrically with 2 parameters -- call them i and j. Therefore, in order to draw a three dimensional surface, we need three equations:

x = fx(i, j)

y = fy(i, j)

z = fz(i, j)

y = fy(i, j)

z = fz(i, j)

We then plot these three equations into three matrices, using the indices of the arrays for our i and j values:

A good example of how to use parameterized equations is converting from spherical to Cartesian coordinates and drawing a sphere. The equation for a sphere in spherical coordinates is quite simplistic:

To parameterize this we will simply convert the equation into Cartesian coordinates using the standard transform:

Running q from 0 to 2p and f from -p to +p generates all of the points in the sphere. Hence this is the Cartesian form of a sphere parameterized in q and f. However, we don't really want all the points in the sphere, (since that would take an infinite amount of time to plot!). Rather, we would like a few representative points from which we can draw the sphere. These we will place into our arrays. Remember that i and j, the indices of the arrays will ultimately become our parameters, so we must also convert q and f to *i* and *j*.

First, we must choose an array size to contain the points. This is largely a decision of how many points are deemed necessary to sufficiently represent the structure. For this example, I will choose arrays that are 25 by 25 elements. Therefore, both i and j will run from 0 to 24. Using this, we must map q to *i* and f to* j* (or vice-versa).

and

Note that for , , and for , . Therefore, rewriting the equations:

With these three equations, we can fill three matrices with values for *x, y*, and *z*based on the parameters *i* and *j*, which are the indices of the values in the array.

Then all that is left is to plot the arrays:

Another way of thinking about the 3-D parametric surface graph is to think of the arrays as one 2-D array of 3-D vectors, (rather than three 2-D arrays of scalars). With this concept, we can easily see that the 3-D graph draws a rectangle for each set or four adjacent elements of the array. In the pictures below, we see a portion of this sphere's data array represented as a 2-D array of 3-D vectors. Notice that every adjacent cell generates a rectangle on the 3-D representation.