5. Drawing wire-frame surfaces with isoparametric curves

It is usually best to have the surface be the image of a rectangle, even if the equations would make sense for all $t,u$. Let's say the rectangle is $a \leq t \leq b$, $c \leq u \leq d$. The choice of $a,b,c,d$ will depend upon the example.

A simple approach, the one used by Mathematica, is to imagine grid lines in the rectangle, compute the grid points where they cross, compute the images of these points under    P$$, and connect them the same way the grid is connected. If your grid points are dense enough, this gives a good picture, provided that you have the capability of surface shading, hidden-line removal, or both.

If you don't have surface shading or hidden-line removal, then you have a problem: If you use too many grid lines, the plot will be too dense and will obscure the shape of the surface. If you use too few grid lines, your surface will look jagged instead of smooth, as in Figure .

Figure: A surface drawn with coarse line segments

book/10dir/coarse.eps

The solution is a modified method: Use only a few grid lines, but use more points on each line than just the grid intersections. Compute the images of all these points and connect them with line segments the same way they are connected in the grid. The image of each grid line will then look like a curve-an isoparametric curve (curve consisting of points all with one value of $t$ or all with one value of $u$), as in Figure .

Figure: Recommended grid points to use

book/10dir/good.eps

Observe the specifications in this example: In the $t$-direction there are four intervals, each subdivided into five parts; in the $u$ direction, there are three intervals, each subdivided into four parts. In a program, these four numbers (4, 5, 3, 4) should be treated as four named constants or variables, so that you can easily choose appropriate values and change them later. The grid boundaries should also be named constants or variables.

To find the actual points used, use this idea: To subdivide an interval $[a,b]$ into $n$ pieces, the subdivision points are $a + {\frac{\displaystyle k}{\displaystyle n}} (b-a)$, for $k = 0,\dots, n$. Notice that you must loop from $0$ to $n$; if you go from $0$ to $n-1$ or $1$ to $n$ parts of your picture will be missing.

(To invent this formula, imitate the formula for a line segment: $a + t (b-a)$ for $0 \leq t \leq 1$ describes all numbers in the interval. Now take $t = {\frac{\displaystyle k}{\displaystyle n}}$ to step across the interval.)

If you are using a language such as PASCAL or C with a program that plots straight line segments, then to generate the points and connect them, use two double loops. For the horizontal grid lines, you'll need an outer loop for $u$ values corresponding to the heights of the lines and an inner loop for $t$ values corresponding to all the points on a horizontal line. For the vertical grid lines, you'll need another double loop of the same sort.

For each $(t,u)$ point you compute, you can apply    P$$ to it immediately and plot the value; there is no need to store values computed. (Points where grid lines meet will be computed twice, but that's OK. By plotting the value is meant either writing out the value in    E$^ 3$ to send to a plotting program or else doing the projection to two dimensions yourself and sending the result to a plotting program.)

For some surfaces, the images of the horizontal grid lines will be straight lines. In this case, it often looks better to use just the horizontal grid lines (or just the vertical ones, if their images are straight). In any case, if the image of the grid line on the surface happens to be a straight line, the endpoints of the grid line are already enough; there is no need to use the fine subdivision.

If there is no easy way to remove hidden lines, the plot may look messy if your surfaces have large parts that should be hidden. An easy way to avoid such messiness is to show less of a surface than you could-say, a hemisphere instead of a sphere. For some surfaces, it may be sufficient merely to view them from a good angle.

In Example 2, it can be effective to show not only the surface, but the original grid as well, drawn in the $x,y$-plane. Axes could also be included.

Of course, it isn't necessary to use the grid pattern. Sometimes some other pattern or picture in the parameter domain can be effective.