6. Introduction to Surfaces

When comparing mathematics in two and three dimensions, there are many similarities. Very often, the techniques used in the simpler two-dimensional case easily extend to cover three dimensions. Some of the curve representations presented in the previous sections easily extend to three dimensions and can therefore represent surfaces.

When creating a curve, we used a single parametric dimension, defined points within this dimension then used this to create our curve. For a surface, we need two orthogonal parametric dimensions of points. These form a rectangular mesh. At any point in parametric space, we use two blending functions, one in each parametric direction. For every knot defined, we calculate the Cartesian product of the two blending functions and this is the weight given to that knot. The sum of all the weights will still be one as it was for a curve.

The most commonly used methods of representing curved surfaces in computing are by Bézier surfaces and B-spline surfaces, and this tutorial is limited to these types.

A note on using the VRML worlds

A VRML world is a text file containing a description of the world. This file is in a computer language and subsequently a VRML browser is needed to decode it. Several browsers exist, though this software has only been tested using Blaxxun Contact, which is available free from http://www.blaxxun.com. Versions are available for Windows 95/98 and NT.

Using a VRML world is fairly intuitive, though a brief introduction is still warranted. Your view is that of an avatar. You can use the cursor keys to walk around or use the compass displayed at the bottom of the screen in Blaxxun Contact. You can not walk through walls or climb over large objects. When you enter these worlds, you will see a table in front of you. The outlined cube above the table is the 3D graph. The knots of the surface are red spheres and the surface itself is flat on the table.

To move a knot, simply move the pointer over it, hold the mouse button down and drag it. Depending on your computer system, the reaction may be sluggish; be patient. By moving a knot upwards, the surface should reveal itself. This is the wire mesh representation. Click on the blue button on the side of the table. Now you can see the solid representation. Click the blue button again and both representations are visible. Clicking again returns to the mesh only display.

The slider to the right of the blue button controls the size of the mesh. Move it to the left for a faster but less accurate display; move it to the right for a more accurate but slower display. Be careful: if your computer struggles with the default setting, moving it to a higher level could keep your browser busy for a minute or two!

The knots can be moved anywhere within the cube. They can only be moved in one plane at a time, either the x-y or z-y planes (where y is vertical). The two buttons to the left of the blue button represent two these two perpendicular planes. The knot will move in the same plane as that indicated by the green button.

To clear all the points and start again, click reload on your browser. To leave the world, click back or type in a new internet address.

Previous: B-Spline curves
Next: Bezier surfaces