The Descend project is now officially reactivated and yesterday I committed the current version of the prototype into SVN repository. The UI is very basic, but it does its job of drawing a 3D surface based on mathematical equations. So far it consist of three important parts:
A parametric surface is described by a function V(p, q), where V is the vector describing the location of a point in 3D space and p and q are parameters (usually in [0, 1] or some other range). Obviously the surface consists of an infinite number of points, so we must calculate a number of samples and join the resulting points into triangles. This process is called tessellation. If the triangles are small enough, the Phong shading will create an illusion that the surface is smooth and curved.
The only difficulty is to determine what does "small enough" mean. Some surfaces are flat or almost flat and need just a few triangles to look good. Other surfaces are very curved and require thousands of triangles. In practice most surfaces are flatter is some areas and more curved in other areas. Take a sphere for example. It's curvature is the same everywhere, but we must remember that our samples are not distributed uniformly on its surface. Imagine a globe: meridians are located much closer to each other near the poles than near the equator. So in practice the distance between two samples located near the equator is greater and the surface needs to be divided into more triangles. This way, the size of all triangles will be more or less the same. Without adaptive tessellation, triangles would be closely packed near the pole and very large near the equator.
The tessellation algorithm works by first calculating four points at the corners of the surface. Wait, where does a sphere have corners? Just unwrap it mentally into a rectangular map, transforming it from (x, y, z) space into the (p, q) space. This gives us a square divided diagonally into two triangles. Then we calculate a point in the middle of the diagonal and divide each triangle into two smaller triangles. This process can be repeated recursively until the desired quality is reached.
How to measure the quality? The simplest method is to calculate the distance between the "new" point and the line that we are attempting to divide. The greater the distance, relatively to the length of the line, the more curved the surface. If this distance is smaller than some threshold value, we simply assume that the point lays on the line and discard it. The smaller the threshold, the more accurate the tessellation and the more triangles we get.
Unfortunately there are situations when this gives wrong results. If the curvature of the surface between two points resembles a sinusoid, then the third point in between appears to be located very near the line drawn between those two points. The tessellation algorithm will assume that the surface is not curved in this area. This produces very ugly artifacts.
So I came up with a method which produces much more accurate results. In order to render the surface with lighting, we need to calculate normal vectors at every point. For the Phong shading to look nice, those normals must be calculated very accurately. So two more points are calculated at a very small distance from the original one and the resulting triangle is used to calculate the normal. Note that the angle between normals is a very accurate measure of the curvature. An algorithm which compares the angle between the normals of two endpoints and the normal of the "new" point with a threshold angle can handle situations like the above much better. It's also more computationally expensive, because we must calculate three samples before we can decide if the point is rejected or not.
Of course this method can also be fooled in some specific cases, but in combination with the first one it works accurately in most cases. Experimentation shows that the threshold angle of 5° gives excellent results for every reasonable surface I was able to come up with.
In practice we also have to introduce the minimum and maximum number of divisions. Up to a certain point we simply keep dividing the grid into smaller triangles without even measuring the curvature, because otherwise the results would be very inaccurate. And since the curvature may be infinite in some points, we also must have some upper limit.
Final notes: Adaptive tessellation of parametric surfaces is the subject of many PhD dissertations and my algorithm is very simplistic, but it's just fast and accurate enough for the purposes of Descend. Also it should not be confused with adaptive tessellation of displacement mapping, which is a different concept.