Index Buffer

Most surfaces consist of connected triangles, whoch means that most vertices are shared between several triangles. It is not efficient to store three vertices per triangle: Much more memory would be consumed and all 3D transformations would needlessly be computed for each identical copy of a vertex. Instead, store each vertex only once and tell the graphics card how to "connect the dots". This is done through an index buffer: an array of integers giving the number of the vertices in the order in which they are to be connected, i.e., three numbers per triangle. See demo.

Reflection Models

Real surfaces can display extraordinarily complex light reflection behavior: Think for instance of perls or of holograms. In basic computer graphics, the reflective behavior of a surface is approximated using four additive components ("Phong model"), see demo:

Emission [Ausstrahlung]: The surface has a certain color, irrespective of light sources
Ambient [Umgebungs-...]: The surface shows a base color irrespective of direction or shadowing.
Diffuse [diffus]: The surface shows matte reflections.
Specular [glänzend] The surface shows glossy highlights. Highlights on highly polished materials have a small diameter.

The diffuse component changes with the surface orientation towards the light. Let n be the normal vector at a surface point, i.e., a vector of length one (unit vector) perpendicular to the surface and pointing outside. Let l be a unit vector pointing from this point to the light source. If f is the angle between both vectors, then the lighting will be proportional to cos(f), which can be computed easily by forming the dot product of n and l.

The specular component changes not only with light direction, but also with viewing direction. Let v be the unit vector pointing from the point on the surface to the viewer. The viewer looks straight into the highlight if we have a mirror-like situation, i.e., n is half-way between l and v. Thus, form the "half vector" h := (l+v)/|l+v|, another unit vector. If h = n, the viewer looks at the center of the highlight. Thus, use the dot product h.n to control the amount of light. However, this expression decays too slowly as the viewer moves away from the center of the highlight. We take the n-th power ("Phong exponent", n = 10...100) to accelerate the decay: (h.n)ⁿ.

Note that the point of maximum brightness of the diffuse component does not coincide with the center of the specular highlight. The latter depends on the position of the viewer, whereas the former does not.

According to physics, the effect of a point light source should decay with the square of the distance between light source and surface. However, this leads to overly dark images because we do not yet take indirect illumination into account in a reasonable manner. Typically, one sticks to linear decay or no decay at all.

Smooth Shading

On standard graphics cards, these lighting computations [Vorsicht: nicht "lightning" = Blitz] happen per vertex. The resulting colors are interpolated across the triangular faces to yield quite smooth shading ("Gouraud interpolation"; demo in DirectX and Cinema 4D). However, any small highlight in the center of a face will be missed; furthermore, sharp highlights reveal the underlying triangular structure. Thus, standard offline renderers use the more expensive "Phong interpolation", in which the normals instead of the colors are interpolated across the faces and the lighting model is evaluated per pixel (demo in DirectX). Using pixel shaders, this can be done on current graphics hardware, too.

Determining the Vertex Normal

Typically, normal vectors for vertices are computed by averaging the normals of the surrounding polygonal faces. The latter normals can easily be found through a vector product. Actually, the Mesh calls of DirectX already contains several helper functions for such computations.

Sometimes, however, we want sharp edges in an otherwise smooth object. Typical 3D software allows you to define such sharp edges for instance through a threshold angle. (Demo) In realtime graphics, this means to duplicate the vertices of the sharp edge: Every vertex may have only one vertex normal, but along sharp edges we need two or more normals. Thus, when smooth shading is switched on, a cube has to be formed from six squares = twelve triangles with 6*4 = 24 Vertices. Here, every geometric vertex is present three times because it belongs to three different faces with different normals.