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To perform diffuse lighting calculations each vertex that defines a triangle must have an associated normal vector. The normal vector defines a direction that is projecting out from the front side of the triangle. The normal vector defines how light will reflect off of a surface defined at the vertex. The normal vector may be 90 degrees to the surface of the triangle, or it might be at some other angle to simulate a curved surface.

It can be shown that the dot product of two vectors is equal to the cosine of the angle between the two vectors divided by the length of the two vectors. In code format, this means that:

The dot product of two vectors is defined as the sum of the products of their associated terms. 3D vectors are normally stored as arrays, where (v[0], v[1], v[2]) is the values of the vector. Therefore, the dot product of vectors v0 and v1 is:

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The example WebGL program above was based on a “point light source”. If you had a different type of light source, such as a sun light source, the shader programs would have to be changed because the definition of your light source would change, but the fundamental math would be the same.

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As you experiment with the demonstration program, please make sure you observe the following characteristics of diffuse reflection.

The diagram to the right labels the pieces needed to calculate diffuse reflection. We need to calculate the angle between the vertex’s normal vector and a vector pointing at the light source from the vertex. This angle is labeled “theta” in the diagram.

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If both v0 and v1 are normal vectors that have a length of 1, the dot product gives the cosine of the angle without any division.

Examine the plot of a cosine curve to the right. Notice that when the angle is zero, the cosine of zero is 1.0. As the angle increases, the cosine of the angle curves to zero. When the angle is 90 degrees, the cosine of 90 is 0.0. This is lambert’s cosine law. The cosine values are treated as percentages of color. When the angle is zero, cos(0) is 1.0, and you get 100% color. When the angle is 90 degrees, cos(90) is zero and you get 0% color. When the angle becomes greater than 90 or less than -90 the cosine goes negative. This is an indication that the front side of the triangle is pointing away from the light source. You can’t have a negative percentage of light, so we clamp the cosine of the angle to values between 0.0 and 1.0.

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Light that directly strikes an object and then reflects in all directions is called “diffuse” light. The amount of light reflection is determined by the angle between the light ray and the surface normal vector. In Physics, Lambert’s cosine law provides an equation for calculating diffuse color.

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