11.1 - Introduction to Surface Properties¶

Overview of Lessons¶

In the previous chapter you learned how to model light and its reflection from surfaces. Isn’t it amazing how much better the renderings looked when the color of the pixels was based on reflected light! However, all of the previous WebGL programs used a single color to represent a surface’s appearance. This causes the surfaces to appear too uniform and “pure.” Real-world objects have more variation over their surfaces. If the surface properties of a model’s triangular mesh are modeled accurately we can approach photo-realism in CGI renderings. (To get exact “photo-realism” you typically need advanced algorithms such as ray tracing and a radiosity lighting model which are beyond the scope of this textbook.)

Surface properties are complex and we can’t cover them all here. However, we will cover the basics:

These properties can be simulated using data, using algorithms, or a combination of the two. We will discuss all three techniques.

Managing the Data¶

Surface properties for a model can be uniform across all surfaces OR the surface properties can vary for each triangle of a triangular mesh. If the surface properties are uniform, the data will be passed to a shader program using uniform variables. If the surface properties are different for each triangle, the data must be stored “per vertex” in a GPU object buffer and accessed using attribute variables.

Surface Property Transformations¶

As in the previous chapter, surface calculations will be performed in “camera space” because:

1. The GPU shader program knows the camera’s location and orientation without updating any uniform variables. (The camera is at the global origin and aligned with the global axes.)
2. Some calculations are simplified with the camera being at the origin.

Therefore, any surface properties that are related to the location or orientation of a model must be transformed into “camera space.” This is straightforward in most cases, but problematic if your transformation includes non-uniform scaling. Note that a camera transformation never includes scaling – only rotation and translation. So the problem arises if your model transformations includes non-uniform scaling.

No Scaling or Uniform Scaling¶

If your model transformation includes translation, rotation, and/or uniform scaling, you can use your camera-model transformation to convert a model’s normal vectors into “camera space”. Vectors have no location, only magnitude and direction. Therefore, a vector’s homogeneous coordinate must be zero so it can’t be translated. The vector <dx,dy,dz> becomes <dx,dy,dz,0> before being multiplied by the camera-model transformation. (As a general rule, never use a projection-camera-model transformation on model normal vectors.)

Non-uniform scaling.

Non-uniform scaling¶

Consider the diagram to the right. If you transform a model using non-uniform scaling, the normal vectors become skewed and no longer point in the correct direction. This simple example shows that the model transformation used on the vertices of a model will produce invalid normal vectors in scene space.

Non-uniform scaling.

Let’s derive a transformation that will transform the normal vectors correctly – assuming non-uniform scaling has been applied to a model. In the diagram to the right, assume that the vector, p, lies in the plane defined by the top face. The angle between p and the face’s normal vector, n, must be 90 degrees. Therefore, the dot product of p and n, must be zero. That is, dot_product(p,n) === 0.

Let M be the transformation that contained the non-uniform scaling of the model. Therefore p' = M * p. We need to solve for a different transformation, S, that will transform n into n' such that dot_product(p',n') === 0. Therefore, dot_product(M * p, S * n) === 0.

You can reorder the multiplication of a matrix times a vector by taking the transpose of each one like this:

a
e
i
m
b
f
j
n
c
g
k
o
d
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p
*x
y
z
0
=x
y
z
0
*a
b
c
d
e
f
g
h
i
j
k
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m
n
o
p
Eq2

Therefore, dot_product(p^T * M^T, S * n) === 0.

Since dot_product(p,n) === 0, M^T * S must be equal to the identity matrix. We can solve for S using this fact:

M^T * S === I

(M^T) ^-1 * M^T * S === (M^T) ^-1 * I

S === (M^T) ^-1

S === (M^-1) ^T (See the matrix identities below.)

Glossary¶

surface properties

Characteristics of a triangle that determines how light is reflected from its surface.

Basic Matrix Identities¶

The following properties of matrix operations are true (1), where A and B are square matrices, A^T is the transpose of A, and A^-1 is the inverse of A.

• (A*B)^T = B^T * A^T
• (A^T)^T = A
• (A+B)^T = A^T + B^T
• (A*B)^-1 = B^-1 * A^-1
• (A^T)^-1 = (A^-1)^T

Self Assessment¶