Notes for Tuesday April 14 class -- Bicubic surface patches

I will be filling in more info in these notes, but I wanted to post this now to get you started with the material as soon as possible.

Crossing one Bezier cubic curve with another to create a bicubic patch

As we discussed in class, we can evaluate a Bezier bicubic surface patch over (u,v), which has a cross section for any fixed u and for any fixed v, by doing the following math:
spline value at (u,v) ← u³ u² u 1 ● -1 3 -3 1
3 -6 3 0
-3 3 0 0
1 0 0 0 ● P_0,0 P_0,1 P_0,2 P_0,3
P_1,0 P_1,1 P_1,2 P_1,3
P_2,0 P_2,1 P_2,2 P_2,3
P_3,0 P_3,1 P_3,2 P_3,3 ● -1 3 -3 1
3 -6 3 0
-3 3 0 0
1 0 0 0 ● v³
v²
v
1
Note that our key points matrix contains 16 key values for each of the x, y and z coordinates.
Therefore, a total of 48 floating point numbers is required to create a single Bezier bicubic surface patch.

Catmull Rom splines

A Catmull Rom spline is purely interpolating: The spline goes through all of the key points.
In a Catmull Rom spline with n cubic segments, consider segment i, which goes between key point P_i and key point P_i+1. We can then use the previous key point P_i-1 to control the slope at the beginning of the cubic segment, and the following key point P_i+2 to control the slope at the end of the cubic segment.
To convert those four key points into an interpolating spline between P_i and P_i+1, we use the Catmull Rom spline basis matrix:
a
b
c
d ←
-1/2
1
-1/2
0 3/2
-5/2
0
1 -3/2
2
1/2
0 1/2
-1/2
0
0
● P_i-1
P_i
P_i+1
P_i+2

Adding noise to lighting

In class we modified the fragment shader by adding the line shown in blue to create more interesting specular reflection lighting by using the noise function.

    // DO PHONG SHADING

    vec3 color = ambient;
    for (int i = 0 ; i < nl ; i++) {
       vec3 R = 2. * dot(uLDir[i], N) * N - uLDir[i];
       float d = max(0., dot(uLDir[i], N));
       float s = pow(max(0., dot(E, R)), p);
       color += uLCol[i] * (diffuse * d + specular * s);
    }
    color += specular * .02 * (.1 + noise(2. * N) * N.z);
    if (ta != 0.)
       color *= 1. + ta * noise(vec3(tf * vUV, 0.));
    gl_FragColor = vec4(sqrt(color), 1.0);