Final

1.

The furnace. Given a closed environment where everything has a reflectance of $\rho < 1.0$ and an emission of 1, give the equation for the radiance at any point in the environment. Is an equation possible for the environment between two concentric spheres with reflectances of $\rho_1$ and $\rho_2$? Why or why not?

2.

Write the equation for the form factor from a point x to an object O. Write the equation for the exact radiance at point x due to object O where the radiance function over the object is L. What are the assumptions that are made when using form factors?

3.

What are the practical advantages of Gauss-Seidel over Jacobi for solving the system. (There are two main ones). Is there an advantage to Jacobi? If so, what? Why do people use iterative techniques to solve these systems?

4.

List 3 things that can be done (and why they are done) to clean up models before computing a radiosity solution.

5.

Given a hierarchy with basis functions B^i,j₁ and B^i,j₂ at each node (patch) of the hierarchy (where i is the level and j is the specific patch at that level), how much of the radiance, b^i,j₁ and b^i,j₂, on level i is pushed down to the basis functions of a child j' on level i + 1 (find b^i+1,j'₁ and b^i+1,j'₂)?

Rough Key

1.

First, I was looking for the $1 + \rho + \rho^2 + \cdots = \frac{1}{1 - \rho}$. For the concentric spheres, the correct answer is that there is a simple equation. The reason there is, is that there are two types of points, outer sphere, and inner sphere. All points on the inner sphere can see only outer sphere points. All points on the outer sphere see exactly the same thing, the inner sphere ringed by a band of the outer sphere. So the system is governed by two simple equations for each bounce. Outer sphere gets some (constant for all points) energy from the outer sphere and the rest from the inner sphere. Inner sphere gets all energy from the outer sphere. The exact values are based on the size of the spheres and the form factor from the inner sphere to a point on the outer sphere.

2.

$$F_{x,O} = \int_O \frac{\cos\theta_x \cos\theta_y}{d^2} {\rm vis}(x,y) dy$$

and (In the general case, I should have asked for the direction as well, but assuming diffuse (with $\rho(x) < frac{1}{\pi}$)

$$L(x) = \rho(x) \int_O \frac{\cos\theta_x \cos\theta_y}{d^2} {\rm vis}(x,y) L(y) dy$$

The key assumption I wanted is that L(y) is constant over the source. Other assumptions for form factors are ideal diffuse radiance distribuitions/BRDF's and BRDF uniform over receiving surface.

3.

Gauss-Seidel converges faster and uses less memory (no temporary vector). Jacobi has an exact physical meaning. Each iteration of Jacobi propagates energy one more bounce into the environment. k iterations computes $(I + M + M^2 + \cdots + M^k)L_e$. This is useful if you want exactly k bounces, or if you want a solution where there is no emission or direct lighting (to be added later) which can be done by not adding in the emitted radiance on the last two iterations of Jacobi. It also simplyfies using importance. Non iterative techniques are way too slow and require too much memory.

4.

Merging geometry (edges, vertices, surfaces), removing invalid geometry, creating better initial meshes (discontinuities, aspect ratios), orienting polygons, etc.

5.

$$b^{i,j}_1 \left< B^{i,j}_1 \, , \, \mbox{$\widetilde{B^{i+1,j}_1}... ... + b^{i,j}_2 \left< B^{i,j}_2 \, , \, \mbox{$\widetilde{B^{i+1,j}_1}$ } \right>$$ b^i+1,j'₁ = (28)

$$b^{i,j}_1 \left< B^{i,j}_1 \, , \, \mbox{$\widetilde{B^{i+1,j}_2}... ... + b^{i,j}_2 \left< B^{i,j}_2 \, , \, \mbox{$\widetilde{B^{i+1,j}_2}$ } \right>$$ b^i+1,j'₂ = (29)