Graphics Theory

Question 1

Define a hyperplane in a d-dimensional space.

Answer 1

A hyperplane is a generalized plane; a point in 1D space, a line in 2D space, a plane in 3D space. In any d dimensional space, a hyperplane comprises a d-1 dimensional subspace.

Question 2

What spatial relations between objects are important to rendering accurately?

Answer 2

Object occlusion and object intersection

Question 2

What is the general word for polygons, polyhedra, etc.?

Answer 2

Polytopes

Question 3

Where are visibility orderings used?

Answer 3

In hidden surface removal, shadow computations, ray tracing, beam tracing, and radiosity computations.

Question 4

What is the difference between ray tracing and beam tracing?

Answer 4

Beam tracing replaces rays, which have no thickness, with beams, which have radial thickness. Beam tracing follows a volume of rays dependently.

Question 5

Explain the Painter’s Algorithm in terms of two objects separated by a single plane.

Answer 5

1) The two objects cannot intersect. 2) Give a viewer position, the object on the same side as the viewer can be drawn after and on top of the other object. Perhaps only this last part is the Painter’s Algorithm.

Question 6

What is Lambertian reflectance?

Answer 6

Lambert’s Law states that the intensity of reflective light is

I(p) = I0 * cos (p)

Where p is the polar angle. The azimuthal angle is not a dependency.

Question 8

What is the complexity for computing the relations between n polygons using brute force?

Answer 8

O(n^2)

Question 9

What two computational tools do Binary Space Partitioning Trees provide?

Answer 9

A search structure and a representation of geometry

Question 10

What aspect of object geometry allows BSP trees to compute spatial relationships (for visibility and intersections) in (n log n)?

Answer 10

The objects need to be somewhat removed from each other.