Math 155 - Computer Graphics
Final Exam Study Topics
This is an outline of the topics that can potentially be on the examination. I have
tried to make it complete, but some topics may be omitted from the list. Basically, any
topics covered in class may possibly appear on the exam. You will not be expected to
remember OpenGL command syntax, but you should know the basic command
functionality. The main OpenGL commands you are responsible for are the
matrix transformation commands (glRotatef, glTranslatef, glScalef).
Items 1-5 were already on the midterm, but could also appear on the final examination.
- Linear and affine transformations in 2-space and 3-space.
Linear maps, rigid maps, orientation-preserving maps. Rotations, translations,
generalized rotations (in 2-space).
Matrix representations. Homogeneous coordinates.
Dot products, cross products. Projecting a vector onto a (unit) vector. Computation of the
rotation matrices in 3-space.
- View transformations.
Orthographic and perspective transformations. Perspective. The use of z-buffers for hidden
surface calculations. The A+B/z "fake" distance and its purpose. How are the
values of A and B chosen? Interpolation of z-buffer values.
Projective space. Points and line at infinity. Spherical representation of projective
- Reflection models.
Phong reflection model. Superposition principle and independent treatment of different
frequencies. Multiple light sources. Ambient/diffuse/specular reflection and how they are
calculated in the Phong reflection model. Formulas for the Phong reflection model. The
properties and meanings of the N, L, V, R vectors. The H
vector and its purpose. Directional and positional lights. Data values (material and light
properties) that must be specified for the Phong reflection model.
The Cook-Torrance model will not be on the midterm.
Gourad interpolation. Phong interpolation. Advantages and disadvantages of each.
Linear interpolation between two points, between three points. Barycentric
coordinates. Calculating a point from its barycentric coordinates and vice-versa.
- Texture mapping.
Parametrically defined surfaces (u,v or s,t
coordinates). Copying an image with change in position, size or resolution -- need
to use the "come-from" or "preimage" method. Texture maps,
aliasing problems, level of detail, mipmaps, blending and averaging to avoid aliasing and
- Beziér curves and surfaces.
Control points and interpolation points. Cubic Beziér curves, basis
functions (also called blending functions) are the Bernstein polynomials, convex hull,
initial and ending tangents to Bezier curves. De Casteljau represetation. Beziér curves
of higher degree. Affine invariance. Recursive subdivision.
Geometric continuity. Geometric G1-continuity and usual C1-continuity.
Beziér patches. Affine invariance. Recursive subdivision. Ensuring
geometric continuity and geometric G1-continuity.
The vector space of degree 3 polynomials.