Interactive Computer Graphics Course Information - Midterm Review | EECS 487, Study notes of Electrical and Electronics Engineering

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EECS 487 Fall 08

Mid-term Review

15 Oct 2008

Syllabus

●

Math

●

Lines

●

Triangles

●

Transforms

●

Hidden Surface Elimination

●

Graphics pipeline and shaders

●

Lighting

●

Textures

Lines

●

Explicit equation

●

Implicit equation

●

Parametric equation

●

Mid-point algorithm

●

Clipping

●

Performance issues

Triangles

●

Issues with horizontal/vertical scanning

●

Barycentric coordinates

●

(Anti-)Aliasing

Graphics Pipeline and Shaders

●

Pipeline architecture

●

Shaders

• (^) Vertex shader
• (^) Fragment shader
• (^) Communication

● Application  vertex / fragment shaders

● Application  vertex shader

● Vertex shader  Fragment shader

Lighting

●

OpenGL model for lights and materials

●

Global ambient light

●

Light sources

• (^) Ambient, Diffuse, Specular ●

Materials

• (^) Ambient, Diffuse, Specular, Emissive ●

The lighting equation

Textures

●

Concepts

●

Types

• (^) Image, bump-mapping, environment mapping ●

Mapping

• (^) Forward, inverse (with intermediate surface) ●

Rasterization

• (^) Bilinear interpolation, perspective ●

Magnification / minification

• (^) Simple ops, mipmaps

Math

●

Trigonometry

●

Vectors

●

Matrices

Vectors

●

Scalars: one-dimensional quantities

●

Vectors: magnitude + direction

• (^) Position vectors
• (^) Free vectors ●

Representation depends on coordinate system

• (^) Origin

● Free vectors, position vectors

• (^) Orientation and non-orthogonality

Vector Operations

a⋅b = ∣a∣∣b∣ cos  a×b = ∣a∣∣b∣ sin  c c⊥ a c ⊥ b ●

Dot product

• (^) Component of a vector along another
• (^) Commutative ●

Cross product

• (^) Defined only in 3D
• (^) Vector perpendicular to plane
• (^) Magnitude = area of parallelogram
• (^) Anti-commutative, but associative

Matrix-Vector Multiplication

●

Two views

[

a

a

⋯ a

1 n

a

a

⋯ a

2 n

⋮ ⋮ ⋱ ⋮ a

m

a

m

⋯ a

mn

] [

v 1 v 2 ⋮ v m

]

[ col 1 col 2 ⋯^ coln ]

[

v 1 v 2 ⋮ vm

]

= v 1 col 1  v 2 col 2  ⋯  vn coln

[

row 1 ⋅v row 2 ⋅v ⋮ rowm ⋅v

]

Linear combination of columns

Column of dot products

Some Matrix Properties

●

Invertible if determinant ≠ 0

●

(A B)

T

= B

T

A

T

● (A B)^1 = B^1 A^1

● (AT)^1 = (A^1 )T

●

Solution of system of linear equations

• (^) Multiplication by inverse
• (^) Cramer's rule (Sec. 5.2.5 of FCG text)
• (^) Gaussian elimination

Lines: Issues

●

Implicit form avoids infinities

●

Mid-point algorithm for scan-conversion

• (^) 4 cases based on slope
• (^) Switch end-points (and colors!)
• (^) For slope between 0 and 1:

● Given (x,y) do we set (x+1,y) or (x+1,y+1)?

• (^) Multiply f(x,y) throughout by 2 for integer arithmetic
• (^) Incremental algorithm:

● Applies to colors also!

f  x1, y 1 = f  x , y  x− y 

Triangles

●

Gouraud interpolation

●

Barycentric coordinates derivation

●

Shared-edge arbitration

●

Anti-aliasing

• (^) Render to large buffer, then downsample