CS 551 / 645:
Introductory Computer Graphics
Review for Midterm
David Luebke
3/10/2016
Administrivia
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Hand out assignment 2
Hand in assignment 3
Late day policy:
– 1 late day = due tomorrow at noon
– Subsequent late days add 24 hours each
– Weekends and holidays count
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Compiling with C++
– UNIX C++ compilers: g++ and /bin/CC
– I’ll make sure it works before next assignment
David Luebke
3/10/2016
Midterm Examination
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Midterm is this Thursday (March 9)
Study aids:
– This lecture
– Earlier lectures (available on course page)
– Last semester’s midterm
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See http://www.cs.virginia.edu/~luebke/old_cs551/
But, its not quite the same material
No calculators! (you won’t need them)
David Luebke
3/10/2016
Display Technologies
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Cathode Ray Tubes
– Earliest, still most common graphical display
– Understand the basic mechanism
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Vacuum tube, phosphors, electron beam
– Pros: bright, fairly high-res, leverages TV tech
– Cons: bulky, size-limited, finicky
David Luebke
3/10/2016
Display Technologies
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Vector versus raster display
– Vector: traces lines like an oscilloscope
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Pros: bright, crisp, uniform lines
Cons: wireframe only, flicker for complex scenes
– Raster: fixed scan pattern for electron beam,
intensity controlled by scan-out from frame buffer
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David Luebke
Pros: display solid objects, image complexity limited only
by framebuffer resolution
Cons: discreet sampling (aliasing), memory cost
3/10/2016
Display Technologies
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LCDs
– Understand the basic mechanism
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Polarized light, crystals twist 90º unless excited
Basically a light valve: reflective or transmissive
– Pros: light-weight and thin
– Cons: expensive, high-power (when backlit),
limited in size
David Luebke
3/10/2016
Display Technologies
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Also know:
– Plasma display panels
– Digital Micromirror Devices
David Luebke
3/10/2016
Framebuffers
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Memory array storing image in pixels
Issues: memory speed, size, bus contention
Different types in common use, motivated
mainly by memory cost
– True-Color: 24 bits, 8 per RGB (or 32 bits with )
– Hi-Color: 16 bits (R = 6, G = 6, B = 4)
– Pseudo-Color: 8 bits index into 256-entiry color
lookup table (entries typically 24-bits)
David Luebke
3/10/2016
Mathematical Foundations
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Geometry (2-D, 3-D)
Trigonometry
Vector spaces
– Elements: scalars and vectors
– Operations:
 Addition (identity & inverse)
 Scalar multiplication (distributive rule)
– Linear combinations, dimension, basis sets
– Inner (dot) product, vector (cross) product
David Luebke
3/10/2016
Mathematical Foundations
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Affine spaces
– Elements: points
– Operations:
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Subtraction (point - point = vector)
Addition (vector + point = point)
Matrices
– Linear transforms, vector-matrix multiplication
– Matrix-matrix multiplication
– Composition of linear transforms = matrix
concatenation
David Luebke
3/10/2016
The Rendering Pipeline
Transform
Illuminate
Transform
Clip
Project
Rasterize
Model & Camera
Parameters
David Luebke
Rendering Pipeline
Framebuffer
Display
3/10/2016
The Rendering Pipeline: 3-D
Scene graph
Object geometry
Result:
Modeling
Transforms
• All vertices of scene in shared 3-D “world” coordinate system
Lighting
Calculations
• Vertices shaded according to lighting model
Viewing
Transform
• Scene vertices in 3-D “view” or “camera” coordinate system
Clipping
Projection
Transform
David Luebke
• Exactly those vertices & portions of polygons in view frustum
• 2-D screen coordinates of clipped vertices
3/10/2016
Geometric Transforms
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Modeling transforms:
object coordinates  world coordinates
Viewing transform:
world coordinates  eye coordinates
– eye coordinates == camera coordinates == view
coordinates
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Projection transform:
eye coordinates  2-D screen coordinates
David Luebke
3/10/2016
Geometric Transforms
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Understand homogeneous coordinates
– [x, y, z, w]T == (x/w, y/w, z/w)
– Allows us to capture translation and projection as
matrices
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Know your 4x4 Euclidean transform matrices:
– Translation, scale, rotation about X, Y, Z
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Understand rotation about an arbitrary axis
Understand order of composition for matrices
– In OpenGL: using column vectors as points 
order from right to left
David Luebke
3/10/2016
Perspective Projection
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Geometry of the perspective projection:
View
plane
X
P (x, y, z)
x’ = ?
(0,0,0)
Z
d
David Luebke
3/10/2016
Perspective Projection
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Desired result:
x' x
 ,
d z
dx
x
x' 

,
z
z d
David Luebke
y' y

d z
dy
y
y' 

, zd
z
z d
3/10/2016
Perspective Projection
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A matrix that accomplishes this:
1
0
Mperspective  
0

0
David Luebke
0
1
0
0
0 1
0 1d
0

0
0

0
3/10/2016
Rasterizing Lines
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Review McMillan’s great java-enabled lecture
First stab: slope-intercept + symmetry
A case study in optimization
– Special case boundary conditions if necessary
– Optimize inner loops
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Incremental update using DDA (biggest win)
Low-level tricks: integer arithmetic, compare to 0, etc.
Culmination: Bresenham’s algorithm
– Be aware of diminishing returns and
readability/portability tradeoffs
David Luebke
3/10/2016
Rasterizing Triangles
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Triangles are nice to deal with because they
are always planar and always convex
Triangle rasterization techniques:
–
–
–
–
David Luebke
REYES: recursive subdivision of primitive
Warnock: recursive subdivision of screen
Edge walking
Edge equations
3/10/2016
Rasterizing Triangles
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Edge walking:
–
–
–
–
Draw edges vertically
Fill in horizontal spans for each scanline
Interpolate colors down edges
At each scanline, interpolate
edge colors across span
– Pros:

Fast: touch only lit pixels,
touch pixels only once
– Cons:

David Luebke
Finicky: lots of special cases,
hard to get just right
3/10/2016
Rasterizing Triangles
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Edge Equations
– Equation of a line defines two half-spaces
– Triangle can be represented as intersection of
three half-spaces:
David Luebke
3/10/2016
Rasterizing Triangles
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Basic algorithm:
– Walk pixels in bounding box
– Evaluate three edge equations
– If all are greater than zero, shade pixel
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Issues:
– Computing edge equations: numerical precision
– Interpolating parameters (i.e., color): just like
another edge equation (why?)
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Optimizing the algorithm
– Like line rasterization: DDA, early termination, etc.
David Luebke
3/10/2016
Rasterizing General Polygons
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Parity test:
– Starting outside polygon,
count edges crossed.
A
– Odd = inside,
even = outside
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D
B
H
C
G
I
E
Big cost: testing every edge F
against every pixel
Solution: the active edge table algorithm
– Sort edges by Y
– Keep a list of edges that intersect current
scanline, sorted by their X-intersection w/ scanline
David Luebke
3/10/2016
Clipping Lines
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Cohen-Sutherland Algorithm
– Clip 2-D line segments to rectangular viewport
– Designed for rapid trivial accept & trivial reject ops
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David Luebke
4-bit outcodes divide screen into 9 regions
Bitwise operations determine whether to accept, reject,
or intersect with a viewport edge & recurse
May require multiple iterations
3/10/2016
Clipping Polygons
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Clipping polygons fundamentally more
difficult
– Polygons can gain or lose edges
– Concave polygons can even multiply
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Sutherland-Hodgman Algorithm
– Simplify by divide-and-conquer: consider each
clipping plane individually
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David Luebke
Input: polygon as ordered list of vertices
Output: polygon as ordered list of vertices
Lends itself to pipelined hardware implementation
3/10/2016
Clipping Polygons
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Sutherland-Hodgman Algorithm
– Know the details:
Point-plane test
Line-plane intersection
Rules:
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inside outside
inside
outside
inside outside
p
s
p
p output
David Luebke
s
i output
p
inside outside
p
s
s
no output
i output
p output
3/10/2016
Clipping in 3-D
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Problem: clipping under perspective must
happen before homogeneous divide
– Solution 1: clip to hither plane in eye coordinates,
then multiply by projection matrix, then do
homogeneous divide
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Better: transform to canonical perspective coordinates to
simplify clipping
– Solution 2: clip after projection (must clip all 4
homogeneous coordinates)
– Solution 3 (ugly but common): clip to hither & yon
before projection, clip to 2-D viewport after
projection and divide
David Luebke
3/10/2016
Color
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Rods and cones
Cones and color perception
– Metamers
– 3-D color: X, Y, and Z; CIE color space
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Gamma correction
David Luebke
3/10/2016
Lighting
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Definitions: illumination, lighting, shading
Illumination:
– Direct versus indirect
– Light properties: geometry, spectrum
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Common simplifications: ambient, directional, and point
– Surface material: geometry, reflectance,
microstructure
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Common simplification: Phong lighting
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David Luebke
Diffuse (Lambertian) reflection: incoming light reflected
equally in all directions, proportional to N • L
Specular reflection: approximate falloff with (V • R)nshiny
3/10/2016
Lighting
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Putting it all together: the Phong lighting model
I total  ka I ambient 
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#lights

i 1

 
I i  kd Nˆ  Lˆ  k s Vˆ  Rˆ


nshiny


Note: evaluate per light, per color component
Common simplification: constant V (viewer
infinitely far away)
David Luebke
3/10/2016
Shading
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Where to apply lighting calculations?
– Once per face: flat shading
– Once per vertex, interpolate resulting color:
Gouraud shading
– Once per pixel, interpolating normal vectors from
vertices: Phong shading
Flat shading
David Luebke
Phong Shading
3/10/2016