1. CS 551 / 645: Introductory Computer Graphics
Review for Midterm
David Luebke /30/2018

2. Administrivia 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
Compiling with C++
UNIX C++ compilers: g++ and /bin/CC
I’ll make sure it works before next assignment
David Luebke /30/2018

3. Midterm Examination Midterm is this Thursday (March 9) Study aids:
This lecture
Earlier lectures (available on course page)
Last semester’s midterm
See
But, its not quite the same material
No calculators! (you won’t need them)
David Luebke /30/2018

4. Display Technologies Cathode Ray Tubes
Earliest, still most common graphical display
Understand the basic mechanism
Vacuum tube, phosphors, electron beam
Pros: bright, fairly high-res, leverages TV tech
Cons: bulky, size-limited, finicky
David Luebke /30/2018

5. Display Technologies Vector versus raster display
Vector: traces lines like an oscilloscope
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
Pros: display solid objects, image complexity limited only by framebuffer resolution
Cons: discreet sampling (aliasing), memory cost
David Luebke /30/2018

6. Display Technologies LCDs Understand the basic mechanism
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 /30/2018

7. Display Technologies Also know: Plasma display panels
Digital Micromirror Devices
David Luebke /30/2018

8. Framebuffers 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 /30/2018

9. Mathematical Foundations
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 /30/2018

10. Mathematical Foundations
Affine spaces
Elements: points
Operations:
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 /30/2018

11. The Rendering Pipeline
Transform
Illuminate
Clip
Project
Rasterize
Model & Camera Parameters
Rendering Pipeline
Framebuffer
Display
David Luebke /30/2018

12. The Rendering Pipeline: 3-D
Scene graph Object geometry
Result:
All vertices of scene in shared 3-D “world” coordinate system
Vertices shaded according to lighting model
Scene vertices in 3-D “view” or “camera” coordinate system
Exactly those vertices & portions of polygons in view frustum
2-D screen coordinates of clipped vertices
Modeling Transforms
Lighting Calculations
Viewing Transform
Clipping
Projection Transform
David Luebke /30/2018

13. Geometric Transforms
Modeling transforms: object coordinates  world coordinates
Viewing transform: world coordinates  eye coordinates
eye coordinates == camera coordinates == view coordinates
Projection transform: eye coordinates  2-D screen coordinates
David Luebke /30/2018

14. Geometric Transforms Understand homogeneous coordinates
[x, y, z, w]T == (x/w, y/w, z/w)
Allows us to capture translation and projection as matrices
Know your 4x4 Euclidean transform matrices:
Translation, scale, rotation about X, Y, Z
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 /30/2018

15. Perspective Projection
Geometry of the perspective projection:
P (x, y, z) X Z
View plane d
(0,0,0)
x’ = ?
David Luebke /30/2018

16. Perspective Projection
Desired result:
David Luebke /30/2018

17. Perspective Projection
A matrix that accomplishes this:
David Luebke /30/2018

18. Rasterizing Lines 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
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 /30/2018

19. Rasterizing Triangles
Triangles are nice to deal with because they are always planar and always convex
Triangle rasterization techniques:
REYES: recursive subdivision of primitive
Warnock: recursive subdivision of screen
Edge walking
Edge equations
David Luebke /30/2018

20. Rasterizing Triangles
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:
Finicky: lots of special cases, hard to get just right
David Luebke /30/2018

21. Rasterizing Triangles
Edge Equations
Equation of a line defines two half-spaces
Triangle can be represented as intersection of three half-spaces:
A1x + B1y + C1 < 0
A2x + B2y + C2 < 0
A3x + B3y + C3 < 0
A1x + B1y + C1 > 0
A3x + B3y + C3 > 0
A2x + B2y + C2 > 0
David Luebke /30/2018

22. Rasterizing Triangles
Basic algorithm:
Walk pixels in bounding box
Evaluate three edge equations
If all are greater than zero, shade pixel
Issues:
Computing edge equations: numerical precision
Interpolating parameters (i.e., color): just like another edge equation (why?)
Optimizing the algorithm
Like line rasterization: DDA, early termination, etc.
David Luebke /30/2018

23. Rasterizing General PolygonsB C D E F G I H
Parity test:
Starting outside polygon, count edges crossed.
Odd = inside, even = outside
Big cost: testing every edge 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 /30/2018

24. Clipping Lines Cohen-Sutherland Algorithm
Clip 2-D line segments to rectangular viewport
Designed for rapid trivial accept & trivial reject ops
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
David Luebke /30/2018

25. Clipping Polygons Clipping polygons fundamentally more difficult
Polygons can gain or lose edges
Concave polygons can even multiply
Sutherland-Hodgman Algorithm
Simplify by divide-and-conquer: consider each clipping plane individually
Input: polygon as ordered list of vertices
Output: polygon as ordered list of vertices
Lends itself to pipelined hardware implementation
David Luebke /30/2018

26. Clipping Polygons Sutherland-Hodgman Algorithm Know the details:
Point-plane test
Line-plane intersection
Rules:
inside
outside
inside
outside
inside
outside
inside
outside p p s s p s p s
p output
i output
no output
i output p output
David Luebke /30/2018

27. Clipping in 3-D
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
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 /30/2018

28. Color Rods and cones Cones and color perception Gamma correction
Metamers
3-D color: X, Y, and Z; CIE color space
Gamma correction
David Luebke /30/2018

29. Lighting Definitions: illumination, lighting, shading Illumination:
Direct versus indirect
Light properties: geometry, spectrum
Common simplifications: ambient, directional, and point
Surface material: geometry, reflectance, microstructure
Common simplification: Phong lighting
Diffuse (Lambertian) reflection: incoming light reflected equally in all directions, proportional to N • L
Specular reflection: approximate falloff with (V • R)nshiny
David Luebke /30/2018

30. Lighting Putting it all together: the Phong lighting model
Note: evaluate per light, per color component
Common simplification: constant V (viewer infinitely far away)
David Luebke /30/2018

31. Shading 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
Phong Shading
David Luebke /30/2018