1. Graphics Review Geometry, Color & Shading Brad Tennis Leslie Wu 10.24.06

2. Topics ● Foundations of 2D graphics  Coordinate systems  Matrices  Transformations  Algorithms  Splines

3. Topics cont. ● Color  Physical Basis, Representation  Reflection Models  Shading

4. Coordinate Systems Cartesian coordinates (x, y) note: (x h, y h, h) with h = 0 corresponds to point at infinity Homogeneous coordinate (x h, y h, h) where x = x h / h y = y h / h

5. Basic definitions ● Affine transformation:  Parallel lines map to parallel lines  Finite points map to finite points  Examples in 2D: ● Translation, rotation, scaling, reflection, shear

6. Transformation Matrices ● Translation ● Rotation ● Scaling

7. Matrix Properties ● Translation is additive ● Rotation is additive ●...both are commutative ● Rotation matrices are orthogonal

8. Algorithms ● Clipping  Interior = saves inside region  Exterior = saves parts outside of region ● Intersection  Point in triangle ● Can be computed using cross-products (see handout)

9. Splines ● Spline curve  Composite curve made out of polynomial sections  Satisfies continuity conditions at section boundaries Control points Control graph / characteristic polygon

10. Continuity ● Parametric continuity  Zero-order (C 0 ) : curves meet  First-order (C 1 ) : first parametric derivatives same  Second-order (C 2 ): 1 st and 2 nd derivatives same ● Geometric continuity  Zero-order (G 0 ) : same as C 0  First-order (G 1 ) : 1 st derivatives proportional  Second-order (G 2 ) : 1 st and 2 nd derivatives proportional

11. Bezier Splines ● Properties  Curve is polynomial of degree (# control points – 1)  Can be defined recursively ● For example, to plot midpoint of Bezier of degree two

12. Physical Basis Humans can distinguish roughly 10 million colors The eye has three different types of cones The response of each peaks at either 420, 534 or 564 nm http://en.wikipedia.org/wiki/Image:Srgbspectrum.png

13. Additive Color Combination of emitters of different wavelengths RGB Color (e.g., monitors) http://en.wikipedia.org/wiki/Image:RGB_illumination.jpg

14. Subtractive Color Combination of absorbers of different wavelengths CMYK Color (e.g., printers) http://en.wikipedia.org/wiki/Image:Synthese-.svg

15. Color Representation Each light is modeled as the summation of three monochromatic lights (RGB) Each object has a collection of material properties which determine how it reflects light

16. Diffuse Reflection Lambert’s Law: The reflection from a perfectly matte surface is proportional to the dot product of the surface normal and light vector i diff = (nl)m diff ⊗ s diff

17. Specular Reflection Not all materials are perfectly matte, and shiny objects tend to reflect primarily along the reflected light vector i spec = (vr) m shi m spec ⊗ s spec

18. Ambient Reflection Sometimes light reaches an object indirectly (e.g., from bouncing off walls in the scene) i amb = m amb ⊗ s amb

19. Emission Sometimes a surface might represent a light Note that emissive surfaces do not light other objects i em = m em

20. Putting it Together The materials model used by most graphics systems isn’t necessarily physically accurate However, it provides a enough control to the programmer for most any effect i tot = i diff + i spec + i amb + i em

21. Transparency Typically implemented with a color buffer Unless objects are rendered back to front, pixels can get incorrect colors c new = α src c src + (1-α src )c dest http://en.wikipedia.org/wiki/Image:Butterfly_transparent.jpg

22. Shading Techniques Flat Gouraud Phong http://www.cs.cmu.edu/~ph/nyit/facet_gouraud_phong.jpeg

23. Shading Comparison Flat shading is very fast, but looks blocky Phong shading looks great, but is difficult to compute Gouraud shading offers a balance of speed and quality