Graphics Review Geometry, Color

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Graphics ReviewGeometry,Color Shading

• Brad Tennis
• Leslie Wu
• 10.24.06

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Topics

• Foundations of 2D graphics
• Coordinate systems
• Matrices
• Transformations
• Algorithms
• Splines

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Topics cont.

• Color
• Physical Basis, Representation
• Reflection Models
• Shading

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Coordinate Systems

• Cartesian coordinates
• (x, y)
• note (xh, yh, h) with h 0
• corresponds to point at infinity

• Homogeneous coordinate
• (xh, yh, h)
• where
• x xh / h
• y yh / h

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Basic definitions

• Affine transformation
• Parallel lines map to parallel lines
• Finite points map to finite points
• Examples in 2D
• Translation, rotation, scaling, reflection, shear

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Transformation Matrices

• Translation
• Rotation

• Scaling

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Matrix Properties

• Translation is additive
• Rotation is additive
• ...both are commutative
• Rotation matrices are orthogonal

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Algorithms

• Clipping
• Interior saves inside region
• Exterior saves parts outside of region
• Intersection
• Point in triangle
• Can be computed using cross-products (see handout)

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Splines

• Spline curve
• Composite curve made out of polynomial sections
• Satisfies continuity conditions at section
  boundaries

Control graph / characteristic polygon
Control points
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Continuity

• Parametric continuity
• Zero-order (C0) curves meet
• First-order (C1) first parametric derivatives
  same
• Second-order (C2) 1st and 2nd derivatives same
• Geometric continuity
• Zero-order (G0) same as C0
• First-order (G1) 1st derivatives proportional
• Second-order (G2) 1st and 2nd derivatives
  proportional

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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

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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/ImageSrgbspectrum.pn
g
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Additive Color

• Combination of emitters of different wavelengths
• RGB Color (e.g., monitors)

http//en.wikipedia.org/wiki/ImageRGB_illuminatio
n.jpg
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Subtractive Color

• Combination of absorbers of different wavelengths
• CMYK Color (e.g., printers)

http//en.wikipedia.org/wiki/ImageSynthese-.svg
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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

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Diffuse Reflection

• Lamberts Law The reflection from a perfectly
  matte surface is proportional to the dot product
  of the surface normal and light vector

idiff (nl)mdiff ? sdiff
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Specular Reflection

• Not all materials are perfectly matte, and shiny
  objects tend to reflect primarily along the
  reflected light vector

ispec (vr)mshi mspec ? sspec
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Ambient Reflection

• Sometimes light reaches an object indirectly
  (e.g., from bouncing off walls in the scene)

iamb mamb ? samb
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Emission

• Sometimes a surface might represent a light
• Note that emissive surfaces do not light other
  objects

iem mem
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Putting it Together

• The materials model used by most graphics systems
  isnt necessarily physically accurate
• However, it provides a enough control to the
  programmer for most any effect

itot idiff ispec iamb iem
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Transparency

• Typically implemented with a color buffer
• Unless objects are rendered back to front, pixels
  can get incorrect colors

http//en.wikipedia.org/wiki/ImageButterfly_trans
parent.jpg
cnew asrccsrc (1-asrc)cdest
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Shading Techniques
Flat Gouraud Phong
http//www.cs.cmu.edu/ph/nyit/facet_gouraud_phong
.jpeg
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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