CS 395: Adv' Computer Graphics

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CS 395 Adv. Computer Graphics

• Overview
• Parametric Surfaces
• Watt Chapter 3 readings
• Jack Tumblin
• jet_at_cs.northwestern.edu

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Curves and Surfaces

• Basic Problem
• Polygons are easy, fast, renderable, BUT
• Polygons meshes are not smoothno derivatives
  poor silhouettes, reflections...
• Polygons can only approximate curves,
• Polygons are less compact
• Previous methods metal/wood 'splines' ...
• (see Farin book)

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What's a Parametric Curve?

• Vary one or more 'parameter' to explore a curve
  or surface
• Example parametric circle, in z1 plane
• x(u) Rcos(u)
• y(u) Rsin(u)
• z(u) 1

z
y
x
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Background

• Many Historical Parametric Curve Makers
• Lissajous Curves, http//kosmoi.com/Science/Mathe
  matics/Graphs/Encyclo/
• Spirographs, http//math.dartmouth.edu/dlittle/
  java/SpiroGraph/
• Harmonographs, http//astronomy.swin.edu.au/pbou
  rke/curves/harmonograph/
• Epicycles, etc.http//www.astronomynotes.com/hist
  ory/epicycle.htm

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Background

• Few found use in design until computers
• Paul DeCastlejau (1950s, Citroen)
• Pierre Bezier (1960s, Renault)
• 70's, 80's explosion of Comp. Geometry
• GREAT results now faded as research area

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OUTLINE

• Historical Parametrics transcendentals
• in CG mostly polynomial
• Key Idea 1 blending points...

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OUTLINE

• Key Idea 2 Linear Interpolation, Nesting
• Paul DeCastlejau (1950s, Citroen)
• Pierre Bezier (1960s, Renault)
• http//www.ibiblio.org/e-notes/Splines/Bezier.htm
• How can we connect multiple Bezier curves?
• How can we make a Bezier surface?

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Efficient!
9 unique Bezier Patches (some
were mirrored around z axis total is ?17?)
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Digital Image a 2D Grid of Numbers

• NO intrinsic meaninguse it for anything
• reflectance, transparency, illumination, normal
  direction, material, velocity...

v
v
u
u
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OUTLINE

• Key Idea 3 Generalize Blending Fcns., in
  Matrix form
• Uniform B-splines
• Other Basis Functions
• Non-uniform? 'Duplicate Control Pts'
• http//www.ibiblio.org/e-notes/Splines/Bezier.htm

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Useful Goals

• Continuity are all derivatives smooth?
• w.r.t. parameters w.r.t. space
• Global / Local Control move 1 control pt does
  entire curve change?
• Convex Hull is curve within its control pts?
• Interpolatingdoes curve touch desired pts?
• Affine Invariant Projective Invariant
• transform control pts, then draw curve, OR
  draw curve, then transform, SAME result!

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Useful Goals

• Invertible find ray-surface intersection in
  3D (for rendering, shading) in u,v parameters
  (for texture, etc.) find surface-surface
  intersection in 3D (for 'trimming', fairing,
  etc.) in u,v parameters

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Further Sources

• Endless books on curves and surfacesG. Farin,
  "Curves and Surfaces for CAGD" (recommended most
  rigorous complete)
• On-line tutorials, Java Applets