Curves

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Curves Surfaces
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Today

• Motivation
• Limitations of Polygonal Models
• Gouraud Shading Phong Normal Interpolation
• Some Modeling Tools Definitions
• Curves
• Surfaces / Patches
• Subdivision Surfaces

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Limitations of Polygonal Meshes

• Planar facets ( silhouettes)
• Fixed resolution
• Deformation is difficult
• No natural parameterization (for texture mapping)

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Can We Disguise the Facets?
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Gouraud Shading

• Instead of shading with the normal of the
  triangle, shade the vertices with the average
  normal and interpolate the color across each face

Illusion of a smooth surface with smoothly
varying normals
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10K facets
1K facets
10K smooth
1K smooth
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Better, but not always good enough

• Still low, fixed resolution (missing fine
  details)
• Still have polygonal silhouettes
• Intersection depth is planar (e.g. ray
  visualization)
• Collisions problems for simulation
• Solid Texturing problems
• ...

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Some Non-Polygonal Modeling Tools
Extrusion
Surface of Revolution
Spline Surfaces/Patches
Quadrics and other implicit polynomials
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Questions?
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Today

• Motivation
• Curves
• What's a Spline?
• Linear Interpolation
• Interpolation Curves vs. Approximation Curves
• Bézier
• BSpline (NURBS)
• Surfaces / Patches
• Subdivision Surfaces

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Definition What's a Spline?

• Smooth curve defined by some control points
• Moving the control points changes the curve

Interpolation
Bézier (approximation)
BSpline (approximation)
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Interpolation Curves / Splines
www.abm.org
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Interpolation Curves

• Curve is constrained to pass through all control
  points
• Given points P0, P1, ... Pn, find lowest degree
  polynomial which passes through the points x(t)
  an-1tn-1 .... a2t2 a1t a0 y(t)
  bn-1tn-1 .... b2t2 b1t b0

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Cubic Bézier Curve

• de Casteljau's algorithm for constructing Bézier
  curves

t
t
t
t
t
t
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Cubic BSplines

• 4 control points
• Locally cubic
• Curve is not constrained to pass through any
  control points

A BSpline curve is also bounded by the convex
hull of its control points.
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Cubic BSplines
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Cubic BSplines

• Can be chained together
• Better control locally (windowing)

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Connecting Cubic BSpline Curves

• Whats the relationship between
• the of control points, and
• the of cubic BSpline subcurves?

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BSpline Curve Control Points
Default BSpline
BSpline with Discontinuity
BSpline which passes through end points
Repeat interior control point
Repeat end points
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Bézier is not the same as BSpline
Bézier
BSpline
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Bézier is not the same as BSpline

• Relationship to the control points is different

Bézier
BSpline
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Converting between Bézier BSpline
original control points as Bézier
new BSpline control points to match Bézier
new Bézier control points to match BSpline
original control points as BSpline
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Converting between Bézier BSpline

• Using the basis functions

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NURBS (generalized BSplines)

• BSpline uniform cubic BSpline
• NURBS Non-Uniform Rational BSpline
• non-uniform different spacing between the
  blending functions, a.k.a. knots
• rational ratio of polynomials (instead of
  cubic)

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Questions?
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Today

• Motivation
• Spline Curves
• Spline Surfaces / Patches
• Tensor Product
• Bilinear Patches
• Bezier Patches
• Subdivision Surfaces

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Bilinear Patch
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Bicubic Bezier Patch
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Editing Bicubic Bezier Patches
Curve Basis Functions
Surface Basis Functions
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Bicubic Bezier Patch Tessellation

• Assignment 8 Given 16 control points and a
  tessellation resolution, create a triangle mesh

resolution5x5 vertices
resolution11x11 vertices
resolution41x41 vertices
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Modeling with Bicubic Bezier Patches

• Original Teapot specified with Bezier Patches

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Modeling Headaches

• Original Teapot model is not "watertight"inters
  ecting surfaces at spout handle, no bottom, a
  hole at the spout tip, a gap between lid base

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Today

• Review
• Motivation
• Spline Curves
• Spline Surfaces / Patches
• Subdivision Surfaces

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Chaikin's Algorithm
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Doo-Sabin Subdivision
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Next Tuesday (no class Thursday!)

• Animation I
• Particle Systems