Parametric Surfaces

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Parametric Surfaces

• January 16, 2003
• Stephen Gordon

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Outline

• Introduction
• Fundamentals
• Parametric Curves
• Bézier
• B-Spline
• Parametric Surfaces
• Usage
• Applications
• Current Trends

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What are the Parametric Advantages?

• Provides exact analytical representation
• Allows 3D shape editing
• More economical

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Why Backseat to Polygon Mesh?

• Extensive mathematics
• Overkill for many applications
• Currently experiencing an evolution.

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Where is it Used?

• CAD interactive design
• Representing real objects
• Entertainment
• Movies
• Video games

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Fundamentals Bézier Curves

• Pierre Bézier created UNISURF in 1960s for
• automotive design at Renault.

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Fundamentals Bézier Curves
P0, P1, P2 are control points. Q(t) is
interpolated between 0 and 1.
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Fundamentals B-Spline Curves

• Generalization of Bézier Curve
• Sequence of control points that guarantee
  continuity.

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Bézier Vs. B-Spline

• Bézier
• Less computation
• B-Spline
• Exhibits non-localness, result smoother
• Multiple curve segments not necessary

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Bézier Patches

• Combine two Bézier curves to create a surface

16 control points
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Bézier Patches

• Great for single patch surfaces
• Problems with multi-patch surfaces
• Cracking can occur
• If adjacent patches are tessellated to different
  levels
• To prevent, common edges must have matching
  tangent vectors

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The Utah Teapot
Bézier 32 patches x 16 control points/patch
288 vertices Polygon Mesh 2048 vertices
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B-Spline Patches

• Combination of 2 B-Spline curves
• 16 control points necessary

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Bézier Vs. B-Spline 2

• Bézier
• Less computation
• B-Spline
• Exhibits non-localness, result smoother
• Multiple curve segments not necessary

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What are Some Bézier Applications?

• Rough collision detection
• Contained within convex hull of control points

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What are Some Bézier Applications?

• Terrain rendering
• Very good compression
• Maintain constant frame rate

Quake III uses Bézier patches to render the
demonic tongue
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More Terrain Rendering

• Shots below from SSX
• Demonstrate versatility of Bézier patches

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How are Models Created?

• Cross-sectional / linear axis design
• Provides symmetry
• Example Vase

Profile
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How Else?

• Control polyhedron design
• Modify control point and 8 neighbors
• Continuity is maintained
• Fine control
• Control scale by subdividing
• Coarse control
• Global deformation by changing curve shape

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How Else?

• Surface fitting
• Fit curves to 3D surface data points
• Create curve network through interpolation

Dense polygon mesh With curve network
Action figure
B-Spline Model
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What About Bézier Triangles?

• Similar to Bézier patches
• Not as straightforward
• Used to form N-Patches

Control Points of Cubic Bézier Triangle
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So What are N-Patches?

• A triangular Bézier surface
• Adds detail to existing polygon mesh models
• Better surface lighting
• More realistic silhouette edges
• Improves shape cheaply

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Why are They Useful?

• Hardware support
• Graphics cards can
• Enable/disable NPs
• Determine level of tessellation

A more advanced technique curved NP Triangles are
applied to these id Software models
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Recap

• Parametric surface advantages
• Provides exact analytical representation
• Allows 3D shape editing
• More economical