Parametric Surfaces

January 16, 2003Stephen Gordon

Outline

• Introduction• Fundamentals

– Parametric Curves• Bézier• B-Spline

• Parametric Surfaces– Usage– Applications

• Current Trends

What are the Parametric Advantages?

• Provides exact analytical representation

• Allows 3D shape editing• More economical

Why Backseat to Polygon Mesh?

• Extensive mathematics• Overkill for many applications

Currently experiencing an evolution.

Where is it Used?

• CAD interactive design– Representing real objects

• Entertainment– Movies– Video games

Fundamentals – Bézier Curves

Pierre Bézier created UNISURF in 1960’s for

automotive design at Renault.

Fundamentals – Bézier Curves

P0, P1, & P2 are control points.

Q(t) is interpolated between0 and 1.

Fundamentals – B-Spline Curves

• Generalization of Bézier Curve

• Sequence of control points that guarantee continuity.

Bézier Vs. B-Spline

• Bézier– Less computation

• B-Spline– Exhibits non-localness, result

smoother– Multiple curve segments not

necessary

Bézier Patches

• Combine two Bézier curves to create a surface

16 control points

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

The Utah Teapot

Bézier: 32 patches x 16 control points/patch= 288 vertices

Polygon Mesh = 2048 vertices

B-Spline Patches

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

Bézier Vs. B-Spline 2

• Bézier– Less computation

• B-Spline– Exhibits non-localness, result

smoother– Multiple curve segments not

necessary

What are Some Bézier Applications?

• Rough collision detection– Contained within convex hull of control

points

What are Some Bézier Applications?

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

Quake III uses Bézier patchesto render the demonic tongue

More Terrain Rendering

• Shots below from SSX– Demonstrate versatility of Bézier patches

How are Models Created?

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

Profile

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

How Else?

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

Action figureDense polygon meshWith curve network

B-Spline Model

What About Bézier Triangles?

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

Control Points ofCubic Bézier Triangle

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

Why are They Useful?

• Hardware support– Graphics cards can:

• Enable/disable NP’s• Determine level of

tessellation

A more advanced techniquecurved NP Triangles are appliedto these id Software models:

Recap

Parametric surface advantages:• Provides exact analytical

representation• Allows 3D shape editing• More economical