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Subdivision Surfaces: Smooth Limits of Complex Geometry, Lecture notes of Topology

The concept of subdivision surfaces, a technique used to create smooth curves and surfaces from arbitrary control meshes. the problems with NURBS patches, the definition and properties of subdivision surfaces, and various subdivision schemes such as Catmull-Clark and Loop. Examples of subdivision in action are provided through the analysis of Pixar's Geri's Game. The document also discusses the classification of subdivision schemes and their differences.

What you will learn

  • What are the advantages of using subdivision surfaces over traditional NURBS patches?
  • What are the different types of subdivision curve schemes?
  • What are the advantages of using T-Splines over NURBS?
  • How does the Doo-Sabin subdivision scheme apply to polygonal meshes?
  • How does the Catmull-Clark subdivision scheme differ from Loop subdivision?

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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Subdivision Surfaces

Geometric ModelingGeometric

Modeling

-^

Sometimes need more than polygon meshes

-^

Sometimes

need more than polygon meshes

  • Smooth surfaces
    • Traditional geometric modeling used NURBS

N

if^

ti^

l B S li

  • Non uniform rational B-Spline– Demo

SubdivisionSubdivision “Subdivision defines a smooth curve or surface asSubdivision defines a smooth curve or surface as

the limit of a sequence of successiverefinements”refinements

Subdivision SurfacesSubdivision

Surfaces

•^

Generalization of

spline

curves / surfaces

•^

Generalization

of spline curves / surfaces

  • Arbitrary control meshes– Successive refinement (subdivision)– Converges to smooth limit surface

g

  • Connection between splines and meshes

5

Example: Geri

’s Game (Pixar)

•^

Subdivision used for

Example: Geri s Game (Pixar) •^

Subdivision

used for

  • Geri’s hands and head – Clothing– Tie and shoes

Example: Geri

’s Game (Pixar)

Example: Geri s Game (Pixar)^ Woody’s hand (NURBS)

Geri’s hand (subdivision) 8

Example: GamesExample:

Games

Supported in hardware in DirectX 11Supported

in hardware in DirectX 11

Subdivision CurvesSubdivision

Curves

Given a control polygon...Given

a control polygon...

...find a smooth curve related to that polygon.

ApproximatingApproximating

ApproximatingApproximating

Splitting step: split each edge in two

ApproximatingApproximating Start over ...Start

over ...

ApproximatingApproximating^ ...splitting......splitting...

ApproximatingApproximating ...and so on......and

so on...

ApproximatingApproximating

If the rule is designed carefully...If

the rule is designed carefully...

... the control polygons will converge to a

smooth limit curve!smooth limit curve!