Subdivision Surfaces: Smooth Limits of Complex Geometry, Study notes of Geometry

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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!