Subdivision Subdivision Bezier Curves (Recall) Subdivision using - - PowerPoint PPT Presentation

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Jul 03, 2023 278 likes •669 views

Subdivision Subdivision Bezier Curves (Recall) Subdivision using control points yields the curve Curves Bezier Curves Subdivision b 2 b 1 Curve P(t) 0 t 1 is subdivided into two curves Q(u) 0 u 1 R(v) 0

SLIDE 1

Subdivision

SLIDE 2

Subdivision

Bezier Curves (Recall) Subdivision using control points yields the curve

SLIDE 3

Curves

Bezier Curves

Subdivision

b1 b2 b3 b0 Q(u) R(v) 1 v R(v) 1 u Q(u) curves two into subdivided is 1 t P(t) Curve ≤ ≤

≤

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Curves

Bezier Curves

Subdivision

b1 b2 b3=d3 b0=c0 Q(u) R(v) c3=d0 c1 d1 c2 d2 Using de Casteljau Algorithm c0=b0 c1=(b0+b1)/2 c2=(b0+2b1+b2)/4 c3=(b0+3b1+3b2+b3)/8 d0=c3 d1=(b1+2b2+b3)/4 d2=(b2+b3)/2 d3=b3

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Curves

Bezier Curves

Subdivision

d3 c0 Q(u) R(v) c3=d0 c1 d1 c2 d2

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Subdivision (Curves)

Idea:

[Zorin and Schröder,2000]

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Subdivision (Surfaces)

Idea:

[Zorin and Schröder,2000]

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Subdivision

[DeRose et al.,1998]

Limit Surface

Input Mesh - Output refined (continuous) Mesh

Mi = f(Mi) S = lim(Mi)

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Subdivision

Do away with the explicit parametric representation
Base a curve or surface solely on its control points

and their connectivity.

Provide a simple mechanism which produces a

larger, more refined set of control points from the current set.

Iterate refinement until the appropriate level of detail

is achieved.

SLIDE 10

Subdivision

Set of rules S that take a curve as input and

produce a more highly refined curve as output

Recursively applying S yields a sequence of

curves which should converge to some limit shape Curves

SLIDE 11

Subdivision

Rules

Typically chosen to be linear combinations of

neighboring vertices

Rules usually depend only on local topology of

shape

4 3 8 1 8 1 2 1 2 1

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Subdivision

Curves S S S For Example

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Example

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Example

2 1 2 1

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Example

8 1 4 3 8 1

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Example

2 1 2 1

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Example

8 1 8 1 4 3

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Example

2 1 2 1

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Example

8 1 8 1 4 3

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Example

2 1 2 1

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Example

8 1 8 1 4 3

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Example

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Example

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Example

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Subdivision

Chaiken’s Algorithm

[Real Time Rendering]

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Subdivision

Chaiken’s Algorithm

[Real Time Rendering] Approximation using initial control points Basically follow ¾ ¼ rule C1 continuous Limiting curve is quadratic B-spline Extension for surfaces: Doo-Sabin Subdivision

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Subdivision

[Real Time Rendering]

Interpolating Curve: Dyn et. al 1987

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Subdivision

[Real Time Rendering]

Interpolating Curve: Dyn et. al 1987

P2ik+1= Pik P2i+1k+1= (1/2 + w) (Pik + Pi+1k) – w (Pi-1k + Pi+2k) w: tension parameter

SLIDE 29

Subdivision: Surfaces

[Zorin and Schröder,2000]

Loop Subdivision

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Subdivision: Surfaces

[Zorin and Schröder,2000]

Loop Subdivision Rules

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Subdivision: Surfaces

[Zorin and Schröder,2000]

Loop Subdivision Rules

Edge Mask Vertex Mask New Vertex Updated Vertex

SLIDE 32

Subdivision: Surfaces

[Zorin and Schröder,2000]

Catmull Clark Subdivision

Introduce new points
At face centre
At mid edges
Adjust position of original points
Repeat until sufficient details

SLIDE 33

Subdivision: Surfaces

Face

Catmull Clark Subdivision

Edge Update

SLIDE 34

Subdivision: Surfaces

[Zorin and Schröder,2000]

Catmull Clark Loop Some Results

SLIDE 35

Subdivision: Surfaces

[Zorin and Schröder,2000]

Catmull Clark Loop Some Results

SLIDE 36

Subdivision: Surfaces

[Zorin and Schröder,2000]

Manipulation/Deformation

SLIDE 37