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Um pouco mais sobre modelos de objetos. Ray Path Categorization. Ray Path Categorization . Nehab, D.; Gattass, M. Proceedings of SIBGRAPI 2000, Brazil, 2000, pp. 227-234. Ray Path Categorization.   -. Curvas e Superfícies. modelagem paramétrica. y. y'. x'. x.

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ray path categorization
Ray Path Categorization

Ray Path Categorization.Nehab, D.; Gattass, M.Proceedings of SIBGRAPI 2000, Brazil, 2000, pp. 227-234.

curvas e superf cies

Curvas e Superfícies

modelagem paramétrica

requisitos redu o da varia o
Requisitos: Redução da Variação

polinômio de grau elevado

requisitos amostragem uniforme

Finalizando:

Formulação matemática tratável

Requisitos: Amostragem Uniforme

Dsn

Ds3

Ds2

Ds4

Ds1

DsiDsj

solu o

t=1

t=0

t=0

t=0

t=1

t=1

u1

u0

u2

un

Solução

Curva representada por partes através de polinômios de grau baixo (geralmente 3)

t=1

t=0

Parametrização

curvas de b zier

z

V2

Vn-1

t=1

V1

Vn

t=0

V3

P(t)

V0

y

x

Curvas de Bézier

P. de Casteljau, 1959 (Citroën)

P. de Bézier, 1962 (Renault) - UNISURF

Forest 1970: Polinômios de Bernstein

onde:

pol. Bernstein

coef. binomial

b zier c bicas
Bézier Cúbicas

z

V1

V3

P(t)

V0

V2

y

x

polin mios c bicos de bernstein

B0,3

B1,3

1

(1-t)3

3(1-t)2t

3

0

0

1

1

t

t

B3,3

B2,3

1

3(1-t)t2

t3

-3

0

1

t

B0,3 +B1,3 +B2,3 +B3,3

0

1

t

1

0

1

t

Polinômios Cúbicos de Bernstein
subdivision curves and surfaces
Subdivision Curves and Surfaces
  • Subdivision curves
    • The basic concepts of subdivision.
  • Subdivision surfaces
    • Important known methods.
    • Discussion: subdivision vs. parametric surfaces.
corner cutting1
Corner Cutting

3 : 1

1 : 3

corner cutting8

A control point

The limit curve

The control polygon

Corner Cutting
the 4 point scheme15
The 4-point scheme

A control point

The limit curve

The control polygon

subdivision curves
Subdivision curves
  • Non interpolatory subdivision schemes
  • Corner Cutting
  • Interpolatory subdivision schemes
  • The 4-point scheme
basic concepts of subdivision
Basic concepts of Subdivision
  • A subdivision curve is generated by repeatedly applying a subdivision operator to a given polygon (called the control polygon).
  • The central theoretical questions:
    • Convergence: Given a subdivision operator and a control polygon, does the subdivision process converge?
    • Smoothness: Does the subdivision process converge to a smooth curve?
subdivision schemes for surfaces
Subdivision schemes for surfaces
  • A Control net consists of vertices, edges, and faces.
  • In each iteration, the subdivision operator refines the control net, increasing the number of vertices (approximately) by a factor of 4.
  • In the limit the vertices of the control net converge to a limit surface.
  • Every subdivision method has a method to generate the topology of the refined net, and rules to calculate the location of the new vertices.
triangular subdivision
Triangular subdivision

Works only for control nets whose faces are triangular.

New vertices

Old vertices

  • Every face is replaced by 4 new triangular faces.
  • The are two kinds of new vertices:
  • Green vertices are associated with old edges
  • Red vertices are associated with old vertices.
loop s scheme

1

3

3

1

Loop’s scheme

Every new vertex is a weighted average of the old vertices. The list of weights is called the subdivision mask or the stencil.

A rule for new red vertices

A rule for new green vertices

1

1

1

1

1

n- the vertex valency

the limit surface
The limit surface

The limit surfaces of Loop’s subdivision have continuous curvature almost everywhere.

the butterfly scheme

-1

-1

2

8

8

-1

2

-1

The Butterfly scheme

This is an interpolatory scheme. The new red vertices inherit the location of the old vertices. The new green vertices are calculated by the following stencil:

the limit surface1
The limit surface

The limit surfaces of the Butterfly subdivision are smooth but are nowhere twice differentiable.

quadrilateral subdivision
Quadrilateral subdivision

Works for control nets of arbitrary topology. After one iteration, all the faces are quadrilateral.

Old vertices

New vertices

Old face

Old edge

  • Every face is replaced by quadrilateral faces.
  • The are three kinds of new vertices:
  • Yellow vertices are associated with old faces
  • Green vertices are associated with old edges
  • Red vertices are associated with old vertices.
catmull clark s scheme

1

Step 1

Step 2

1

First, all the yellow vertices are calculated

1

Then the green vertices are calculated using the values of the yellow vertices

1

1

1

1

1

1

Catmull Clark’s scheme

Step 3

Finally, the red vertices are calculated using the values of the yellow vertices

1

1

1

1

1

1

1

1

1

1

n- the vertex valency

the limit surface2
The limit surface

The limit surfaces of Catmull-Clarks’s subdivision have continuous curvature almost everywhere.