Um pouco mais sobre modelos de objetos

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

Requisitos: Independência de eixos

x

y

x'

y'

Requisitos: Valores Múltiplos

x

y

Requisitos: Controle Local

x

y

Requisitos: Redução da Variação

polinômio de grau elevado

Requisitos: Continuidade Variável

Requisitos: Versatilidade

Requisitos: Amostragem Uniforme

s1

s2

s3

s4

sn

si sj

Formulação matemática tratávelFinalizando:

Solução

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

zzzz

yyyy

xxxx

dtctbtatz

dtctbtaty

dtctbtatx

23

23

23

)(

)(

)(

globaluuu

ou

localt

n,

1,0

0

t=0

t=1

Parametrização

t=0 t=1 t=0 t=1 t=0 t=1

u0 u1 u2 un

Curvas de Bézier

P. de Casteljau, 1959 (Citroën)P. de Bézier, 1962 (Renault) - UNISURFForest 1970: Polinômios de Bernstein

iinni tt

i

ntB

)1()(,

n

iini VtBtP

0, )()(

x

P(t)

y

z

t=0

t=1

V0

V1

V2

V3

Vn-1

Vn onde:

)!(!

!

ini

n

i

n

coef. binomial

pol. Bernstein

Bézier Cúbicas

30033,0 )1()1(

0

3)( ttttB

x

P(t)

3

03, )()(

iii VtBtP

y

z

V0

V1

V2

V3

tttttB 21133,1 )1(3)1(

1

3)(

22233,2 )1(3)1(

2

3)( tttttB

33333,3 )1(

3

3)( ttttB

i

i tB )(3, 1)1( 3 tt

33

22

12

03 )1(3)1(3)1()( VtVttVttVttP

Polinômios Cúbicos de Bernstein

1

10 t

B0,3

(1-t)3

3

10 t

B1,3

3(1-t)2t

1

10 t

B3,3

t3

10 t

B2,3

3(1-t) t2

-3

1

10 t

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

Propriedades da Bézier Cúbica

33

22

12

03 )1(3)1(3)1()( VtVttVttVttP

33

22

12

02 )1(63)1(3)1(6)1(3)( VtVtttVtttVttP

dt

d

0)0( VP

3)1( VP

10 33)0( VVPdt

d

32 33)1( VVPdt

d

x

P(t)

y

z

V0

V1

V2

V3

R(0)

R(1)

Controle da Bézier Cúbica

Redução de n=3 para n=2

33

22

12

03 )1(3)1(3)1()( VtVttVttVttP

101

0 )1()( VtVttV

211

1 )1()( VtVttV

3212 )1()( VtVttV

12

211

10

2 )1(2)1()( VtVttVttP

32

2

21102

)1(

)1()1(2)1()1()(

VtVtt

VtVtttVtVtttP

)(10 tV

)(11 tV )(1

2 tV

Bezier n=2

Redução de n=2 para n=1

11

10

20 )1()( VtVttV

12

211

10

2 )1(2)1()( VtVttVttP

12

11

11

10 )1()1()1()( VtVttVtVtttP

Bezier n=1

12

11

21 )1()( VtVttV

11

20)1()( VtVttP

10V

11V

12V

20V

21V

)(tP

Cálculo de um Ponto

10V

11V

12V

20V

21V

)(tP

0V

1V

2V

3V

11V

10V

12V

20V

21V )(tP

(1-t)

t

)()()1()( 1,11,, tBttBttB ninini Mostre que:

0V

1V 2V

3V

Subdivisão de Bézier Cúbicas

3

2

1

0

3

2

1

0

1331

0242

0044

0008

8

1

V

V

V

V

V

V

V

V

L

L

L

L

3

2

1

0

3

2

1

0

8000

4400

2420

1331

8

1

V

V

V

V

V

V

V

V

R

R

R

R

101 2

1

2

1VVV L

. . .LV1

H

00 VV L

1V

2V

LV2

RL VV 03

RV1

RV2

33 VV R

Construção de uma Bezier

u=1/2

P(1/2)

Introduction toSubdivision Surfaces

Adi Levin

Subdivision Curves and Surfaces

• Subdivision curves– The basic concepts of subdivision.

• Subdivision surfaces– Important known methods.– Discussion: subdivision vs. parametric

surfaces.

Corner Cutting

Corner Cutting

1 : 33

: 1

Corner Cutting

Corner Cutting

Corner Cutting

Corner Cutting

Corner Cutting

Corner Cutting

Corner Cutting

The control polygon

The limit curveA control point

The 4-point scheme

The 4-point scheme

The 4-point scheme

1 :

1

1 :

1

The 4-point scheme

1 :

8

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The 4-point scheme

The control polygon

The limit curveA control point

Subdivision curves

Non interpolatory subdivision schemes

• Corner Cutting

Interpolatory subdivision schemes

• The 4-point scheme

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

• 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

Works only for control nets whose faces are triangular.

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.

Old verticesNew vertices

Loop’s scheme

3 3

1

1

1

1

1

1

1

nw

n

n

nwn

22cos2340

64

n - the vertex valency

A rule for new red vertices A rule for new green vertices

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

The original control net

After 1st iteration

After 2nd iteration

After 3rd iteration

The limit surface

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

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:

-1

-1

-1

-1

88

2

2

The original control net

After 1st iteration

After 2nd iteration

After 3rd iteration

The limit surface

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

Quadrilateral subdivision

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

Every face is replaced by quadrilateral faces.The are three kinds of new vertices:

• YellowYellow vertices are associated with old facesfaces• Green vertices are associated with old edges• Red vertices are associated with old vertices.

Old vertices New vertices

Old edge

Old face

Catmull Clark’s scheme

1

1

11

1

First, all the yellow vertices are calculated

Step 1

1 1

1

1

Then the green vertices are calculated using the values

of the yellow vertices

Step 2

11

11

1

11

1

1

nw

Finally, the red vertices are calculated using the values

of the yellow vertices

Step 3

)2( nnwn

n - the vertex valency

1

The original control net

After 1st iteration

After 2nd iteration

After 3rd iteration

The limit surface

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