Um pouco mais sobre modelos de objetos

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

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 s 1 s 2 s 3 s 4 s n s i s j Formulação matemática tratável Finalizando:

Solução Curva representada por partes através de polinômios de grau baixo (geralmente 3) t=0 t=1 Parametrização t=0 t=1t=0 t=1 t=0 t=1 u0u0 u1u1 u2u2 unun

Curvas de Bézier P. de Casteljau, 1959 (Citroën) P. de Bézier, 1962 (Renault) - UNISURF Forest 1970: Polinômios de Bernstein x P(t) y z t=0 t=1 V0V0 V1V1 V2V2 V3V3 V n-1 VnVn onde: coef. binomial pol. Bernstein

Bézier Cúbicas x P(t) y z V0V0 V1V1 V2V2 V3V3

Polinômios Cúbicos de Bernstein t B 0,3 (1-t) t B 1,3 3(1-t) 2 t t B 3,3 t3t3 1 0 t B 2,3 3(1-t) t t B 0,3 + B 1,3 + B 2,3 + B 3,3

Propriedades da Bézier Cúbica x P(t) y z V0V0 V1V1 V2V2 V3V3 R(0) R(1)

Controle da Bézier Cúbica

Redução de n=3 para n=2 Bezier n=2

Redução de n=2 para n=1 Bezier n=1

Cálculo de um Ponto (1-t) t Mostre que:

Subdivisão de Bézier Cúbicas...

Construção de uma Bezier u=1/2 P(1/2)

Introduction to Subdivision 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

1 : 3 3 : 1

Corner Cutting

The control polygon The limit curve A control point

The 4-point scheme

1 : 1

The 4-point scheme 1 : 8

The 4-point scheme

The control polygon The limit curve A 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 vertices New vertices

Loops scheme n - the vertex valency A rule for new red verticesA 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 Loops 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:

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: Yellowfaces Yellow vertices are associated with old faces Green vertices are associated with old edges Red vertices are associated with old vertices. Old vertices New vertices Old edge Old face

Catmull Clarks scheme First, all the yellow vertices are calculated Step Then the green vertices are calculated using the values of the yellow vertices Step Finally, the red vertices are calculated using the values of the yellow vertices Step 3 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-Clarkss subdivision have continuous curvature almost everywhere.