Apresentação em tema: "Um pouco mais sobre modelos de objetos. Ray Path Categorization Ray Path Categorization. Nehab, D.; Gattass, M. Proceedings of SIBGRAPI 2000, Brazil,"— Transcrição da apresentação:
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
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 33 1 1 1 1 1 1 1 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 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 1 1 1 1 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 1 1 1 1 1 1 1 1 1 Finally, the red vertices are calculated using the values of the yellow vertices Step 3 n - the vertex valency 1