05jet - Eurographics Symposium on Geometry Processing(2006...

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Eurographics Symposium on Geometry Processing (2006) Konrad Polthier, Alla Sheffer (Editors) A C 2 Polar Jet Subdivision K. Karˇciauskas 0 and A. Myles 1 and J. Peters †1 0 University of Vilnius 1 University of Florida Abstract We describe a subdivision scheme that acts on control nodes that each carry a vector of values. Each vector defines partial derivatives, referred to as jets in the following and subdivision computes new jets from old jets. By default, the jets are automatically initialized from a design mesh. While the approach applies more generally, we consider here only a restricted class of design meshes, consisting of extraordinary nodes surrounded by triangles and otherwise quadrilaterals with interior nodes of valence four. This polar mesh structure is appropriate for surfaces with the combinatorial structure of objects of revolution and for high valences. The resulting surfaces are curvature continuous with good curvature distribution near extraordinary points. Near extraordinary points the surfaces are piecewise polynomial of degree (6,5), away they are standard bicubic splines. Categories and Subject Descriptors (according to ACM CCS) : I.3.5 [Computer Graphics]: Curve, Surface, Solid, and Object Representations 1. Introduction We present a subdivision algorithm for a restricted class of meshes as depicted in Figure 1 . This polar mesh struc- ture [ KP06b ], while very special, is natural for meshes with the combinatorial structure of surfaces of revolution and lo- cally, for many-sided blends and vertices of high valence. Often, for example for reflective surfaces such as inside car headlights, exactly these high-valence blends require good curvature distribution. However, for standard subdivi- sion schemes, high valence leads to visibly poor shape as illustrated in Figure 2 .Indeed, standard schemes have been shown to generate saddles even though the initial control net has a convex triangulation [ KPR04 ]. While it is notoriously difficult to argue that a scheme results in high-quality sur- faces, or even to define high quality, the proposed scheme does not exhibit the high-valence flaws observed for stan- dard schemes. Polar jet subdivision has the following properties. 1. Linear, stationary, affine invariant refinement of a control structure. 2. Control nodes of arbitrary valence. 3. Generates curvature continuous surfaces that supported by NSF DMI-0400214 and CCF-0430891 Figure 1: A polar design mesh: the extraordinary vertex is surrounded by triangles. All quadrilaterals have nodes of valence four. 4. do not have obvious shape limitations for high valence and 5. can be represented as a sequence of polynomial pieces of degree (6,5). While the user manipulates a design mesh , the scheme re- fines vectors, called 2-jets , that are associated with nodes.
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