Localized-hierarchy surface splines (LeSS)

Abstract

An explicit spline representation of smooth free-form surfaces is combined with a hierarchy of meshes to form the basis of an interactive sculpting environment. The environment offers localized hierarchical modeling at different levels of detail, direct surface manipulation, change of connectivity for extrusion and to form holes and bridges, and built-in tangent continuity across the surface where wanted. The free-form surface is represented and can be exported either in NURBS form or as cubic triangular Bezier patches. Key characteristics of the approach are: (1) mesh pieces and surface pieces are related by strictly local averaging rules; (2) refinement rules depend only on direct, coarser-level ancestors and not on adjacent submeshes or patches; (3) submeshes at different levels look alike. The underlying data structure is a single winged-edge structure with additional pointers to support the hierarchy. Multiply refined regions may be directly adjacent to unrefined regions, and mesh fragments at different levels of refinement can be connected. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations,Splines; I.3.6 [Computer Graphics]: Methodology and Techniques—Graphics data structures and data types 1 Motivation, results and prior work Conventional modeling systems require precise specification of final features, e.g., distance in datum planes, positioning of control vertices with respect to one another, etc. Besides requiring a certain mindset not always compatible with creative, exploratory design, this is modeling at the highest level of refinement: a change in the overall shape of the model entails modifying all detail, however small, associated with the area of change. Not only is this time-consuming, but repeated detail work puts an undue strain on creative, exploratory design. Parameterized modeling, which allows initially precisely instantiated specifications to be varied, has been used in commercial solid modeling systems. A complementary paradigm that supports changes at different levels in the design gonzalez@acm.org yjorg@cise.ufl.edu zSupported by NSF National Young Investigator award 9457806-CCR history is hierarchical refinement. Levels express the designer’s intent by encoding the structure and dependence of geometric features. Several recent modeling environments are based on hierarchical modeling [10, 7, 21, 13]. Localized-hierarchy Surface Splines (LeSS) extend this modeling paradigm by combining localized mesh hierarchies with the ability to change topology by adding a handle or punching a hole. LeSS automatically maintains tangent continuity across the surface where wanted and defines the surface in standard form either as bicubic NURBS or cubic triangular Bezier patches. Non-standard properties of LeSS are: 1. The surface is represented explicitly in piecewise polynomial or spline form, and not as surface patches offset from ancestor surfaces. Finer-level patches replace coarse-level patches. Changes are propagated from coarse to fine inside an underlying mesh hierarchy: fragments (pieces with boundary) of finer meshes are represented as offsets in local coordinate frames of coarser meshes. This has several implications: The evaluation of the spline patches does not require traversal of the hierarchy and is numerically stable. Explicit bounding boxes and convex hulls for the surface are directly available and the moments of the enclosed object can be efficiently computed. The influence of changing coarser levels is transmitted by local reevaluation of the mesh hierarchy. Local topology changes are possible because there is no direct 1-1 correspondence of the coarse to the finer-level surface. 2. LeSS is based on a strictly localized hierarchy: the default position of a finer level mesh fragment before local editing is determined by local averaging rules that depend only on those coarser-level nodes from which the mesh nodes of the fragment are offset and not on adjacent mesh or patch information. This simplifies the data structures since no searches for neighbors at different levels of refinement are necessary. Repeatedly-refined patches can be placed directly next to unrefined patches. (Since the surface representation is explicit, the resolution of ‘crack problems’ where refined and unrefined patches meet can be delegated to the spline display routine.) 3. The representation is homogeneous: control structures everywhere, even across levels, look the same. Together with locality this homogeneity allows connecting arbitrary submeshes. This is useful since not all parts of a complex object must enter a strict dependence relation with all other parts – the designer’s choice of dependence can be only local and later decisions, say to join parts or create a hole (Figure 9), are not prevented by a rigid, global hierarchy.