Complex polyhedral objects are often constructed from simpler polyhedral objects using CSG, booleans and various blend operations that produce fillets and chamfers. Skinning and shrink wrapping techniques provide alternatives to generating a smooth composite object from individual object parts. This paper presents an effective and general technique for incrementally building complex polyhedral objects from simpler parts. We provide a procedural implicit function definition for a region of a polyhedral object that is star-shaped with respect to a skeletal point, called a blend center. We extend this definition to provide a single implicit function definition for an arbitrary polyhedral object, where every region is star-shaped with respect to a proximal blend center, chosen from an arbitrary set of blend centers. This allows the application of implicit function based modeling techniques in constructing transition surfaces between arbitrary polyhedral object parts. Generalizations of our approach allow the algebraic combination of arbitrarily shaped and positioned, disjoint object parts, and is thus a significant superset of CSG, booleans, and other blending techniques. At the same time original detail and character of object parts are preserved in regions where they are not involved in constructing transition surfaces or do not interact with other object parts. A complete implemetation of the presented concepts show polyhedral implicit primitives to be a general and efficient technique for building complex polyhedral objects from a modular set of polyhedral object parts.

/pdf/joining-polyhedral-objects-using-implicitly-defined-surfaces-33ludjuqq1.pdf

Joining polyhedral objects using implicitly defined surfaces

Recent research has demonstrated the usefulness of implicit surfaces for modeling geometric objects. The interactive design of such surfaces has not, however, received the same attention as has the design of parametric surfaces. Principally this is due to the difficulty of controlling the shape of implicit surfaces while displaying the changes quickly enough for use within an interactive design environment. This paper describes progress towards interactive control of implicit surfaces and introduces new techniques useful to the designer.

Interactive techniques for implicit modeling.

A graphics system based on quadrics

This paper presents a unified approach for incorporating free-form solids in bilateral Brep and CSG representation schemes, by resorting to low-degree (quadratic, cubic) algebraic surface patches. We develop a general CSG solution that represents a free-form solid as a boolean combination of a direct term and a complicated delta term. This solution gives rise to the trunctet-subshell conditions, under which the delta term computation can be obviated. We use polyhedral smoothing to construct a Brep consisting of quadratic algebraic patches that meet with tangent-plane continuity, such that the trunctet-subshell conditions are guaranteed automatically. This guarantee is not currently available for cubic patches. The general CSG solution thus applies whenever trunctet-subshell conditions are violated, e.g. sometimes for cubic patches or sometimes for patches of any degree that are subject to shape control operations. Manifold solids of arbitrary topology can be represented in our dual representation system. Ensuing CSG constructs are parallel processed on the RayCasting Engine to support a wide range of solid modeling applications, including general sweeping, Minkowski operations, NC machining, and touch-sense probing.

Free-Form Modeling in Bilateral Brep and CSG Representation Schemes

This paper presents a novel method for the subdivision of surfaces of revolution. We develop a new technique for approximating the genertrix by a series of pairs of conic sections. By using an error estimate based on convex combination, an efficient least-squares approach is proposed that yields near-optimal fitting. The resulting surface approximation is shown to be more efficient than other tessellation methods in terms of the number of fitting segments. This in turn allows us to implement efficient and robust algorithms for such surfaces. In particular, novel intersection techniques based on the proposed subdivision method are introduced for the two most fundamental types of intersections – line/surface and surface/surface intersections. The experimental results show that our method outperforms conventional methods significantly in both computing time and memory cost.

http://repository.ust.hk/ir/bitstream/1783.1-1982/1/BiConic.pdf

Biconic subdivision of surfaces of revolution and its applications in intersection problems

This paper is concerned with free-form surface constructions using implicit quadrics. More specifically, we are interested in the following problem: given a polyhedron with triangular facets and tangent planes prescribed at its vertices, fit a smooth (tangent-plane continuous), implicit, and piecewise quadric surface through the vertices of the polyhedron so that the surface is tangent to the prescribed tangent plane at each vertex. We show that in order to solve this problem without splitting the facets of the polyhedron, the prescribed tangent planes must satisfy a condition, and under this condition we give a local scheme for constructing the smooth piecewise quadric surface. Using this scheme, we can represent arbitrary shapes by quadric primitives. The implementation results are reported.

Representation of arbitrary shapes using implicit quadrics

Some implicit spline surface schemes require filling two-sided holes [Dahmen 1989; Guo 1993] and a blending technique of Warren (1987) can be used to fill two-sided holes in quadric spline surfaces. This note shows that there are important cases where the blending technique does not apply. These cases are identified and an applicatble technique is proposed.

A note on two-sided holes in implicit quadric spline surfaces

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Representation of arbitrary shapes using implicit quadrics

A note on two-sided holes in implicit quadric spline surfaces