A Method of Generating Subdivision Surfaces with Planar Hexagon Corner-Cutting
Research on subdivision surfaces has achieved significant results and has been applied in various fields,such as smooth surface modeling in animation and architectural geometric design.Among them,constructing various feasible subdivision schemes is a fundamental work in the research of subdivision surfaces.Hexagon is a polygon that is more compatible with a nature and elegant appearance.Some scholars have proposed subdivision schemes based on hexagons.But,these hexagon-based subdivision schemes either have poor smoothness and computational complexity or produce non-coplanar vertices in the facet.There are also problems of discontinuity,self-intersection,or inability to subdivide at extraordinary points/faces,especially triangular facets.However,the planarity of the mesh faces is as important as smoothness in some applications,for example in architecture.It is worth looking forward to a convenient planar hexagonal subdivision scheme.On the other hand,the construction of subdivision curves can be perfectly summarized as a sequence of corner cutting operations,which is geometrically intuitive,flexible,convex-preserving,and computationally stable.The non-uniform corner cutting subdivision curve method proposed by Gregory and Qu(referred to as the G-Q algorithm)is a classic method.With this method,for any initial polyline on the plane,by sequentially cutting all the corners formed by the polyline(the cutting parameters satisfy the corresponding smoothness conditions),a C1 smooth limit curve can be obtained.Each segment of the initial polyline has a point preserved on the limit curve,and the tangent line of the limit curve at that point is the initial polyline at that point.This method can be extended to describe the generation of spatial surfaces as follows:for any convex polyhedron in space,by sequentially cutting all the corners of the polyhedron,a C1 smooth limit surface can be obtained.Each initial face has a point preserved on the limit surface,and the tangent plane of the limit surface at that point is the initial face at that point.However,no method has yet achieved this generalization.In this paper,we propose a planar hexagonal subdivision scheme called Planar Honeycomb Subdivision(PHS)for any convex polyhedral.Through only corner-cutting by planes,PHS can generate interpolatory and convexity preserving smooth surfaces,and every new facet produced in the subdivision procedure is a planar hexagon.We present PHS's quasilinear four-point scheme and its subdivision matrix with an obvious geometric significance.The algorithm is simple,viable,and stable.We analyze PHS's convergence and smoothness,give the conditions for generating C1 surfaces and its proof.We observed that the conventional method using a constant shrinking rate of the polygons is ineffective to generate smoother surfaces for PHS and put forward a new idea to determine the corner-cutting parameters with the unified form.The new method is effective even for triangles and refrains from the peculiarities due to the extraordinary faces/vertices.We also give methods for generating a natural boundary or a flat boundary,generalize PHS to arbitrary orientable topology meshes,and discuss a method for generating sharp features in degenerate polygons.In the paper,there are several examples compared with the classical subdivision surfaces.The results show that the new method is valuable and possesses some advantages.
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