Abstract To overcome the well‐known shape deficiencies of bi‐cubic subdivision surfaces, Evolving Guide subdivision (EG subdivision) generalizes C2 bi‐quartic (bi‐4) splines that approximate a sequence of piecewise polynomial surface pieces near extraordinary points. Unlike guided subdivision, which achieves good shape by following a guide surface in a two‐stage, geometry‐dependent process, EG subdivision is defined by five new explicit subdivision rules. While formally only C1 at extraordinary points, EG subdivision applied to an obstacle course of inputs generates surfaces without the oscillations and pinched highlight lines typical for Catmull‐Clark subdivision. EG subdivision surfaces join C2 with bi‐3 surface pieces obtained by interpreting regular sub‐nets as bi‐cubic tensor‐product splines and C2 with adjacent EG surfaces. The EG subdivision control net surrounding an extraordinary node can have the same structure as Catmull‐Clark subdivision: two rings of 4‐sided facets around each extraordinary nodes so that extraordinary nodes are separated by at least one regular node.

Abstract The idea of improving multi‐sided piecewise polynomial surfaces, by explicitly prescribing their behavior at a central surface point, allows for decoupling shape finding from enforcing local smoothness constraints. Quadratic‐Attraction Subdivision determines the completion of a quadratic expansion at the central point to attract a differentiable subdivision surface towards bounded curvature, with good shape also in‐the‐large.

For control nets outlining a large class of topological polyhedra, not just tensor-product grids, bi-cubic polyhedral splines form a piecewise polynomial, first-order differentiable space that associates one function with each vertex. Akin to tensor-product splines, the resulting smooth surface approximates the polyhedron. Admissible polyhedral control nets consist of quadrilateral faces in a grid-like layout, star-configuration where n ≠ 4 quadrilateral faces join around an interior vertex, n -gon configurations, where 2n quadrilaterals surround an n -gon, polar configurations where a cone of n triangles meeting at a vertex is surrounded by a ribbon of n quadrilaterals, and three types of T-junctions where two quad-strips merge into one. The bi-cubic pieces of a polyhedral spline have matching derivatives along their break lines, possibly after a known change of variables. The pieces are represented in Bernstein-Bézier form with coefficients depending linearly on the polyhedral control net, so that evaluation, differentiation, integration, moments, and so on, are no more costly than for standard tensor-product splines. Bi-cubic polyhedral splines can be used both to model geometry and for computing functions on the geometry. Although polyhedral splines do not offer nested refinement by refinement of the control net, polyhedral splines support engineering analysis of curved smooth objects. Coarse nets typically suffice since the splines efficiently model curved features. Algorithm 1032 is a C++ library with input-output example pairs and an IGES output choice.

Abstract Point‐Augmented Subdivision (PAS) replaces complex geometry‐dependent guided subdivision, known to yield high‐quality surfaces, by explicit subdivision formulas that yield similarly‐good limit surfaces and are easy to implement using any subdivision infrastructure: map the control net d augmented by a fixed central limit point C, to a finer net (d̃,C) = M(d,C), where the subdivision matrix M is assembled from the provided stencil Tables. Point‐augmented bi‐cubic subdivision improves the state of the art so that bi‐cubic subdivision surfaces can be used in high‐end geometric design: the highlight line distribution for challenging configurations lacks the shape artifacts usually associated with explicit iterative generalized subdivision operators near extraordinary points. Five explicit formulas define Point‐augmented bi‐cubic subdivision in addition to uniform B‐spline knot insertion. Point‐augmented bi‐cubic subdivision comes in two flavors, either generating a sequence of C2‐joined surface rings (PAS2) or C1‐joined rings (PAS1) that have fewer pieces.

Abstract Polyhedral modeling and re‐meshing algorithms use T‐junctions to add or remove feature lines in a quadrilateral mesh. In many ways this is akin to adaptive knot insertion in a tensor‐product spline, but differs in that the designer or meshing algorithm does not necessarily protect the consistent combinatorial structure that is required to interpret the resulting quad‐dominant mesh as the control net of a hierarchical spline – and so associate a smooth surface with the mesh as in the popular tensor‐product spline paradigm. While G‐splines for multi‐sided holes or generalized subdivision can, in principle, convert quad‐dominant meshes with T‐junctions into smooth surfaces, they do not preserve the two preferred directions and so cause visible shape artifacts. Only recently have n‐gons with T‐junctions (T‐gons) in unstructured quad‐dominant meshes been recognized as a distinct challenge for generalized splines. This paper makes precise the notion of locally quad‐dominant mesh as quad‐meshes including τ‐nets, i.e. T‐gons surrounded by quads; and presents the first high‐quality G‐spline construction that can use τ‐nets as control nets for spline surfaces suitable, e.g., for automobile outer surfaces. Remarkably, T‐gons can be neighbors, separated by only one quad, both of T‐gons and of points where many quads meet. A τ‐net surface cap consists of 16 polynomial pieces of degree (3,5) and is refinable in a way that is consistent with the surrounding surface. An alternative, everywhere bi‐3 cap is not formally smooth, but achieves the same high‐quality highlight line distribution.

Abstract Converting quadrilateral meshes to smooth manifolds, guided subdivision offers a way to combine the good highlight line distribution of recent G‐spline constructions with the refinability of subdivision surfaces. This avoids the complex refinement of G‐spline constructions and the poor shape of standard subdivision. Guided subdivision can then be used both to generate the surface and hierarchically compute functions on the surface. Specifically, we present a C2 subdivision algorithm of polynomial degree bi‐6 and a curvature bounded algorithm of degree bi‐5. We prove that the common eigenstructure of this class of subdivision algorithms is determined by their guide and demonstrate that their eigenspectrum (speed of contraction) can be adjusted without harming the shape. For practical implementation, a finite number of subdivision steps can be completed by a high‐quality cap. Near irregular points this allows leveraging standard polynomial tools both for rendering of the surface and for approximately integrating functions on the surface.

T-junctions occur where surface strips start or terminate. This paper develops a new way to create smooth piecewise polynomial free-form spline surfaces from quad-meshes that include T-junctions. All mesh nodes are interpreted as control points of GT-splines, that is, geometrically smoothly joined piecewise polynomials. GT-splines are akin to and compatible with B-splines and cover simple T-junctions by two polynomial pieces of degree bi-4 and more complex ones by four such patches. They complement multi-sided surface constructions in generating free-form surfaces with adaptive layout. Since GT-splines do not require a global coordination of knot intervals, GT-constructions are easy to deploy and can provide smooth surfaces with T-junctions where T-splines cannot have a smooth parameterization. GT-constructions display a uniform highlight linedistribution on input meshes where alternatives, such as Catmull-Clark subdivision, exhibit oscillations.

Abstract 'Class A surface’ is a term in the automotive design industry, describing spline surfaces with aesthetic, non‐oscillating highlight lines. Tensor‐product B‐splines of degree bi‐3 (bicubic) are routinely used to generate smooth design surfaces and are often the de facto standard for downstream processing. To bridge the gap, this paper explores and gives a concrete suggestion, how to achieve good highlight line distributions for irregular bi‐3 tensor‐product patch layout by allowing, along some seams, a slight mismatch of normals below the industry‐accepted tolerance of one tenth of a degree. Near the irregularities, the solution can be viewed as transforming a higher‐degree, high‐quality formally smooth surface into a bi‐3 spline surface with few pieces, sacrificing formal smoothness but qualitatively retaining the shape.

Abstract The paper develops a rational bi‐cubic G2 (curvature continuous) analogue of the non‐uniform polynomial C2 cubic B‐spline paradigm. These rational splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, in one by default smoothly‐connected structure. The versatility of this new tool for processing exact geometry is illustrated by conceptual design from basic shapes.

We complete and bring together two pairs of surface constructions that use polynomial pieces of degree (3,3) to associate a smooth surface with a mesh. The two pairs complement each other in that one extends the subdivisionmodeling paradigm, the other the NURBS patch approach to free-form modeling. Both Catmull-Clark [3] and polar subdivision [7] generalize bi-cubic spline subdivision. Together, they form a powerful combination for smooth object design: while Catmull-Clark subdivision is more suitable where few facets join, polar subdivision nicely models regions where many facets join, as when capping extruded features. We show how to easily combine the meshes of these two generalizations of bi-cubic spline subdivision. A related but different generalization of bi-cubic splines is to model non-tensor-product configurations by a finite set of smoothly connected bi-cubic patches. PCCM [12] does so for layouts where Catmull-Clark would apply. We show that a single NURBS patch can be used where polar subdivision would be applied. This spline is singularly parametrized, but, using a novel technique, we show that the surface is C1 and has bounded curvatures.

We describe and analyze a subdivision scheme that generalizes bicubic spline subdivision to control nets with polar structure. Such control nets appear naturally for surfaces with the combinatorial structure of objects of revolution and at points of high valence in subdivision meshes. The resulting surfaces are C 2 except at a finite number of isolated points where the surface is C 1 and the curvature is bounded.