Polyhedral design is a well-established paradigm for creating complex free-form shapes by smoothing control polyhedra. Several approaches exist, including recursive subdivision and direct algorithms. We introduce a direct method where multi-sided and multi-connected, ribbon-based patches are stitched together. While traditional methods are based on control polyhedra with only convex faces, our approach can handle concave angles and multiple hole loops, as well; moreover, the free-form patchwork has exactly the same topological structure as the input control polyhedron. The algorithm consists of three basic steps: (i) computing an auxiliary polyhedron, (ii) defining a general topology curve network and determining cross-derivative (ribbon) data along the patch boundaries, (iii) computing Generalized B-spline surfaces. Furthermore, using a shape parameter, a family of models with the same topological structure can be generated. Several test examples are provided to demonstrate the capabilities of this new method.

We show that the radials of log-aesthetic curves are generalized Archimedean spirals. Examining the logarithmic curvature histogram reveals that these radials have an inherent similarity to the associated log-aesthetic curves, and can also be used as computationally inexpensive approximants. Different kinds of fits are proposed and discussed through examples. A possible generalization of log-aesthetic curves based on generalized Archimedean spirals is also explored.

Ribbon-based multi-sided parametric patches represent a special group of free-form surfaces. While ribbons with boundary and cross-derivative constraints can sufficiently define patches, modifying their interior – in addition – may be required, as well. We introduce a new technique to add interior control structures for Generalized B-spline (GBS) patches defined over multi-connected curved domains, where the sum of the blending functions produces a weight deficiency. A local parameterization and a parametric MAT (medial axis transform) structure is computed within the domain, which leads to a family of quad structures, called templates. We define blend functions by distributing weight deficiency amongst the vertices of the template and describe algorithms to locate related control points in 3D, yielding Template GBS patches that tightly approximate the input ribbons. As a final step, a modified representation is created, called Hybrid GBS, that exactly interpolates the given input ribbons and inherits the interior control structure of an arbitrarily chosen Template GBS patch. • Interior control structure for multi-sided Generalized B-spline patches. • Support for multiply connected domains. • Quadrilateral templates from parametric medial axis. • Special blending functions associated with the template geometry. • Surfaces based on approximating and exact ribbons.

This work investigates ribbon-based multi-sided surfaces that satisfy positional and cross-derivative constraints to ensure smooth transitions with adjacent tensor-product and multi-sided surfaces. The influence of cross-derivatives, crucial to surface quality, is studied within Kato’s transfinite surface interpolation instead of control point-based methods. To enhance surface quality, the surface is optimized using cost functions based on curvature metrics. Specifically, a Gaussian curvature-based cost function is also proposed in this work. An automated optimization procedure is introduced to determine rotation angles of cross-derivatives around normals and their magnitudes along curves in Kato’s interpolation scheme. Experimental results using both primitive (e.g., spherical) and realistic examples highlight the effectiveness of the proposed approach in improving surface quality.

A state-of-the-art survey is presented on various formulations of multi-sided parametric surface patches, with a focus on methods that interpolate positional and cross-derivative information along boundaries.

Generalized Bézier patches with curved domains can represent complex, multi-sided surfaces, but do not provide explicit control over the interior of the surface, as they are defined by means of side-based ribbons. In this paper we extend this representation by proposing a uniform, intuitive control structure, based on templates – a collection of quadrilaterals that covers and affects the 3D shape. It is constructed based on a variant of the Medial Axis Transform (MAT) that uses the local parameterization of the domain. For a given patch a hierarchical sequence of 2D templates can be defined, each determining the topology of the corresponding 3D control structure. First we introduce templates, then present the way of associating biparametric Bernstein blend functions with the control points. Next we describe how to position the control points of the MAT skeleton and the remaining interior control points, while ribbons are preserved. Finally we show a few examples that demonstrate the method and discuss the pros and cons of the approach.

We investigate genuinely multi-sided patches that interpolate ribbon surfaces along their boundaries. Recent works suggest defining patches over parametric domains with curved boundaries and hole loops, where the domain mimics the shape of the surface to be constructed. Cross-derivatives of the input are interpreted with respect to this curved parametric domain, but it is an open question how to initialize and modify these vector functions. We propose algorithms to set the cross-derivatives of multi-sided patches defined by Bézier and B-spline ribbons. Boundaries and surface constraints are inherited from adjacent patches, and our goal is to define a nice surface while ensuring smooth (G1) connections. We exploit that ribbon parameterizations induce ‘proportional’ cross-derivative magnitudes in 3D, and express cross-derivatives as the combination of vector functions and appropriately chosen scalar functions. Continuity constraints imply complex relations between the control points of the ribbons, so their direct modification is not feasible. Instead we suggest a constrained editing technique based on control vectors that significantly simplifies this task.

Abstract The $$\kappa $$ κ -curve is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or $$\epsilon \kappa $$ ϵ κ -curves. $$\kappa $$ κ -curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bézier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using $$\epsilon \kappa $$ ϵ κ -curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.

We propose a new surface representation, the Generalized B-spline (GBS) patch, that combines ribbon interpolants given in B-spline form. A GBS patch can connect to tensor-product B-spline surfaces with arbitrary Gm continuity. It supports ribbons not only along the perimeter loop, but also around holes in the interior of the patches. This is a follow-up paper of a recent publication (Várady et al., 2020) that described multi-sided Bézier surfaces over curved multi-sided domains. While the fundamental concept is retained, several new details have been elaborated. The weighting functions are modified to be products of B-spline and Bernstein basis functions, multiplied by rational terms. A new local parameterization method is introduced using harmonic functions, that handles periodic hole loops, as well. Interior shape control is adapted to the B-spline representation of the ribbons. Several examples illustrate the capabilities of the proposed scheme. Our implementation is based on a computationally efficient discretization.

A new multi-sided, control point based surface formulation is proposed, extending the principles of the Generalized Bézier patch published by Várady et al. (2016). The surface is constructed from side interpolants given in Bézier form. The boundary curves may have different degrees, and the cross-derivatives can ensure arbitrary G-continuity with adjacent patches. The representation is capable of handling boundary curves with high curvature variations and concave angles. The surfaces are C∞-continuous and may have holes in their interior. The most distinctive feature of the patch is that it is defined over a planar domain with curved boundaries that mimic the shape of the 3D boundary curves. Local parameters, derived from harmonic barycentric coordinates, are associated with each side of the domain. The control points are multiplied by Bernstein functions and additional rational terms that provide the same degree of freedom for shape design as their quadrilateral counterparts. The paper introduces the full construction of Curved Domain (CD) Bézier patches including domain generation, parameterization, basis functions, and methods for additional interior shape control. Specific problems related to CD surfaces are also discussed with many examples that demonstrate the peculiarity and usefulness of the scheme.

I-patches represent a family of implicit multi-sided surfaces. Similarly to functional splines, each boundary curve of the patch is defined as the intersection of a primary and a bounding surface, both given in implicit form, and the patch can connect to the primaries with arbitrary geometric continuity. Following the publication of Várady et al. [1], this paper elaborates the basic formulation in more detail, and introduces several interesting features, including a distance-based surface interpretation, consistent orientation of the primaries, setting shape parameters, and handling various special cases. Implicit multi-sided patches are primarily used for connecting simple implicit surfaces, such as planes, cylinders, spheres etc., however, I-patches are also capable of modeling complex free-form shapes. We show constructions for producing setback vertex blends with conic boundaries and patchworks defined by control polyhedra. We discuss the benefits and limitations of the representation through several examples.

Abstract A new n‐sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross‐derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi‐sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points—other than those specified by the user—are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G1continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.

Abstract The basic idea of curve network‐based design is to construct smoothly connected surface patches, that interpolate boundaries and cross‐derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G 1 ) continuity between the adjacent patches, curvature continuous connections (G 2 ) may also be required. Examples include special curve network configurations with supplemented internal edges, “master‐slave” curvature constraints, and general topology surface approximations over meshes. The first step is to assign optimal surface curvatures to the nodes of the curve network; we discuss different optimization procedures for various types of nodes. Then interpolant surfaces called parabolic ribbons are created along the patch boundaries, which carry first and second derivative constraints. Our construction guarantees that the neighboring ribbons, and thus the respective transfinite patches, will be G 2 continuous. We extend Gregory's multi‐sided surface scheme in order to handle parabolic ribbons, involving the blending functions, and a new sweepline parameterization. A few simple examples conclude the paper.

The surfaces of complex free-form objects can typically be modeled by a hierarchy of primary surfaces, connecting surfaces and corner patches. Within the context of digital shape reconstruction, these surfaces simultaneously approximate measured data points, satisfy fairness criteria and adhere to continuity constraints according to their dependencies. A new framework algorithm is introduced to perfect existing B-spline surfaces; the algorithm alternates --- in a stepwise manner --- between continuity constraint satisfaction and fairing by the remaining degrees of freedom. Some well-known fairing methods are adapted to this framework. Constrained fairing for n-sided corner patches, composed of quadrilaterals, is also briefly discussed. A few examples illustrate the results.