Jörg Peters's research while affiliated with University of Florida and other places
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Publications (153)
Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for control nets that merge parameter lines. Additionally, untilrecently,no...
Digital Elevation Models (DEMs) are crucial for modeling and analyzing terrestrial environments, but voids in DEMs can compromise their downstream use. Diff-DEM is a self-supervised method for filling DEM voids that leverages a Denoising Diffusion Probabilistic Model (DDPM). Conditioned on a void-containing DEM, the DDPM acts as a transition kernel...
Introduction:
The Toolkit for Illustration of Procedures in Surgery (TIPS) is an open source virtual reality (VR) laparoscopic simulation-based training environment with force feedback. The TIPS-author is a content creation interface that allows a surgeon educator (SE) to assemble new laparoscopic training modules. New technology enables safety ru...
Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n≠4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic C1 splines is evaluated as a tool for solving elliptic higher-order partial differential equati...
For control nets outlining a large class of topological polyhedra, not just tensor-product grids, bi-cubic polyhedral spline s 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 co...
PurposeSoft-tissue manipulations, such as collecting, stretching or tearing tissue, are a common component of surgery. When too much force is applied, these manipulations result in a residual plastic deformation that surgeons should be aware of and that should be modeled by surgical simulation.Methods
Many tissues, vessels and organs can be modeled...
Smooth spline surfaces can now be built with polyhedral control nets rather than just grid-like tensor-product control nets. However, irregularities such as T-junctions, multi-sided facets, and n-valent vertices need to be sufficiently separated. Automatically generated quad-dominant meshes, and meshes created by designers unaware of the requiremen...
Scaffold-like surfaces, ranging from large-scale trusses to engineered micro-structures, are often sketched via repeating patterns of nodes and edges. Offsetting these graphs turns them into meshes for which a smoothly-rounded scaffold surface ’skin’ needs to be locally generated on the fly for production or analysis. We focus on minimal single-val...
This survey of piecewise polynomial surface constructions for filling multi-sided holes in a smooth spline complex focusses on a class of hybrid constructions that, while heterogeneous, combines all the practical advantages of state-of-the-art for modelling and analysis: good shape, easy implementation and simple refinability up a pre-defined level...
Multi-sided faces arise in polyhedral modeling through introduction of features not aligned with the regular grid structure, e.g. when trimming off corners or merging primary shapes. A standard first step is to split the n-gon into n quadrilaterals that join at the n-gon’s centroid. A canonical example is the ‘face point rule’ of Catmull–Clark subd...
Collecting, stretching and tearing soft tissue is common in surgery. These repeated deformations have a plastic component that surgeons take into consideration and that surgical simulation should model. Organs and tissues can often be modeled as curved cylinders or planes, offset orthogonally to form thick shells. A pair of primary directions, e.g....
When the full-scale storing and retrieving of volumetric models is cost prohibitive, intersection queries require intelligent access to pieces generated on demand. To conform to a given curved outer shape without clipping, such models are often the result of a non-linear free-form deformation applied to a geometrically simpler, canonical model. The...
Traditionally the approach to filling n-sided holes differs substantially for bicubic C2-splines vs biquadratic C1-splines due to the ‘primal’ vs ‘dual’ interpretation of the control net that emphasises either n-valent vertices or n-sided facets. Here we propose a construction that unites the treatment of both types of spline surfaces, notably for...
A mesh is locally quad-dominant (lqd) if all non-4-sided facets are surrounded by quadrilaterals. Lqd meshes allow for irregular nodes where n≠4 quads meet and for multi-sided facets, called T-gons, that end quad-strips and so adjust mesh density.
This paper introduces a new class of bi-cubic (bi-3) Geometric T-joint (GT) splines whose control nets...
Geometrically smooth spline surfaces, generalized to include n-sided facets or configurations of n≠4 quads, can exhibit a curious lack of additional degrees of freedom for modelling or engineering analysis when refined.
This paper establishes a minimal polynomial degree for smooth constructions of multi-sided surfaces that guarantees more flexibili...
A polar configuration is a node surrounded by m triangles. Polar configurations are common to cap off cylinders and spheres. When the triangles, interpreted as quadrilaterals with one edge collapsed, are surrounded by a quad-strip then the extended polar configuration qualifies as part of a locally quad-dominant (lqd) mesh. Recent constructions, re...
Querying and interacting with models of massive material micro-structure requires localized on-demand generation of the micro-structure since the full-scale storing and retrieving is cost prohibitive. When the micro-structure is efficiently represented as the image of a canonical structure under a non-linear space deformation to allow it to conform...
C1 splines over box-complexes generalize C1 degree 3 (cubic) tensor-product splines. A box-complex is a collection of 3-dimensional boxes forming an unstructured hexahedral mesh that can include irregular points and irregular edges where the layout deviates from the tensor-product grid layout. For example, an edge shared and enclosed by five boxes...
Refinement of a space of splines should yield additional degrees of freedom for modeling and engineering analysis, both along boundaries and in the interior. Yet such additional flexibility fails to materialize for multi-sided G2 surface constructions when the polynomial degree is too low.
This paper establishes a tight lower bound on the polynomia...
Multi-sided facets in polyhedral models and meshes serve to connect regular sub-meshes (star-configurations) and to start or end quad-strips (T-configurations). Using the polyhedral mesh as control net, recursive subdivision algorithms often yield poor shape for these non-quad configurations. Polynomial surface constructions such as geometrically s...
This paper introduces Corner-Sharing Tetrahedra (CoSTs), a minimalist, constraint-graph representation of micro-structure. CoSTs have built-in structural guarantees, such as connectivity and minimal rigidity. CoSTs form a space, fully accessible via local operations, that is rich enough to design regular or irregular micro-structure at multiple sca...
A sequence of C2-connected nested subdivision rings of polynomial degree bi-4 can be made to follow a guide surface and completed by a tiny finite cap to serve as a refinable surface representation for design and analysis (Karčiauskas and Peters, 2018) . This raises the question, both of academic and practical interest, how much and at what cost to...
Enriching tensor-product B-spline control nets by allowing T-gons (where strips of quadrilaterals start or end) and irregular nodes (where n≠4 quadrilaterals meet) reduces the requirements on quad-meshing and increases the flexibility for polyhedral design with associated smooth surfaces. This paper introduces a family of piecewise polynomial, geom...
Consider n space curves c˜i(t) i∈Zn, meeting at a vertex. The geodesic network interpolation problem is to determine n surface pieces (patches) xi(u,v) that are each internally C², surround the vertex and each interpolate curves c˜i and c˜i+1 so that the curves are geodesics of the resulting surface. This paper proves that together three local cons...
Compared to G
k
continuity, C
k
continuity simplifies the construction of functions on surfaces and their refinement, e.g. to solve differential equations on the surface. The new class of almost everywhere parametrically C2 free-form surfaces provide such a parameterization. For example, a new bi-6 construction combines a fast-contracting C2 guided...
State-of-the-art representations of volumetric multi-scale shape and structure can be classified into three broad categories: continuous, continuous-from-discrete, and discrete representations. We propose modeling micro-structure with a class of discrete Corner-Sharing Tetrahedra (CoSTs). CoSTs can represent bar-joint, tensegrity, line-incidence, a...
Tools of subdivision theory allow constructing surfaces that are effectively C² and have a good highlight line distribution also near irregularities where more or fewer than four quadrilateral patches meet. Here, effectively C² means that transitions that are only first-order smooth are confined to a tiny multi-sided cap. The cap can be chosen smal...
To be directly useful both for shape design and a thin shell analysis, a surface representation has to satisfy three properties: (1) be compatible with CAD surface representations, (2) yield generically a good highlight distribution, and (3) offer a refinable space of functions on the surface. Here we propose a new construction, based on a number o...
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 u...
Lower bounds on the generation of smooth bi-cubic surfaces imply that geometrically smooth ($G^1$) constructions need to satisfy conditions on the connectivity and layout. In particular, quadrilateral meshes of arbitrary topology can not in general be covered with $G^1$ -connected B\'ezier patches of bi-degree 3 using the layout proposed in [ASC17]...
To date, singularly-parameterized surface constructions suffer from poor highlight line distributions, ruling them out as a surface representation of choice for primary design surfaces. This paper explores graded, many-piece, everywhere C1 singularly-parameterized surface caps that mimic the shape of a high-quality guide surface. The approach illus...
Converting quad meshes to smooth manifolds, guided subdivision offers a way to combine the good highlight line distributions of recent G-spline constructions with the refinability of subdivision surfaces. 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...
For two high-quality piecewise polynomial geometrically smooth ( ) surface constructions, explicit functions are derived that form the basis of a functions space on the surfaces. The spaces are refinable and nested, i.e. the functions can be re-represented at a finer level. By choosing all basis functions to be first order smooth a maximal set of d...
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 c...
Objective:
The study assesses user acceptance and effectiveness of a surgeon-authored virtual reality (VR) training module authored by surgeons using the Toolkit for Illustration of Procedures in Surgery (TIPS).
Methods:
Laparoscopic adrenalectomy was selected to test the TIPS framework on an unusual and complex procedure. No commercial simulati...
Scaffold surfaces bound geometric structures that have a dual characterization as a curve network and a solid. A subset of scaffold surfaces can be modeled with minimal single-valence (MSV) meshes, i.e. meshes consisting of vertices of a single irregular valence two of which are separated by exactly one regular, 4-valent vertex. We present an algor...
Polycube G-splines form a 2-manifold guided by a mesh of quadrilateral faces such that at most six quads meet at each vertex. In particular, this replicates the layout of the quad faces of a polycube. Polycube G-splines are piecewise bicubic and polycube G-spline surfaces are almost everywhere tangent-continuous (G1) based on rational linear repara...
Building on a result of U. Reif on removable singularities, we construct bi-3 splines that may include irregular points where less or more than four tensor-product patches meet. The resulting space complements PHT splines, is refinable and the refined spaces are nested, preserving for example surfaces constructed from the splines. As in the regular...
Enabling surgeon-educators to themselves create virtual reality (VR) training units promises greater variety, specialization, and relevance of the units. This paper describes a software bridge that semi-automates the scene-generation cycle, a key bottleneck in authoring, modeling, and developing VR units. Augmenting an open source modeling environm...
This paper addresses a gap in the arsenal of curvature continuous tensor-product spline constructions: an algorithm is provided to fill n-sided holes in C2 bi-3 spline surfaces using n patches of degree bi-6. Numerous experiments illustrate good highlight line and curvature distribution on the resulting surfaces.
Geometrically continuous (Gk) constructions naturally yield families of finite elements for isogeometric analysis (IGA) that are Ck also for non-tensor-product layout. This paper describes and analyzes one such concrete C1 geometrically generalized IGA element (short: gIGA element) that generalizes bi-quadratic splines to quad meshes with irregular...
'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 g...
The discontinuous Galerkin (dG) method outputs a sequence of polynomial pieces. Post-processing the sequence by Smoothness-Increasing Accuracy-Conserving (SIAC) convolution not only increases the smoothness of the sequence but can also improve its accuracy and yield superconvergence. SIAC convolution is considered optimal if the SIAC kernels, in th...
Quad meshes can be interpreted as tensor-product spline control meshes as long as they form a regular grid, locally. We present a new option for complementing bi-3 splines by bi-4 splines near irregularities in the mesh layout, where less or more than four quadrilaterals join. These new generalized surface and IGA (isogeometric analysis) elements h...
For many design applications, where multiple primary surface pieces meet, the distribution of curvature is more important than formally achieving exact curvature continuity. For parametric spline surfaces, when constructing a multi-sided surface cap, we demonstrate a strong link between the uniform variation of the re-parameterization between (boun...
We explore a class of polynomial tensor-product spline surfaces on 3-6 polyhedra, whose vertices have valence n=3 or n=6. This restriction makes it possible to exclusively use rational linear transition maps between the pieces: transitions between the bi-cubic tensor-product spline pieces are either C1 or they are G1 (tangent continuous) based on o...
(geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched constructions for geometry an...
Recently, it was shown that a bi-cubic patch complex with n-sided holes can be completed into a curvature-continuous (G(2)) surface by n-sided caps of degree bi-5 that offer good and flexible shape (Karciauskas and Peters, 2013). This paper further explores the space of n-sided caps of degree bi-5 but focuses on functionals to set degrees of freedo...
Shape artifacts, especially for convex input polyhedra, make Doo and Sabin’s generalization of bi-quadratic (bi-2) subdivision surfaces unattractive for general design. Rather than tuning the eigenstructure of the subdivision matrix, we improve shape by adding a point and enriching the refinement rules. Adding a guiding point can also yield a polar...
Current strategies for real-time rendering of trimmed spline surfaces re-approximate the data, pre-process extensively or introduce visual artifacts. This paper presents a new approach to rendering trimmed spline surfaces that guarantees visual accuracy efficiently, even under interactive adjustment of trim curves and spline surfaces. The technique...
This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson’s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, partic...
Biquadratic (bi-2) splines are the simplest choice for converting a regular quad meshes into smooth tensor-product spline surfaces. Existing methods for blending three, five or more such bi-2 spline surfaces using surface caps consisting of pieces of low polynomial degree suffer from artifacts ranging from flatness to oscillations. The new construc...
This paper presents new univariate linear non-uniform interpolatory subdivision constructions that yield high smoothness, C
3 and C
4, and are based on least-degree spline interpolants. This approach is motivated by evidence, partly presented here, that constructions based on high-degree local interpolants fail to yield satisfactory shape, especial...
The definition of a B-spline is extended to unordered knot sequences. The added flexibility implies that the resulting piecewise polynomials, named U-splines, can be negative and locally linearly dependent. It is therefore remarkable that linear combinations of U-splines retain many properties of splines in B-spline form including smoothness, polyn...
We present a framework for deriving non-uniform interpolatory subdivision algorithms closely related to non-uniform spline interpolants. Families of symmetric non-uniform interpolatory 2n-point schemes of smoothness C^n^-^1 are presented for n=2,3,4 and even higher order, as well as a variety of non-uniform 6-point schemes with C^3 continuity.
This paper presents a non-uniform cubic C2C2 spline framework that unifies three scenarios for incorporating data from basic curves, such as spirals and conics. In the first scenario, no parameterization of the basic curves is available, only well-spaced samples; in the second, a parameterization is available but cannot be used directly in a spline...
We show how to automatically join, into one unified spline surface, C2C2 tensor-product bi-cubic NURBS and G2G2 bi-cubic rational splines. The G2G2 splines are capable of exactly representing basic shapes such as (pieces of) quadrics and surfaces of revolution, including tori and cyclides. The main challenge is to transition between the differing f...
In this paper we present the first comprehensive study and analysis on different sketch-based mesh cutting approaches. To compare a representative number of state-of-the-art sketch-based mesh cutting methods, we conduct a large scale user study which ...
A curved or higher-order surface, such as spline patch or a Bézier patch, is rendered pixel-accurate if it displays neither polyhedral artifacts nor parametric distortion. This paper shows how to set the evaluation density for a patch just finely enough so that parametric surfaces render pixel-accurate in the standard graphics pipeline. The approac...
Prescribing a network of curves to be interpolated by a surface model is a standard approach in geometric design. Where n curves meet, even when they afford a common normal direction, they need to satisfy an algebraic condition, called the vertex enclosure constraint, to allow for an interpolating piecewise polynomial C1 surface. Here we prove the...
A polar configuration is a triangle fan in a quad-dominant mesh; it allows for many mesh lines to join at a single polar vertex. This paper shows how a single tensor-product spline of degree (3,6) can cap a polar configuration with a C2 surface. By design, this C2 polar spline joins C2 with surrounding bi-3 tensor-product splines and thereby comple...
We develop a rational biquadratic G1 analogue of the non-uniform C1 B-spline paradigm. These G1 splines can exactly reproduce parts of multiple basic shapes, such as cyclides and quadrics, and combine them into one smoothly-connected structure. This enables a design process that starts with basic shapes, re-represents them in spline form and uses t...
We develop a class of rational, G2-connected splines of degree 3 that allow modeling multiple basic shapes, such as segments of conics and circle arcs in particular, in one structure. ► This can be used, for example, to have portions of a control polygon exactly reproduce segments of the shapes while other portions blend between these primary shape...
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...
Root lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions, due to their highly symmetric structure. One root lattice, the Cartesian grid, is almost exclusively used since it matches the coordinate grid; but it is less efficient than other root lattices. Box-splines, on the other hand, generalize tens...
Converting a quadrilateral input mesh into a C1 surface with one bi-3 tensor-product spline patch per facet is a classical challenge. We give explicit local averaging formulas for the spline control points. Where the quadrilateral mesh is not regular, the patches have two internal double knots, the least number and multiplicity to always allow for...
To broaden the use of simulation for teaching, in particular of new procedures and of low-volume procedures, we propose an environment and workflow that allows surgeon-educators create teaching modules. Our challenge is to make the simulation tools accessible, modifiable and sharable by users with moderate computer and VR experience. Our contributi...
We derive box-spline quasi-interpolants for root lattices based on the treatment in the book Box Splines by de Boor, Höllig and Riemenschneider [1].
We derive box-spline quasi-interpolants for root lattices based on the treatment in the book Box Splines by de Boor, Höllig and Riemenschneider [1].
Sampling and reconstruction of generic multivariate functions is more efficient on non-Cartesian root lattices, such as the BCC (Body-Centered Cubic) lattice, than on the Cartesian lattice. We introduce a new n x n generator matrix A* that enables, in n variables, efficient reconstruction on the non-Cartesian root lattice A(n)* by a symmetric box-s...
We exhibit the essentially unique projective linear (rational linear) reparameterization for constructing C<sup>s</sup> surfaces of genus g>0. Conversely, for quadrilaterals and isolated vertices of valence 8, we show constructively for s=1,2 that this map yields a projective linear spline space for surfaces of genus greater or equal to 1. This est...
The promise of modeling by subdivision is to have simple rules that avoid cumbersome stitching-together of pieces. However, already in one variable, exactly reproducing a variety of basic shapes, such as conics and spirals, leads to non-stationary rules that are no longer as simple; and combining these pieces within the same curve by one set of rul...
Graphs of pairwise incidences between collections of rigid bodies occur in many practical applications and give rise to large algebraic systems for which all solutions have to be found. Such pairwise incidences have explicit, simple and rational parametrizations that, in principle, allow us to partially resolve these systems and arrive at a reduced...
A key problem when interpolating a network of curves occurs at vertices: an algebraic condition called the vertex enclosure con- straint must hold wherever an even number of curves meet. This paper recasts the constraint in terms of the local geometry of the curve network. This allows formulating a new geometricconstraint, related to Euler's Theore...
We discuss the problem of fitting a curve or surface to given measurement data. In many situations, the usual least-squares approach (minimization of the sum of squared norms of residual vectors) is not suitable, as it implicitly assumes a Gaussian distribution ...
Subdivision surfaces provide a compact representation for smooth surfaces that facilitate modeling and animation. They have widespread application in the movie industry, and there's a natural desire to use them also in real-time applications. This course presents theoretical results, implementations, applications, and future research directions. To...
For use in real-time applications, we present a fast algorithm for converting a quad mesh to a smooth, piecewise polynomial
surface on the Graphics Processing Unit (GPU). The surface has well-defined normals everywhere and closely mimics the shape
of Catmull–Clark subdivision surfaces. It consists of bicubic splines wherever possible, and a new cla...
Popular subdivision algorithms like Catmull-Clark and Loop are C-2 almost everywhere, but suffer from shape artifacts and reduced smoothness exactly near the so-called "extraordinary vertices" that motivate their use. Subdivision theory explains that inherently, for standard stationary subdivision algorithms, curvature-continuity and the ability to...
a) e 3 (b) e 4 (c) e 5 (d) e 3 : x 2 + y 2 (e) e 4 : x 2 − y 2 (f) e 5 : 2xy n = 6: e 3 , e 4 , e 5 ∈ span{e 2 1 , e 1 e 2 , e 2 2 } n → ∞: e 3 , e 4 , e 5 ∈ span{e 2 1 , e 1 e 2 , e 2 2 } + O(1 4 n) Figure 1: Making C 2 subdivision work with degree bi-3. (left: a,b,c) Stationary degree bi-3 subdivision cannot be C 2 at a central point of fixed val...
This paper derives strong relations that boundary curves of a smooth complex of patches have to obey when the patches are computed by local averaging. These relations restrict the choice of reparameterizations for geometric continuity.In particular, when one bicubic tensor–product B-spline patch is associated with each facet of a quadrilateral mesh...
We assemble triangular patches of total degree at most eight to form a curvature continuous surface. The construction illustrates how separation of local shape from representation and formal continuity yields an effective construction paradigm in partly underconstrained scenarios. The approach localizes the technical challenges and applies the spli...
To repeatedly evaluate linear combinations of box-splines in a fast and stable way, in particular along knot planes, the box-spline
is converted to and tabulated as piecewise polynomial in BB-form (Bernstein–Bézier-form). We show that the BB-coefficients
can be derived and stored as integers plus a rational scale factor and derive a hash table for...
We describe a fully interactive, low-overhead and robust peritoneum representation allowing for probing and cutting. The peritoneum implementation has been tested within a surgical illustration environment.
![Fig. 3. A 6-5 patch x l : [0 .. 1] 2 → R 3 of degree 6 in the periodic...](profile/Kestutis-Karciauskas-2/publication/221403843/figure/fig1/AS:305593115004928@1449870484200/A-6-5-patch-x-l-0-1-2-R-3-of-degree-6-in-the-periodic-and-degree-5-in-the-radial_Q320.jpg)
Determining the least m such that one m×m bi-cubic macro-patch per quadrilateral offers enough degrees of freedom to construct a smooth surface by local operations regardless of the vertex valences is of fundamental interest; and it is of interest for computer graphics due to the impending ability of GPUs to adaptively evaluate polynomial patches a...
Surface constructions of polynomial degree (3,3) come in four flavours that complement each other: one pair extends the subdivision paradigm, the other the NURBS patch approach to free-form modeling. The first pair, Catmull–Clark subdivision and Polar subdivision (Catmull, E., Clark, J., 1978. Recursively generated B-spline surfaces on arbitrary to...
We introduce and analyze an efficient reconstruction algorithm for FCC-sampled data. The reconstruction is based on the 6-direction box spline that is naturally associated with the FCC lattice and shares the continuity and approximation order of the triquadratic B-spline. We observe less aliasing for generic level sets and derive special techniques...
Polyhedral meshes consisting of triangles, quads, and pentagons and polar configurations cover all major sampling and modeling scenarios. We give an algorithm for efficient local, parallel conversion of such meshes to an everywhere smooth surface consisting of low-degree polynomial pieces. Quadrilateral facets with 4-valent vertices are 'regular' a...
This paper gives an overview of two recent techniques for high-quality surface constructions: polar layout and the guided approach. We demonstrate the challenge of high-quality surface construction by examples since the notion of surface quality lacks an overarching theory. A key ingredient of high-quality constructions is a good layout of the surf...
We convert any quad manifold mesh into an at least C1 surface consisting of bi-cubic tensor-product splines with localized pertur- bations of degree bi-5 near non-4-valent vertices. There isone poly- nomial piece per quad facet, regardless of the valence of the ver- tices. Particular care is taken to derive simple formulas so that the surfaces are...
Citations
... Since infinite refinement is not practical, if the finest resolution, for example to avoid pixel dropout at the extraordinary points, is not known, one can cover the remaining hole with a tiny piecewise smooth bi-4, respectively bi-3 cap construction from the literature, e.g. [KP23b]. For n = 3, 5, . . . ...
Reference: Quadratic‐Attraction Subdivision
... For all practical purposes, it is sensible to assume C 1 when W 2,2 is needed. refinement property, and the generalization to the important 3d case seems to be practicable [Pet20,YP22]. By contrast, trivariate analogues of the competing approaches subdivision and geometric continuity appear to be much harder to develop, see [DPR + 23] for a discussion of this issue. ...
... Subdivision generalizations of bi-2 splines consist of an infinite sequence of nested (contracting) bi-2 polynomial surface rings. Ref. [3] has visible artifacts already in the first ring, Augmented Subdivision presented in [4] improves the shape by following a carefully chosen central guide point and polyhedral-net splines [35] combine algorithms from [8,18,36] to generalize tensor-product bi-quadratic (bi-2) splines, filling in finitely many polynomial pieces of degree at most bi-3. T-splines [37] address the merging parallel quad strips but typically serve only to refine an existing quad partition; due to their global parameterization requirement, they may not be well-defined for a given T-configuration, (see Figure 2 in [38] and Figure 6 in [39]). ...
... Existing algorithms require these faces to be separated by a frame of quadrilaterals. Mitigation strategies range from ad hoc designer intervention, to an improved Doo-Sabin refinement step [3,4], to special re-meshing rules for T 0 -and T 1 -locations, [5] (see Figure 1e). The drawbacks of these mitigations are both an increase in the number of patches and a decrease in the surface quality. ...
... The shapes of G 2 constructions of degree bi-7 [25] or degree bi-6 [26], and lower-degree tangent-continuous splines [27][28][29][30][31][32][33] are empirically measured via highlighted line distribution [34]. FC surfaces fill irregularities in a C 1 bi-quadratic (bi-2) tensor-product surface, which is attractive since bi-2 splines have minimal bi-degree for smoothing quadrilateral meshes. ...
... io/). Due to the pandemic, and for the second year in a row, SMI was convened online, and this year was hosted by Texas on November [14][15][16]2021. ...
... Typically locking depends on the number of degrees of freedom [Frâncu et al. 2021] and disappears when higher order bases are used [Wang et al. 2004]. This makes hexahedral meshes particularly suited for problems where linear elements are used, such as in the interactive simulation of hyper-elastic and plastic phenomena (e.g. in surgical simulation [Gao and Peters 2021]) and in fast transient dynamic phenomena that employ explicit time integration (e.g. crash and impact simulation) [Gravouil et al. 2009] because higher-order basis functions would necessarily demand a reduction of the time step to achieve numerical stability, according to the Courant−Friedrichs−Lewy condition [Courant et al. 1967;Weber et al. 2021]. ...
... Such so-called hole-filling techniques are commonplace in geometric modeling and can also be used to construct smooth spaces for isogeometric analysis, cf. [96][97][98][99][100][101]. We focus here on the simplest possible way of resolving this issue, which is to enforce only C 0 -smoothness near the EVs and G 1 at the EV, namely the Almost-C 1 construction proposed in [11]. ...
... Section 4 provides exampledriven critical assessment and discussion of variants as well a comparison to FC 3 . For rotationally symmetric scenarios that permit regular layout or less contraction, FC 3 8 is shown to be at least as well-shaped as regular bi-2 splines and the surfaces presented in [8]. [8,9] and two-strip contraction via (c) cascading triangles, (d) T 0 -gon + T 1 -gon and (e) refinement according to [5]; (f) △ 2 -net with triangulated gray core as a generalization of (c,d), see [6]. ...
... Such so-called hole-filling techniques are commonplace in geometric modeling and can also be used to construct smooth spaces for isogeometric analysis, cf. [96][97][98][99][100][101]. We focus here on the simplest possible way of resolving this issue, which is to enforce only C 0 -smoothness near the EVs and G 1 at the EV, namely the Almost-C 1 construction proposed in [11]. ...
