The paper generalizes the classical C1 cubic Clough–Tocher spline space over a triangulation to C1 spaces of any degree higher that three. It shows that the considered spaces can be equipped with a ...basis consisting of non-negative locally supported functions forming a partition of unity and demonstrates the applicability of the basis in the context of the finite element method. The studied spaces have optimal approximation power and are defined by enforcing additional smoothness inside the triangles of the triangulation where the Clough–Tocher splitting is used. Locally, over each triangle of the triangulation, the splines are expressed in the Bernstein–Bézier form, which enables one to take the full advantage of the geometric properties and computational techniques that come with such a representation. Solving boundary problems with Galerkin discretization is thus relatively straightforward and is illustrated with several examples.
Despite many advances in dentistry, no objective and quantitative method is available to evaluate gingival shape. The surface curvature of the optical scans represents an unexploited possibility. The ...present study aimed to test surface curvature estimation of intraoral scans for objective evaluation of gingival shape.
The method consists of four main steps, i.e., optical scanning, surface curvature estimation, region of interest (ROI) definition, and gingival shape analysis. Six different curvature measures and three different diameters were tested for surface curvature estimation on central (n = 78) and interdental ROI (n = 88) of patients with advanced periodontitis to quantify gingiva with a novel gingival shape parameter (GS). The reproducibility was evaluated by repeating the method on two consecutive intraoral scans obtained with a scan-rescan process of the same patient at the same time point (n = 8).
Minimum and mean curvature measures computed at 2 mm diameter seem optimal GS to quantify shape at central and interdental ROI, respectively. The mean (and standard deviation) of the GS was 0.33 ± 0.07 and 0.19 ± 0.09 for central ROI using minimum, and interdental ROI using mean curvature measure, respectively, computed at a diameter of 2 mm. The method's reproducibility evaluated on scan-rescan models for the above-mentioned ROI and curvature measures was 0.02 and 0.01, respectively.
Surface curvature estimation of the intraoral optical scans presents a precise and highly reproducible method for the objective gingival shape quantification enabling the detection of subtle changes. A careful selection of parameters for surface curvature estimation and curvature measures is required.
•G1 interpolation scheme for motion data with cubic PH biarcs is presented.•The length of the center trajectory is prescribed in advance.•The solution is given in a closed form and depends on four ...shape parameters.•The twist of the Euler–Rodrigues frame is minimized.•The spline construction is provided.
In this paper the G1 interpolation scheme for motion data, i.e., interpolation of data points and rotations at the points, with cubic PH biarcs is presented. The rotational part of the motion is determined by the Euler–Rodrigues frame which matches the given boundary positions. In addition, the length of the biarc is prescribed. It is shown that the interpolant exists for any data and any chosen length greater than the difference between the interpolation points. The interpolant is given in a closed form and depends on some free shape parameters, which are determined so that the curve is of a nice shape and the twist of the Euler–Rodrigues frame is minimized. The spline construction is provided and numerical examples that confirm the derived theoretical results are included.
•A new construction of C1 continuous edge B-spline functions over two triangles is presented.•The non-negativity of the derived splines is established provided the two triangles form a strictly ...convex quadrilateral.•The introduced concept of edge B-splines is applied to construct a new basis for the Argyris type splines over triangulations.•The basis functions are locally supported and non-negative C1 splines forming a partition of unity.•The usability and stability of the basis is demonstrated on solving different least square problems.
Given two triangles in a planar domain sharing an edge and forming a convex quadrilateral, it is shown how to construct a non-negative basis for C1 splines that restrict to polynomials of a total degree higher than one on each of the triangles. The representation may be seen as a generalization of the Bernstein–Bézier form of a spline on every separate triangle, and the main challenge in its development is the construction of basis functions associated with the common edge. This novel concept is aimed to be used in assembling B-spline-like bases for C1 splines on triangulations, as it is demonstrated for Argyris type splines of degree higher than five on triangulations with flippable edges.
In this paper interpolation methods for the construction of quartic splines on triangulations refined with 10-splits are proposed. After examining the C2 macro-structure on a single triangle in terms ...of the Bernstein–Bézier representation, three methods that can be applied on general triangulations are developed. The methods make use of Hermite interpolation data prescribed at the vertices and (optionally) on the edges of the triangulation. The first two approaches lead to splines that are C2 continuous on triangles and C1 continuous across the interior edges of the triangulation. The third method gives rise to splines with overall C2 continuity, which is an exceptionally high order of smoothness for splines of degree four, but comes at the cost of solving a global system of linear equations. The derived results are accompanied with a few numerical examples that show an interesting behavior of splines in dependence of interpolated data.
•Quartic splines on triangulations refined with 10-splits are studied.•A construction of C2 continuous quartic spline over one triangle with a 10-split is presented.•Three different interpolation schemes on triangulations are proposed.•Theoretical results are illustrated with numerical examples.
In this paper the problem of constructing spatial G2 continuous Pythagorean-hodograph (PH) spline curves, that interpolate points and frame data, and in addition have the prescribed arc-length, is ...addressed. The interpolation scheme is completely local and can be directly applied for motion design applications. Each spline segment is defined as a PH biarc curve of degree 7 satisfying super-smoothness conditions at the biarc’s joint point. The biarc is expressed in a closed form with additional free parameters, where one of them is determined by the length constraint. The selection of the remaining free parameters is suggested, that allows the existence of the solution of the length interpolation equation for any prescribed length and any ratio between norms of boundary tangents. By the proposed automatic procedure for computing the frame and velocity quaternions from the first and second order derivative vectors, the paper presents a direct generalization of the construction done for planar curves to spatial ones. Several numerical examples are provided to illustrate the proposed method and to show its good performance, also when a spline construction in considered.
The paper considers the macro-element splitting technique that refines every triangle of the initial triangulation into ten smaller triangles. The resulting refinement is an extension of the ...well-known Powell–Sabin 6-split and enables a construction of polynomial C1 splines of degree two interpolating first order Hermite data at the vertices of the initial triangulation. A particular construction, called a balanced 10-split, is presented that allows a numerically stable B-spline representation of such splines. This amounts to, firstly, defining locally supported basis functions for the macro-element space that form a convex partition of unity, and, secondly, expressing the coefficients of the spline represented in this basis by the means of spline values and derivatives at the vertices of the initial triangulation.
