Geometric Hermite interpolation by rational curves of constant width

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.Funding for the other authors not affiliated with BCAM: Grant PID2021-124577NBI00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Project PID2019-104927GB-C21 funded by MCIN/AEI/10.13039/501100011033. Project UJI-B2022-19 funded by Universitat Jaume I. Project CIAICO/2021/180 funded by Generalitat Valenciana

Ellipse-preserving Hermite interpolation and subdivision

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behaviour is the same as the classical cubic Hermite spline algorithm. The same convergence properties---i.e., fourth order of approximation---are hence ensured

Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

Interpolation of equation-of-state data

Aims. We use Hermite splines to interpolate pressure and its derivatives simultaneously, thereby preserving mathematical relations between the derivatives. The method therefore guarantees that thermodynamic identities are obeyed even between mesh points. In addition, our method enables an estimation of the precision of the interpolation by comparing the Hermite-spline results with those of frequent cubic (B-) spline interpolation. Methods. We have interpolated pressure as a function of temperature and density with quintic Hermite 2D-splines. The Hermite interpolation requires knowledge of pressure and its first and second derivatives at every mesh point. To obtain the partial derivatives at the mesh points, we used tabulated values if given or else thermodynamic equalities, or, if not available, values obtained by differentiating B-splines. Results. The results were obtained with the grid of the SAHA-S equation-of-state (EOS) tables. The maximum $lg P$ difference lies in the range from $10^{-9}$ to $10^{-4}$, and $\Gamma_1$ difference varies from $10^{-9}$ to $10^{-3}$. Specifically, for the points of a solar model, the maximum differences are one order of magnitude smaller than the aforementioned values. The poorest precision is found in the dissociation and ionization regions, occurring at $T \sim 1.5\cdot 10^3 - 10^5$ K. The best precision is achieved at higher temperatures, $T>10^5$ K. To discuss the significance of the interpolation errors we compare them with the corresponding difference between two different equation-of-state formalisms, SAHA-S and OPAL 2005. We find that the interpolation errors of the pressure are a few orders of magnitude less than the differences from between the physical formalisms, which is particularly true for the solar-model points.Comment: Accepted for publication in A&