Georg Muntingh

Detecting symmetries of rational plane and space curves

Abstract This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.

Detecting similarity of rational plane curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.

Symmetry detection of rational space curves from their curvature and torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem (Alcazar et al., 2014b). To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage. The paper presents a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric.The method is significantly faster, simpler, and more general than earlier methods addressing similar problems.An analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage are included.

Size distributions of chemically synthesized Ag nanocrystals

Silver nanocrystals made by a chemical reduction of silver salts (AgNO3) by sodium borohydride (NaBH4) were studied using transmission electron microscopy and light scattering simulations. For various AgNO3/NaBH4 molar ratios, the size distributions of the nanocrystals were found to be approximately log-normal. In addition, a linear relation was found between the mean nanocrystal size and the molar ratio. In order to relate the size distribution of Ag nanocrystals of the various molar ratios to the scattering properties of Ag nanocrystals in solar cell devices, light scattering simulations of Ag nanocrystals in Si, SiO2, SiN, and Al2O3 matrices were carried out using MiePlot. These light scattering spectra for the individual nanocrystal sizes were combined into light scattering spectra for the fitted size distributions. The evolution of these scattering spectra with respect to an increasing mean nanocrystal size was then studied. From these findings, it is possible to find the molar ratio for which the co...

A Hermite interpolatory subdivision scheme for C 2 -quintics on the Powell-Sabin 12-split

In order to construct a C 1 -quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme (Dyn and Lyche, 1998). In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally C 3 and globally C 2 . For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.

Detecting Similarities of Rational Space Curves

We provide an algorithm to check whether two rational space curves are related by a similarity, i.e., whether they are equal up to position, orientation and scale. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner bases for the special case of helical curves.

Symbols and exact regularity of symmetric pseudo-splines of any arity

Pseudo-splines form a family of subdivision schemes that provide a natural blend between interpolating schemes and approximating schemes, including the Dubuc–Deslauriers schemes and B-spline schemes. Using a generating function approach, we derive expressions for the symbols of the symmetric m-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact Hölder regularity algebraically as a logarithm of the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, including the symmetric binary, ternary, and quarternary pseudo-spline schemes.

Solar induced growth of silver nanocrystals

The effect of solar irradiation on plasmonic silver nanocrystals has been investigated using transmission electron microscopy and size distribution analysis, in the context of solar cell applications for light harvesting. Starting from an initial collection of spherical nanocrystals on a carbon film whose sizes are log-normally distributed, solar irradiation causes the nanocrystals to grow, with one particle reaching a diameter of 638 nm after four hours of irradiation. In addition some of the larger particles lose their spherical shape. The average nanocrystal diameter was found to grow as predicted by the Ostwald ripening model, taking into account the range of area fractions of the samples. The size distribution stays approximately log-normal and does not reach one of the steady-state size distributions predicted by the Ostwald ripening model. This might be explained by the system being in a transient state.

Symmetry detection of rational space curves

We present a novel, deterministic, and efficient method to detect whether a given rational space curve C is symmetric. The method combines two ideas. On one hand in a similar way to [1], [2], if C is symmetric then the symmetry provides a second parametrization of the curve; furthermore, whenever the first parametrization is proper, i.e. injective except for finitely many parameter values, the latter is also proper and both are related by means of a Mobius transformation [3] that completely determines the symmetry. On the other hand, if C is symmetric then the curvature and torsion of C at corresponding points must coincide. By putting together these two ideas we can give an algorithm to directly find the Mobius transformations defining the symmetries of the curve. From here we can compute these symmetries and its characteristic elements (symmetry axes, symmetry planes, etc.) This completes and improves on an earlier method addressing a similar problem [3]. Keywords Symmetry Detection, Space Curves, Rational Curves References [1] Alcazar J.G. (2014), Efficient detection of symmetries of polynomially parametrized curves, Journal of Computational and Applied Mathematics vol. 255, pp. 715–724. [2] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Similarity of Plane Rational Plane Curves, Journal of Computational and Applied Mathematics, Vol. 269, pp. 1–13. [3] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Symmetries of Rational Plane and Space Curves, to appear in Computer Aided Geometric Design.

Introduction to ShApes, Geometry, and Algebra

We present a brief overview of the ShApes, Geometry and Algebra Initial Training Network, including its motivation and results. After this we provide a preview of this volume, where we briefly describe the flavor of each chapter.