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New Developments in Geometry – Theory and Applications
Computer Aided Geometric Design 47 (2016) 1–2
Contents lists available at ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Editorial
New Developments in Geometry – Theory and Applications The idyllic scenery of the Mühlviertel in the northern part of Upper Austria provided the perfect setting for the 2015 edition of the Conference on Geometry – Theory and Applications, which was held at Schloss Weinberg, Kefermarkt/Austria. As in previous years, this conference was a forum for the exchange of ideas and for reporting progress in the field of Applied Geometry, focusing on Computational Geometry, Computer Aided Geometric Design, Differential Geometry, Robotics and Kinematics, and Isogeometric Analysis. Previous events in this conference series took place in Ljubljana/Slovenija (2013), Vorau/Austria (2007 & 2011), and Plzen/Czech Republic (2009). This volume of Computer Aided Geometric Design contains twelve papers focusing on related results, the majority of which was presented at the conference. The first group consists of two papers that present results for univariate splines. In her paper entitled “Design with Quasi Extended Chebyshev Piecewise Spaces”, Mazure studies a new class of spline spaces and shows these are inverse images of two-dimensional Chebyshev spaces under piecewise generalised derivatives associated with systems of piecewise weight functions. It is also shown that these spaces can produce interesting shape effects. Goldman and Simeonov explore generalized quantum splines. In particular, they investigate the properties of generalized quantum B-spline bases and generalized quantum B-spline curves. They also use a generalization of the blossoming principle to the quantum case. The second group, which contains four submissions, is the largest one. It contains several results related to bivariate splines and surfaces. The first paper in this group, authored by Csima and Szirmai, is devoted to isoptic surfaces of polyhedra in three-dimensional space. So far, most related results are limited to planar objects. In their paper, the authors develop an algorithm to determine the isoptic surface of a polyhedron and apply it to the Platonic solids and to some semi-regular Archimedean solids. Sangalli, Takacs and Vázquez investigate unstructured spline spaces for isogeometric analysis based
http://dx.doi.org/10.1016/j.cagd.2016.08.002 0167-8396/© 2016 Published by Elsevier B.V.
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Editorial
on spline manifolds. They generalize the concept of dual-compatible B-splines, which has been developed for structured T-splines. The third paper in this group, which was contributed by Buffa, Giannelli, Morgenstern and Peterseim, is devoted to the analysis of the complexity of refinement for a class of admissible mesh configurations in the context of hierarchical spline refinement. Analysis-suitable G 1 multi-patch parametrizations for C 1 isogeometric spaces are studied by Collin, Sangalli and Takacs. The authors analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and discuss their approximation power. The special issue contains two papers which focus on algebraic methods. In their paper, Mantzaflaris, Rahkooy and Zafeirakopoulos focus on the efficient computation of the dual space and the directional multiplicity of an isolated point. A completely different topic is addressed in the article by Farre et al., which is devoted to algorithms for detecting dependencies and rigid subsystems for Computer-Aided Design. Kinematics and robotics are classical fields in applied geometry and we are happy that they are represented by three submissions to this special issue. A more theoretical topic from kinematical geometry is addressed by Nawratil in his paper on a quaternion-based approach to equiform kinematics and the line-elements. More precisely, he studies the intimate relationship between instantaneous equiform motions and the geometry of line-elements in E 4 . In his contribution, Pfurner presents a closed form solution for the inverse kinematics problem of a redundant anthropomorphic robot arm. Due to the redundancy, the solution consists of a one parameter set and the degree of freedom provided by the additional joint can be used for various applications, such as avoiding singularities. Last, but not least, we included two submissions containing results about the interesting topic of offset and bisector curves and surfaces. Kozak, Krajnc and Vitrih introduce a quaternion-based approach to polynomial surfaces with Pythagorean normal vectors (PN surfaces). They derive relations between quaternion-valued coefficients of certain bivariate polynomials that allow to construct PN surfaces. An algebro-geometric analysis of the bisectors of two algebraic plane curves is presented by Fioravanti and Sendra. The authors address both the cases of curves given by implicit and parametric representations. We would like to thank the authors for the valuable contributions and the numerous referees for their comments and suggestions. We further thank the Editors-in-Chief, Rida T. Farouki, and Konrad Polthier, the Elsevier publisher, Gail Rodney; and the whole Elsevier CAGD team for the support during the preparation of the special issue.
Udo Hertrich-Jeromin Bert Jüttler Josef Schicho Linz and Vienna, July 2016
Contents lists available at ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Editorial
New Developments in Geometry – Theory and Applications The idyllic scenery of the Mühlviertel in the northern part of Upper Austria provided the perfect setting for the 2015 edition of the Conference on Geometry – Theory and Applications, which was held at Schloss Weinberg, Kefermarkt/Austria. As in previous years, this conference was a forum for the exchange of ideas and for reporting progress in the field of Applied Geometry, focusing on Computational Geometry, Computer Aided Geometric Design, Differential Geometry, Robotics and Kinematics, and Isogeometric Analysis. Previous events in this conference series took place in Ljubljana/Slovenija (2013), Vorau/Austria (2007 & 2011), and Plzen/Czech Republic (2009). This volume of Computer Aided Geometric Design contains twelve papers focusing on related results, the majority of which was presented at the conference. The first group consists of two papers that present results for univariate splines. In her paper entitled “Design with Quasi Extended Chebyshev Piecewise Spaces”, Mazure studies a new class of spline spaces and shows these are inverse images of two-dimensional Chebyshev spaces under piecewise generalised derivatives associated with systems of piecewise weight functions. It is also shown that these spaces can produce interesting shape effects. Goldman and Simeonov explore generalized quantum splines. In particular, they investigate the properties of generalized quantum B-spline bases and generalized quantum B-spline curves. They also use a generalization of the blossoming principle to the quantum case. The second group, which contains four submissions, is the largest one. It contains several results related to bivariate splines and surfaces. The first paper in this group, authored by Csima and Szirmai, is devoted to isoptic surfaces of polyhedra in three-dimensional space. So far, most related results are limited to planar objects. In their paper, the authors develop an algorithm to determine the isoptic surface of a polyhedron and apply it to the Platonic solids and to some semi-regular Archimedean solids. Sangalli, Takacs and Vázquez investigate unstructured spline spaces for isogeometric analysis based
http://dx.doi.org/10.1016/j.cagd.2016.08.002 0167-8396/© 2016 Published by Elsevier B.V.
2
Editorial
on spline manifolds. They generalize the concept of dual-compatible B-splines, which has been developed for structured T-splines. The third paper in this group, which was contributed by Buffa, Giannelli, Morgenstern and Peterseim, is devoted to the analysis of the complexity of refinement for a class of admissible mesh configurations in the context of hierarchical spline refinement. Analysis-suitable G 1 multi-patch parametrizations for C 1 isogeometric spaces are studied by Collin, Sangalli and Takacs. The authors analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and discuss their approximation power. The special issue contains two papers which focus on algebraic methods. In their paper, Mantzaflaris, Rahkooy and Zafeirakopoulos focus on the efficient computation of the dual space and the directional multiplicity of an isolated point. A completely different topic is addressed in the article by Farre et al., which is devoted to algorithms for detecting dependencies and rigid subsystems for Computer-Aided Design. Kinematics and robotics are classical fields in applied geometry and we are happy that they are represented by three submissions to this special issue. A more theoretical topic from kinematical geometry is addressed by Nawratil in his paper on a quaternion-based approach to equiform kinematics and the line-elements. More precisely, he studies the intimate relationship between instantaneous equiform motions and the geometry of line-elements in E 4 . In his contribution, Pfurner presents a closed form solution for the inverse kinematics problem of a redundant anthropomorphic robot arm. Due to the redundancy, the solution consists of a one parameter set and the degree of freedom provided by the additional joint can be used for various applications, such as avoiding singularities. Last, but not least, we included two submissions containing results about the interesting topic of offset and bisector curves and surfaces. Kozak, Krajnc and Vitrih introduce a quaternion-based approach to polynomial surfaces with Pythagorean normal vectors (PN surfaces). They derive relations between quaternion-valued coefficients of certain bivariate polynomials that allow to construct PN surfaces. An algebro-geometric analysis of the bisectors of two algebraic plane curves is presented by Fioravanti and Sendra. The authors address both the cases of curves given by implicit and parametric representations. We would like to thank the authors for the valuable contributions and the numerous referees for their comments and suggestions. We further thank the Editors-in-Chief, Rida T. Farouki, and Konrad Polthier, the Elsevier publisher, Gail Rodney; and the whole Elsevier CAGD team for the support during the preparation of the special issue.
Udo Hertrich-Jeromin Bert Jüttler Josef Schicho Linz and Vienna, July 2016