- Email: [email protected]
Recent trends in theoretical and applied geometry
Computer Aided Geometric Design 31 (2014) 329–330
Contents lists available at ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Editorial
Recent trends in theoretical and applied geometry Classical geometric constructions are essential components in modern geometric design environments, due to their fundamental role in numerical and geometrical algorithms for computer aided geometric design, scientific visualization, interactive geometric modeling, and numerical simulation. In addition, they have the great potential to open different paths of research connected to interdisciplinary topics in the context of diverse geometric applications, not only in geometry processing and computer graphics, but also in a wider range of research areas related to biology and medicine. This special issue focuses on the interaction between the theoretical side of geometric methods, and the applicationoriented side devoted to the analysis and development of computer aided geometric design methods. We received 31 submissions and 21 of them were selected for publication in this special issue after a comprehensive review process. The majority of these papers are related to talks presented at the Conference on Geometry: Theory and Applications (CGTA), which was held at the University of Ljubljana, Slovenia in June 2013. Previous CGTA conferences were held in Vorau, Austria ˇ Czech Republic (2009). Topics of interest for the special issue include results related to the ge(2007 & 2011), and Plzen, ometry of curves and surfaces, computational methods, such as spline and approximation techniques, as well as geometric constructions related to robotics and kinematics. The selected papers can be divided in four different groups as follows. The first group consists of seven papers that present theoretical and applied results for planar and spatial curves. Affine arc length polylines and curvature continuous uniform B-splines studies connections between discrete affine differential geometry and spline theory, including n-dimensional generalizations, and shows how to approximate a planar curve with an affine regular polyline. The notion of infinity branches and approaching curves are introduced in Asymptotic behavior of an implicit algebraic plane curve, together with an algorithm to compare the behavior of two curves at infinity. The problem of Identifying and approximating monotonous segments of algebraic curves using support function representation is addressed in the third paper that presents a new approach for describing the topology of a plane real algebraic curve. The characterization by the support function is also used to decompose and approximate the given curve within certain bounds on the approximation error in terms of Hausdorff distance. Classical curve theory in normed planes provides a comprehensive survey on classical curves in Minkowski planes that revisits known results, points out open problems, and outlines possible directions for future studies in related research fields. The three remaining papers of the first group introduce novel schemes for the solution of Hermite interpolation problems with Pythagorean-hodograph (PH) biarcs and triarcs, as well as related applications to rational rotation minimizing motions. In particular, A fully data-dependent criterion for free angles selection in spatial PH cubic biarc Hermite interpolation presents an effective strategy for an automatic selection of C 1 /G 1 interpolants with desirable shape properties among the family of formal solutions that characterizes the problem. In order to consider analogous C 2 problems with lowdegree PH curves, C 2 Hermite interpolation by Pythagorean-hodograph quintic triarcs discusses a construction based on three arcs of degree five joined together. In both papers the approximation order of the schemes is also studied. Finally, a motion design algorithm that controls not only the path of a moving object but also its orientation along the spatial interpolant of low degree is developed in C 1 interpolation by rational biarcs with rational rotation minimizing directed frames. The second group of six papers is related to the representation and construction of surfaces and their applications in engineering and design. Transition to canonical principal parameters on minimal surfaces describes how to convert a minimal surface into a special canonical parametric form. The latter can be used to easily check whether two given minimal surfaces are just different parametric representations of the same geometric shape. The second paper Surfaces with Pythagorean normals along rational curves studies an extension of Pythagorean normal vector surfaces, where only the offset to the surface along certain curves is rational. This relaxed condition allows for more general surface shapes and is sufficient for practical applications, like numerical-controlled milling. The next three papers deal with subdivision surfaces. A new construction of a triangular macro-element, leading to C 2 -continuous interpolatory limit surfaces over a given triangulation is described in A Hermite interpolatory subdivision scheme for C 2 -quintics on the Powell–Sabin 12-split, along with a subdivision scheme for efficiently evaluating the limit surface and its derivatives. General triangular midpoint subdivision considers a family of subdivision schemes for triangle meshes, based on the refine-and-smooth principle of the Lane–Riesenfeld algorithm, and shows that the limit surfaces are always C 1 -continuous. An algorithm for approximating trimmed NURBS surfaces with Catmull– Clark surfaces and producing gap-free models with C 1 -continuity across shared boundaries is presented in Conversion of http://dx.doi.org/10.1016/j.cagd.2014.09.002 0167-8396/© 2014 Published by Elsevier B.V.
330
Editorial
trimmed NURBS surfaces to Catmull–Clark subdivision surfaces. The last paper in this group, KSpheres – An efficient algorithm for joining skinning surfaces is about a new surface design tool which allows to construct a G 1 -continuous parametric surface that touches a given set of spheres. The third group of five papers is about splines and their suitable application for approximation algorithms, both in the context of data fitting and for the numerical solution of partial differential equations in isogeometric analysis. A diversification approach for the B-spline basis is introduced in Approximation with diversified B-splines. In the bivariate case this leads to an approximation error that can be bounded in terms of pure partial derivatives with a constant depending only on the order of the spline space. The authors also show that similar results cannot be extended to higher dimensions. The paper Spline multiresolution and wavelet-like decompositions presents a multiresolution approach to spline curves, motivated by computer numerical control applications, and proves a general decay estimate for wavelet coefficients together with desirable properties for edge detection and denoising. The next two papers investigate suitable extensions of classical tensor-product spline spaces based on hierarchical constructions. TDHB-splines: the truncated decoupled basis of hierarchical tensor-product splines combines and extends the so-called truncation and decoupling mechanisms to identify a novel hierarchical basis with distinguishing properties. The extension of the truncated hierarchical B-spline basis to hierarchies of spaces that are spanned by generating systems is introduced and analyzed in Adaptively refined multilevel spline spaces from generating systems. This new hierarchical model does not require the linear independence of the underlying generating systems, but it allows to extend the key properties of existing constructions to a more general setting that can also be used to perform local refinement in the presence of features via a hierarchy of hierarchical B-splines. Derivatives of isogeometric functions on n-dimensional rational patches in Rd concludes the third group of papers by considering the properties, as well as the smoothness behavior of derivatives of functions mapped to a NURBS patch using the isoparametric concept. The fourth group of three papers focuses on kinematics and robotics. The first two papers study two special kinds of kinematic mechanism and their properties. A classification of Stewart Gough platforms with self-motions of dimension greater than two is given in On Stewart Gough manipulators with multidimensional self-motions, as well as some results towards the complete analysis of two-dimensional self-motions, which remains an open problem. Overconstrained mechanisms based on planar four-bar-mechanisms presents a novel class of four-bar mechanisms that can be used to define spatial one-parametric motions. Finally, an efficient algorithm for detecting the collisions between a fixed polyhedron and a moving polyhedron, which is an essential ingredient for robot path-planning, physical simulations, and computer graphics applications, is described in Guaranteed collision detection with toleranced motions. We would like to thank the authors for the valuable contributions and the numerous referees for their comments and suggestions. We further thank the journal manager, Vasanth Thomas William; the Editors-in-Chief, Gerald Farin, Rida T. Farouki, Konrad Polthier, and Hartmut Prautzsch; the Elsevier publisher, Gail Rodney; and the whole Elsevier CAGD team for the support during the preparation of the special issue.
Carlotta Giannelli INdAM c/o University of Florence, Italy Kai Hormann Università della Svizzera italiana, Switzerland Emil Žagar University of Ljubljana, Slovenia
Contents lists available at ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Editorial
Recent trends in theoretical and applied geometry Classical geometric constructions are essential components in modern geometric design environments, due to their fundamental role in numerical and geometrical algorithms for computer aided geometric design, scientific visualization, interactive geometric modeling, and numerical simulation. In addition, they have the great potential to open different paths of research connected to interdisciplinary topics in the context of diverse geometric applications, not only in geometry processing and computer graphics, but also in a wider range of research areas related to biology and medicine. This special issue focuses on the interaction between the theoretical side of geometric methods, and the applicationoriented side devoted to the analysis and development of computer aided geometric design methods. We received 31 submissions and 21 of them were selected for publication in this special issue after a comprehensive review process. The majority of these papers are related to talks presented at the Conference on Geometry: Theory and Applications (CGTA), which was held at the University of Ljubljana, Slovenia in June 2013. Previous CGTA conferences were held in Vorau, Austria ˇ Czech Republic (2009). Topics of interest for the special issue include results related to the ge(2007 & 2011), and Plzen, ometry of curves and surfaces, computational methods, such as spline and approximation techniques, as well as geometric constructions related to robotics and kinematics. The selected papers can be divided in four different groups as follows. The first group consists of seven papers that present theoretical and applied results for planar and spatial curves. Affine arc length polylines and curvature continuous uniform B-splines studies connections between discrete affine differential geometry and spline theory, including n-dimensional generalizations, and shows how to approximate a planar curve with an affine regular polyline. The notion of infinity branches and approaching curves are introduced in Asymptotic behavior of an implicit algebraic plane curve, together with an algorithm to compare the behavior of two curves at infinity. The problem of Identifying and approximating monotonous segments of algebraic curves using support function representation is addressed in the third paper that presents a new approach for describing the topology of a plane real algebraic curve. The characterization by the support function is also used to decompose and approximate the given curve within certain bounds on the approximation error in terms of Hausdorff distance. Classical curve theory in normed planes provides a comprehensive survey on classical curves in Minkowski planes that revisits known results, points out open problems, and outlines possible directions for future studies in related research fields. The three remaining papers of the first group introduce novel schemes for the solution of Hermite interpolation problems with Pythagorean-hodograph (PH) biarcs and triarcs, as well as related applications to rational rotation minimizing motions. In particular, A fully data-dependent criterion for free angles selection in spatial PH cubic biarc Hermite interpolation presents an effective strategy for an automatic selection of C 1 /G 1 interpolants with desirable shape properties among the family of formal solutions that characterizes the problem. In order to consider analogous C 2 problems with lowdegree PH curves, C 2 Hermite interpolation by Pythagorean-hodograph quintic triarcs discusses a construction based on three arcs of degree five joined together. In both papers the approximation order of the schemes is also studied. Finally, a motion design algorithm that controls not only the path of a moving object but also its orientation along the spatial interpolant of low degree is developed in C 1 interpolation by rational biarcs with rational rotation minimizing directed frames. The second group of six papers is related to the representation and construction of surfaces and their applications in engineering and design. Transition to canonical principal parameters on minimal surfaces describes how to convert a minimal surface into a special canonical parametric form. The latter can be used to easily check whether two given minimal surfaces are just different parametric representations of the same geometric shape. The second paper Surfaces with Pythagorean normals along rational curves studies an extension of Pythagorean normal vector surfaces, where only the offset to the surface along certain curves is rational. This relaxed condition allows for more general surface shapes and is sufficient for practical applications, like numerical-controlled milling. The next three papers deal with subdivision surfaces. A new construction of a triangular macro-element, leading to C 2 -continuous interpolatory limit surfaces over a given triangulation is described in A Hermite interpolatory subdivision scheme for C 2 -quintics on the Powell–Sabin 12-split, along with a subdivision scheme for efficiently evaluating the limit surface and its derivatives. General triangular midpoint subdivision considers a family of subdivision schemes for triangle meshes, based on the refine-and-smooth principle of the Lane–Riesenfeld algorithm, and shows that the limit surfaces are always C 1 -continuous. An algorithm for approximating trimmed NURBS surfaces with Catmull– Clark surfaces and producing gap-free models with C 1 -continuity across shared boundaries is presented in Conversion of http://dx.doi.org/10.1016/j.cagd.2014.09.002 0167-8396/© 2014 Published by Elsevier B.V.
330
Editorial
trimmed NURBS surfaces to Catmull–Clark subdivision surfaces. The last paper in this group, KSpheres – An efficient algorithm for joining skinning surfaces is about a new surface design tool which allows to construct a G 1 -continuous parametric surface that touches a given set of spheres. The third group of five papers is about splines and their suitable application for approximation algorithms, both in the context of data fitting and for the numerical solution of partial differential equations in isogeometric analysis. A diversification approach for the B-spline basis is introduced in Approximation with diversified B-splines. In the bivariate case this leads to an approximation error that can be bounded in terms of pure partial derivatives with a constant depending only on the order of the spline space. The authors also show that similar results cannot be extended to higher dimensions. The paper Spline multiresolution and wavelet-like decompositions presents a multiresolution approach to spline curves, motivated by computer numerical control applications, and proves a general decay estimate for wavelet coefficients together with desirable properties for edge detection and denoising. The next two papers investigate suitable extensions of classical tensor-product spline spaces based on hierarchical constructions. TDHB-splines: the truncated decoupled basis of hierarchical tensor-product splines combines and extends the so-called truncation and decoupling mechanisms to identify a novel hierarchical basis with distinguishing properties. The extension of the truncated hierarchical B-spline basis to hierarchies of spaces that are spanned by generating systems is introduced and analyzed in Adaptively refined multilevel spline spaces from generating systems. This new hierarchical model does not require the linear independence of the underlying generating systems, but it allows to extend the key properties of existing constructions to a more general setting that can also be used to perform local refinement in the presence of features via a hierarchy of hierarchical B-splines. Derivatives of isogeometric functions on n-dimensional rational patches in Rd concludes the third group of papers by considering the properties, as well as the smoothness behavior of derivatives of functions mapped to a NURBS patch using the isoparametric concept. The fourth group of three papers focuses on kinematics and robotics. The first two papers study two special kinds of kinematic mechanism and their properties. A classification of Stewart Gough platforms with self-motions of dimension greater than two is given in On Stewart Gough manipulators with multidimensional self-motions, as well as some results towards the complete analysis of two-dimensional self-motions, which remains an open problem. Overconstrained mechanisms based on planar four-bar-mechanisms presents a novel class of four-bar mechanisms that can be used to define spatial one-parametric motions. Finally, an efficient algorithm for detecting the collisions between a fixed polyhedron and a moving polyhedron, which is an essential ingredient for robot path-planning, physical simulations, and computer graphics applications, is described in Guaranteed collision detection with toleranced motions. We would like to thank the authors for the valuable contributions and the numerous referees for their comments and suggestions. We further thank the journal manager, Vasanth Thomas William; the Editors-in-Chief, Gerald Farin, Rida T. Farouki, Konrad Polthier, and Hartmut Prautzsch; the Elsevier publisher, Gail Rodney; and the whole Elsevier CAGD team for the support during the preparation of the special issue.
Carlotta Giannelli INdAM c/o University of Florence, Italy Kai Hormann Università della Svizzera italiana, Switzerland Emil Žagar University of Ljubljana, Slovenia