Authors:Aksel Sveier; Adam Leon Kleppe; Lars Tingelstad; Olav Egeland Pages: 1961 - 1976 Abstract: Abstract This paper presents an approach for detecting primitive geometric objects in point clouds captured from 3D cameras. Primitive objects are objects that are well defined with parameters and mathematical relations, such as lines, spheres and ellipsoids. RANSAC, a robust parameter estimator that classifies and neglects outliers, is used for object detection. The primitives considered are modeled, filtered and fitted using the conformal model of geometric algebra. Methods for detecting planes, spheres and cylinders are suggested. Least squares fitting of spheres and planes to point data are done analytically with conformal geometric algebra, while a cylinder is fitted by defining a nonlinear cost function which is optimized using a nonlinear least squares solver. Furthermore, the suggested object detection scheme is combined with an octree sampling strategy that results in fast detection of multiple primitive objects in point clouds. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0759-1 Issue No:Vol. 27, No. 3 (2017)

Authors:Mauricio Cele Lopez Belon; Dietmar Hildenbrand Pages: 2019 - 2033 Abstract: Abstract The usage of Geometric Algebra motors instead of Euclidean vectors for describing the position and orientation of points on a surface has promising applications in Computer Science and Engineering. Common geometric transformations, such as rotations and translations of Euclidean points, are also applicable to motors. However, encoding vertex positions and orientations as motors adds the capability of computing motor interpolation on surfaces. Thanks to that, general curves and surfaces can be generated by a motor interpolation process using different basis functions and parameterizations. In applications, the generated surfaces can be visually manipulated and deformed in a predictable way by changing the motors. In this paper we look inside the theory behind those applications as well as practical details on how Geometric Algebra algorithms can be computed efficiently. We show that geometric deformations can be computed at interactive rates on surface models with millions of vertices using the GPU. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0777-z Issue No:Vol. 27, No. 3 (2017)

Authors:Lars Tingelstad; Olav Egeland Pages: 2035 - 2049 Abstract: Abstract In this paper we present a novel method for nonlinear rigid body motion estimation from noisy data using heterogeneous sets of objects of the conformal model in geometric algebra. The rigid body motions are represented by motors. We employ state-of-the-art nonlinear optimization tools and compute gradients and Jacobian matrices using forward-mode automatic differentiation based on dual numbers. The use of automatic differentiation enables us to employ a wide range of cost functions in the estimation process. This includes cost functions for motor estimation using points, lines and planes. Moreover, we explain how these cost functions make it possible to use other geometric objects in the conformal model in the motor estimation process, e.g., spheres, circles and tangent vectors. Experimental results show that we are able to successfully estimate rigid body motions from synthetic datasets of heterogeneous sets of conformal objects including a combination of points, lines and planes. PubDate: 2017-09-01 DOI: 10.1007/s00006-016-0692-8 Issue No:Vol. 27, No. 3 (2017)

Authors:Huijing Yao; Qiaohong Chen; Xinxue Chai; Qinchuan Li Pages: 2097 - 2113 Abstract: Abstract This paper presents a new method based on geometric algebra for the singularity analysis of 3-degrees of freedom overconstrained 3-RPR planar parallel manipulators. Constraint wrenches acting on the moving platform are obtained using the outer product and dual operations. After the redundant constraint wrenches are identified and removed, a singular polynomial is derived as the coefficient of the outer product of all the non-redundant constraint wrenches. This polynomial provides an overall perspective of the singularity of the 3-RPR parallel manipulators and enables the drawing of 3-dimensional singularity loci, which are important in trajectory planning and workspace design. The main advantage of using geometric algebra is the compact and geometrically intuitive formulation of the singularity polynomial of the 3-RPR parallel manipulators. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0794-y Issue No:Vol. 27, No. 3 (2017)

Authors:Stéphane Breuils; Vincent Nozick; Laurent Fuchs Pages: 2133 - 2151 Abstract: Abstract This paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension (e.g. algebra of dimension 15). The method is compared with the main state of the art geometric algebra implementations, with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0770-6 Issue No:Vol. 27, No. 3 (2017)

Authors:Sabrine Arfaoui; Anouar Ben Mabrouk Pages: 2287 - 2306 Abstract: Abstract In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well known ones of Jacobi and Gegenbauer polynomials when relaxing one of the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rules have been proved. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0788-9 Issue No:Vol. 27, No. 3 (2017)

Authors:Abhijit Banerjee Pages: 2321 - 2332 Abstract: Abstract We investigate the complete analytical solutions of quantum mechanical harmonic and isotonic oscillators formulated in the commutative ring of bicomplex numbers. We obtain the explicit closed form expressions for the excited eigenstates and corresponding energy eigenvalues associated with the harmonic and isotonic oscillator potentials by extending the formalism adopted in Banerjee (Ann Phys 377:493, 2017) recently to find the analytical closed form solutions for ground states. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0772-4 Issue No:Vol. 27, No. 3 (2017)

Authors:Carlos Castro Pages: 2393 - 2405 Abstract: Abstract After a very brief introduction to generalized gravity in Clifford spaces (C-spaces), generalized metric solutions to the C-space gravitational field equations are found, and inspired from the (Anti) de Sitter metric solutions to Einstein’s field equations with a cosmological constant in ordinary spacetimes. C-space analogs of static spherically symmetric metrics solutions are constructed. Concluding remarks are devoted to a thorough discussion about Areal metrics, Kawaguchi–Finsler Geometry, Strings, and plausible novel physical implications of C-space Relativity theory. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0763-5 Issue No:Vol. 27, No. 3 (2017)

Authors:Jamel El Kamel; Rim Jday Pages: 2429 - 2443 Abstract: Abstract In this paper, we establish analogues of Hardy’s and Miyachi’s theorems for the Clifford–Fourier transform. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0791-1 Issue No:Vol. 27, No. 3 (2017)

Authors:Sergio Giardino Pages: 2445 - 2456 Abstract: Abstract A quaternionic analog of the Aharonov–Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics. A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0766-2 Issue No:Vol. 27, No. 3 (2017)

Authors:Önder Gökmen Yıldız; Murat Tosun Pages: 2873 - 2884 Abstract: Abstract In this paper, we study the evolution of non-null curve in n-dimensional Minkowski Space. We express evolution equation of the Frenet frame by matrix equation. We obtain integrability conditions for the evolutions. Finally, we give examples of evolutions. PubDate: 2017-09-01 DOI: 10.1007/s00006-017-0760-8 Issue No:Vol. 27, No. 3 (2017)

Authors:Mehmet Ali Güngör; Ayşe Zeynep Azak Abstract: Abstract In this study, we define the dual complex Fibonacci and Lucas numbers. We give the generating functions and Binet formulas for these numbers. Moreover, the well-known properties e.g. Cassini and Catalan identities have been obtained for these numbers. PubDate: 2017-09-15 DOI: 10.1007/s00006-017-0813-z

Authors:Ioannis Chrysikos Abstract: Abstract Consider a Riemannian spin manifold \((M^{n}, g)\) \((n\ge 3)\) endowed with a non-trivial 3-form \(T\in \Lambda ^{3}T^{*}M\) , such that \(\nabla ^{c}T=0\) , where \(\nabla ^{c}:=\nabla ^{g}+\frac{1}{2}T\) is the metric connection with skew-torsion T. In this note we introduce a generalized \(\frac{1}{2}\) -Ricci type formula for the spinorial action of the Ricci endomorphism \({{\mathrm{Ric}}}^{s}(X)\) , induced by the one-parameter family of metric connections \(\nabla ^{s}:=\nabla ^{g}+2sT\) . This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schrödinger–Lichnerowicz formula related to the square of the Dirac operator \(D^{s}\) , induced by \(\nabla ^{s}\) , under the condition \(\nabla ^{c}T=0\) . In the same case, we provide integrability conditions for \(\nabla ^{s}\) -parallel spinors, \(\nabla ^{c}\) -parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly Kähler manifolds and nearly parallel \(\hbox {G}_2\) -manifolds, in dimensions 5, 6 and 7, respectively. PubDate: 2017-09-12 DOI: 10.1007/s00006-017-0810-2

Authors:Jianquan Liao; Xingmin Li Abstract: Abstract A theory of k-analytic functions on octonions is established. The Cauchy integral formulas, Taylor series and Laurent series for the k-analytic functions are given. Moreover, we obtain the orthogonality relations for the basis of k-analytic functions. PubDate: 2017-09-08 DOI: 10.1007/s00006-017-0807-x

Authors:Pei Dang; Tao Qian; Qiuhui Chen Abstract: Abstract This paper is devoted to studying uncertainty principle of Heisenberg type for signals on the unit sphere in the Clifford algebra setting. In the Clifford algebra setting we propose two forms of uncertainty principle for spherical signals, of which both correspond to the strongest form of uncertainty principle for periodic signals. The lower-bounds of the proven uncertainty principles are in terms of a scalar-valued phase derivative. PubDate: 2017-09-06 DOI: 10.1007/s00006-017-0808-9

Authors:Juan Bory Reyes; Ricardo Abreu Blaya; Marco Antonio Pérez-de la Rosa; Baruch Schneider Abstract: Abstract The Cimmino system offers a natural and elegant generalization to four-dimensional case of the Cauchy–Riemann system of first order complex partial differential equations. Recently, it has been proved that many facts from the holomorphic function theory have their extensions onto the Cimmino system theory. In the present work a Poincaré–Bertrand formula related to the Cauchy–Cimmino singular integrals over piecewise Lyapunov surfaces in \(\mathbb {R}^4\) is derived with recourse to arguments involving quaternionic analysis. Furthermore, this paper obtains some analogues of the Hilbert formulas on the unit 3-sphere and on the 3-dimensional space for the theory of Cimmino system. PubDate: 2017-09-06 DOI: 10.1007/s00006-017-0809-8

Authors:Shi-Fang Yuan; Qing-Wen Wang; Yi-Bin Yu; Yong Tian Abstract: Abstract In this paper, we discuss Hermitian solutions of split quaternion matrix equation \(AXB+CXD=E,\) where X is an unknown split quaternion Hermitian matrix, and A, B, C, D, E are known split quaternion matrices with suitable size. The objective of this paper is to establish a necessary and sufficient condition for the existence of a solution and a solution formulas. Moreover, we provide numerical algorithms and numerical examples to exemplify the results. PubDate: 2017-09-02 DOI: 10.1007/s00006-017-0806-y

Authors:Ali Atasoy; Erhan Ata; Yusuf Yayli; Yasemin Kemer Abstract: Abstract We present a new different polar representation of split and dual split quaternions inspired by the Cayley–Dickson representation. In this new polar form representation, a split quaternion is represented by a pair of complex numbers, and a dual split quaternion is represented by a pair of dual complex numbers as in the Cayley–Dickson form. Here, in a split quaternion these two complex numbers are a complex modulus and a complex argument while in a dual split quaternion two dual complex numbers are a dual complex modulus and a dual complex argument. The modulus and argument are calculated from an arbitrary split quaternion in Cayley–Dickson form. Also, the dual modulus and dual argument are calculated from an arbitrary dual split quaternion in Cayley–Dickson form. By the help of polar representation for a dual split quaternion, we show that a Lorentzian screw operator can be written as product of two Lorentzian screw operators. One of these operators is in the two-dimensional space produced by 1 and i vectors. The other is in the three-dimensional space generated by 1, j and k vectors. Thus, an operator in a four-dimensional space is expressed by means of two operators in two and three-dimensional spaces. Here, vector 1 is in the intersection of these spaces. PubDate: 2017-07-03 DOI: 10.1007/s00006-017-0797-8