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- CUBIC SPLINES IN THE GRASSMANN MANIFOLD GENERATED BY THE DE CASTELJAU ALGORITHMPublication . Pina, FátimaWe present a detailed analysis of the De Casteljau algorithm to gen erate cubic polynomials satisfying certain boundary conditions in the Grassmann manifold, and extend this approach to produce cubic splines that also solve inter polation problems on that manifold.
- Interpolation on the Essential Manifold using RollingPublication . Pina, Fátima
- COMPLETE CONTROLLABILITY OF THE KINEMATIC EQUATIONS DESCRIBING PURE ROLLING OF GRASSMANNIANSPublication . Pina, FátimaThis paper studies the controllability properties of certain nonholo nomic control systems, describing the rolling motion of Grassmann manifolds over the affine tangent space at a point. The control functions correspond to the freedom of choosing the rolling curve. The nonholonomic constraints are imposed by the non-slip and non-twist conditions on the rolling. These systems are proved to be controllable in some submanifold of the group of isometries of the space where the two rolling manifolds are embedded. The constructive proof of controllability is also partially addressed.
- Livro de Atas do IX Encontro Científico da UI&D (ecUI&D´23)Publication . Cadete Pires, Cristina Maria Paulo; Pina, Fátima; Reis, Patrícia; Vieira, Jorge; Pinto dos Reis, Isabel; MONTEIRO, DINIS
- The Role of the Essential Manifold in Data Mining – An Introductory ApproachPublication . Pina, FátimaInterpolating data and the application of data mining tech niques in nonlinear manifolds plays a significant role in different areas of knowledge, ranging from computer vision and robotics, to industrial and medical requests, and these growing number of applications have sparked the research interest of the scientific community to these topics. The Gen eralized Essential manifold, briefly, Essential manifold, consisting of the product of the Grassmann manifold of all k-dimensional subspaces of Rn and the Lie group of rotations in Rn, for instance, plays an impor tant role in the problem of recovering the structure and motion from a sequence of images, also known as stereo matching, which is a crucial problem in image processing and computer vision. A well-known recur sive procedure to generate interpolating polynomial curves in Euclidean spaces is the classical De Casteljau algorithm, which is a simple and pow erful tool widely used in the field of Computer Aided Geometric Design, particularly because it is essentially geometrically based. This algorithm has been generalized to geodesically complete Riemannian manifolds. Thus, having this in mind, in this work we present all the ingredients for a detailed implementation of the generalized De Casteljau algorithm to generate geometric cubic polynomials in the Essential manifold preparing the ground to solve different real interpolation problems in this manifold
- COMPLETE CONTROLLABILITY OF THE KINEMATIC EQUATIONS DESCRIBING PURE ROLLING OF GRASSMANNIANSPublication . Pina, FátimaThis paper studies the controllability properties of certain nonholo nomic control systems, describing the rolling motion of Grassmann manifolds over the affine tangent space at a point. The control functions correspond to the free dom of choosing the rolling curve. The nonholonomic constraints are imposed by the non-slip and non-twist conditions on the rolling. These systems are proved to be controllable in some submanifold of the group of isometries of the space where the two rolling manifolds are embedded. The constructive proof of controllability is also partially addressed.
- Rolling Maps for the Essential ManifoldPublication . Pina, Fátima
- Controllability of the kinematic equations describing pure rolling of GrassmanniansPublication . Pina, FátimaThis paper studies the controllability properties of certain nonholonomic control systems, describing the rolling motion of Grassmann manifolds over the affine tangent space at a point. The control functions correspond to the freedom of choosing the rolling curve. The nonholonomic constraints are imposed by the no-slip and no-twist conditions on the rolling. These systems are proved to be controllable in some submanifold of the group of isometries of the space where the two rolling manifolds are embedded. The constructive proof of controllability is also addressed
- Cubic Splines in the Grassmann ManifoldPublication . Pina, FátimaWe present a detailed implementation of the De Casteljau algorithm to generate cubic splines that solve certain interpolation problems in the Grassmann manifold.