Description
Two objects are said to be similar if they have the same shape regardless of the position and/or the scale. Identifying similar objects is essential in the field of Pattern Recognition. In the case of this project the object to be compared is a space rational curve. From the point of view of a potential user of the algorithm, one can assume that there exists a database of “classical” space curves, so that the user wants to check if the curve he is studying corresponds to an already known/classified space curve. Therefore, the goal of the project is to devise an algorithm for checking whether or not two space rational curves, defined by means of rational parametrizations, are similar. The main strategy is to appropriately extend previous results of the project promoter in collaboration with donor partner, on close problems, namely detecting symmetry in planar and space rational curves and detecting similarity between planar rational curves. Based on previous papers on symmetry computation by the researchers involved in the project, they conjecture that, under certain hypotheses of the parametrizations of the space curves to be compared, these space curves are similar if and only if the parametrizations are related by a Möebius transformation respecting “arc length” in a certain way. The algorithm could be of interest in the Pattern Recognition field and also for computer algebra system users. Donor partner’s Applied Mathematics division has contributed to many problems related to parametric and implicit algebraic varieties, specially planar and space curves and surfaces. The study of rational curves is in the scope of SINTEF, because of its applicability in down-to-earth problems like graphic design, solid modeling and Computer Aided Geometric Design. The current project will help to maintain the contact and the collaboration between the researchers, and therefore between their corresponding institutions.
Summary of project results
The goal of the activity was to extend the results of previous collaboration among Juan Gerardo Alcázar and Georg Muntingh, currently post-doc researcher at SINTF, concerning the problem of checking whether or not two given rational space curves are similar, The same problem for planar curves was successfully solved, and published, jointly with Georg Muntingh in 2014. Before the stay there was a fruitful communication with Georg Muntingh, in order to isolate potential difficulties when trying to generalize to space the techniques used in the planar case. As a result, partners observed that while the general strategy was certainly applicable to the space case, there was a special case, corresponding to helical curves, i.e. curves where curvature and torsion are proportional, that needed a different approach. During the stay they could find a computational method to solve this special case. Furthermore, they also explored another problem somehow related to similarity detection, namely the refinability of Wachspress coordinates. These coordinates, which are bivariate rational functions, are generalizations of barycentric coordinates, defied over triangles, to convex polygons with more than three edges. Similarity comes up in this context when trying to characterize when these functions are refinable. They could complete the analysis of the problem for quadrilaterals, although at the end of the stay they were still uncertain about the relevance of the results. During the stay partners maintained daily working meetings. Related to the results there are two papers in progress. The collaboration among partners will be maintained through the exchange of information via email etc.; visits of Dr. Muntingh in the frame of different grants of the University of Alcalá; and prospect visits of Dr. Alcázar to Oslo, funded either by University of Alcalá either by a Spanish research project.
Summary of bilateral results
The goal of the activity was to extend the results of previous collaboration among Juan Gerardo Alcázar and Georg Muntingh, currently post-doc researcher at SINTF, concerning the problem of checking whether or not two given rational space curves are similar, The same problem for planar curves was successfully solved, and published, jointly with Georg Muntingh in 2014. Before the stay there was a fruitful communication with Georg Muntingh, in order to isolate potential difficulties when trying to generalize to space the techniques used in the planar case. As a result, partners observed that while the general strategy was certainly applicable to the space case, there was a special case, corresponding to helical curves, i.e. curves where curvature and torsion are proportional, that needed a different approach. During the stay they could find a computational method to solve this special case. Furthermore, they also explored another problem somehow related to similarity detection, namely the refinability of Wachspress coordinates. These coordinates, which are bivariate rational functions, are generalizations of barycentric coordinates, defied over triangles, to convex polygons with more than three edges. Similarity comes up in this context when trying to characterize when these functions are refinable. They could complete the analysis of the problem for quadrilaterals, although at the end of the stay they were still uncertain about the relevance of the results. During the stay partners maintained daily working meetings. Related to the results there are two papers in progress. The collaboration among partners will be maintained through the exchange of information via email etc.; visits of Dr. Muntingh in the frame of different grants of the University of Alcalá; and prospect visits of Dr. Alcázar to Oslo, funded either by University of Alcalá either by a Spanish research project.