Projects / Programmes
PH curves and their generalizations
| Code |
Science |
Field |
Subfield |
| 7.00.00 |
Interdisciplinary research |
|
|
| Code |
Science |
Field |
| P170 |
Natural sciences and mathematics |
Computer science, numerical analysis, systems, control |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
PH curve, PN surface, interpolation, polynomial, computer numerical control machine, robotics, arc length, offset, frame
Researchers (1)
| no. |
Code |
Name and surname |
Research area |
Role |
Period |
No. of publicationsNo. of publications |
| 1. |
27559 |
PhD Vito Vitrih |
Mathematics |
Head |
2011 - 2013 |
111 |
Organisations (1)
Abstract
Polynomial Pythagorean-hodograph curves (PH curves) have been introduced in 1990 by Farouki and Sakkalis. They are characterized by the property that their parametric speed, i.e., the derivative of the arc length with respect to the curve parameter, is a polynomial function. Polynomial PH curves form an important class of parametric polynomial curves for which the arc length can be computed exactly and their offsets (parallel curves) are rational curves. This makes them very useful in many practical applications, e.g. in CAD/CAM systems, robotics, animation, NC machining, etc. Although they have less degrees of freedom than general parametric curves, they still admit nice shape properties, which makes them interesting in shape preserving techniques. They are also very appropriate objects for real-time motion control and spatial path planning based on rotation-minimizing frames. The aim of the proposed project is to obtain some new results on PH curves and their generalizations. Initially, research will be focused on some open problems concerning (spatial) polynomial PH curves, in particular some important interpolation schemes, that have not been analysed yet, will be considered. Then particular rational adapted frames of spatial polynomial PH curves will be considered. They play an important role in motion design applications, which is an important field in robotics. More precisely, a scheme for interpolation of G1 Hermite motion data with ERF and RRMF frames of spatial PH curves (of degree 5 and 7) will be analysed. Furthermore, some generalizations of polynomial PH curves will be considered. First, some interpolation schemes with planar rational PH curves will be studied, and secondly interpolation by planar cycloidal PH curves (PHC curves) will be considered. Finally, some PN surfaces, as generalization of PH curves to surfaces, will be discussed. These results can all be applied to robotics, CNC machines, manufacturing technologies and systems, CAD/CAM systems and many other modern computer supported applications. From the scientific point of view, the obtained results will be published in high quality SCI journals and will certainly be internationally respected. They will also contribute to international value of Slovenia.
Significance for science
Pythagorean-hodograph curves and their generalizations form a research field that has received numerous interest of many researchers and engineers around the world in the last 25 years. This is evident from the extensive global bibliography and citations. The results of the project are among the important further steps in developing this research area, particularly study of rational spatial PH curves raises many open questions for further research for all active researchers from this field. In the project results a number of methods (such as solving problems by using the Gram matrix approach, homotopy method ...) have been used, which have previously not been applied to solve similar problems. These methods also give new possibilities how to solve some of the remaining open questions. All project results are published in the first half of the SCI journals in the field, which indicates their scientific validity.
Significance for the country
Curves with Pythagorean hodograph and their generalizations play a very important role in many fields, such as robotics, CNC applications, manufacturing technologies and systems, CAD / CAM systems and other modern computer aided applications. All of these areas have an appreciable impact on quality of life and standard in today's society. With the publication of original scientific papers in reputable journals on the SCI list, with contributions in many international scientific conferences ... Slovenia has gained international recognition in the CAGD community. This also contributes to the promotion of the Slovenian science and Slovenia in general. One invited long-term research visit, carried out in 2012, and some invited lectures given at foreign universities confirm this fact.
Most important scientific results
Annual report
2011,
2012,
final report,
complete report on dLib.si
Most important socioeconomically and culturally relevant results
Annual report
2011,
2012,
final report,
complete report on dLib.si
