V. Znak (Vladimir Ilich Znak) is Senior scientific researcher at The Institute of Computational Mathematics and Mathematical Geophysics, Russia. He received his Ph.D. in 1980 at The Sevastopol instrument-making institute, Sevastopol. His Research interests were in the field of signal processing, estimation of signal parameters and characteristics, computational mathematics, applied statistics, computational technologies, development of algorithms and computer programs, formal logic, etc. he is Author of more than 70 publications, including inventor\'s certificates.

Periodic (harmonic and frequency-modulated – FM) signals are widely used, and an appropriate research can be of interest in different fields of activity. Our purpose is to study the above signals for estimating their parameters and characteristics such as locus of a signal on a time axis and the degree of their presence in noise. Such indications determine the degree of trust to consequent estimations. At the same time, we assume that a signal is recorded at a discrete time t1,…, tN, where ti-1 – ti =
t = const, i=2,…, N.\r\nWe propose to use the cluster analysis for studying periodic signals. According to this, we consider the corresponding approach, the statement of a problem and specify the way of its decision. Finally, we intend to present some results of the data obtained of a model of the FM signal.

Lionel Garnier obtained a PhD thesis on computer science from the University of Burgundy, France in 2004 on the use of Dupin cyclides and supercyclides in geometric modelling. His research interests include shape modelling using algebraic and parametric surfaces, surfaces blending, and geometric constraints based modelling. To simplify the blends of canal surfaces using Dupin cyclides, the space of spheres in the Minkowski-Lorentz space is used. In this space, any canal surface is represented by a curve and a Dupin cyclide is modelled by two conics. Using these spaces, the blend of these surfaces is very easy: it is enough to join two curves. Moreover, the conics can be represented by Bézier curves and mass points.

This poster deals with the Minkowski-Lorentz space and applications in the Computer AidedrnGeometric Design. The Minkowski-Lorentz space generalises to R5 the one used in the relativityrntheory. A bilinear and its quadratic forms states the Minkowski-Lorentz space for R5. For CADrnpurposes, the working set in that space is a unit sphere denoted by λ4. The Minkowski-Lorentz space embeds R3. The spheres and planes in R3 are considered as pointsrnon λ4 in the Minkowski-Lorentz space. Moreover, some quadratic calculation in R3 become linearrnin R5. The pencils of spheres in R3 are modelled by the intersection of a plane with λ4. The result isrna unit circle given three situations. Depending on the space-like plane, light-like plane, time-likernplane, the unit intersection circle looks like an ellipse, two straight lines or a hyperbola. The termsrnlight cone, like-like, space-like, time-like vectors were found by Minkowski. The envelop of a one-parameter family of spheres in R3 defines a canal surface which is a curve onrnλ4. That makes the use of algorithms based on canal surfaces easier than in R3. The Dupin cyclidesrnare well known and particular cases of canal surfaces. On λ4, a Dupin cyclide is represented by tworncircles which look like ellipse(s) or hyperbola or by a circle, which look like an ellipse, and arnparabola isometric to a line. They provide new algorithm in CAD purposes. An example of G1rnjoined Dupin cycles is shown in a schematic seahorse.