Jan Vršek: On rational surfaces with pencils of curves providing Pythagorean surface normals

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Jan Vršek, University of West Bohemia, Plzen, Czech Republic: On rational surfaces with pencils of curves providing Pythagorean surface normals

 

The main aim of our work is to initiate an investigation of rational offsets to surfaces from a new point of view and return this research back to one of the practical applications, namely closer to numerical-control machining. So we will not assume that the rational-offset- (or Pythagorean-normal-) property must be satisfied globally but it is satisfactory if it is fulfilled only along some special curves (trajectories of the sphere milling cutter). These curves are then called curves providing Pythagorean surface normals (shortly PSN curves). The properties of PSN curves can be be studied with help of normal double planes associated to the given surface and equivalent to offsets of algebraic surfaces. It will be shown that PSN curves can occur also on non-PN surfaces, and conversely on PN surfaces we can find curves which are non-PSN. An analysis of simple, special and degenerated components of offsets along PSN curves will be presented and discussed. Especially, we will focus on rational surfaces possessing pencils of PSN curves. As special cases the ruled surfaces and ringed surfaces (both very imported classes of surfaces applicable in technical practise) will be thoroughly studied from this point of view.