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Stożkowe jako zbiory środków sfer przechodzących przez dwa punkty i stycznych do prostej, płaszczyzny bądź sfery

Ochoński S.

Materiały Seminarium Geometrii i Grafiki Inżynierskiej

2000

z. 9

39--49

The paper presents results of studies on sets of centers of spheres containing two different points or common point, and at the same time tangent to the straight line, plane or sphere. The studies proved that these sets are conics and in the case of spheres passing through point and tangent to the plane or sphere – surfaces of revolution in the form of sphere, ellipsoid, paraboloid and double-sheet hyperboloid. Well known definition of parabola, as a set of equally – distant points from fixed point and straight line, has been extended to the remaining nondegenerated conics via exchange straight line into circle. In this work was given also original general definition of nondegenerated conics as a set of equally – distant points from two fixed and coplanar reciprocally passing circles. Only one of these sets can be the straight line. It has been proved that two of these circles define anti-inversion and it could be realised on plane with orthogonal cones with common vertex and height. The results of these studies were also two algorithms of universal kinematic conic constructions which let us determine tangent line with its tangent point. Solutions of two exercises are given as ilustrating examples there.