Chelyabinsk, Chelyabinsk, Russian Federation
In the plane П given curves in two-point contact (first order smoothness). It is shown that for an arbitrary affine mapping plane П to plane П′ attitude q of the radii of curvature of the osculating curves in the touch point not changed. In accordance with theorem Tissot, the nonlinear mapping can be replaced affine in a small neighborhood of a nonsingular point. It is shown that when considering the convergence of the curves with the smoothness of the second and higher orders such substitution may lead to erroneous conclusions. The article considers the special case, when the contact involves not arbitrary curves, and conic section. It is proved that the ratio of the radii of curvature of the osculating conic sections in the touch point does not change under arbitrary projective transformation. To prove the invariance of q are projective correspondence between the beams of the conic sections, beams direct and point rows of the first and second orders. Reviewed five of the beam properties of conic sections, of which there were three lemmas: proactively beam Konik and point number, proactively two beams Konik, a promising line two dot rows of lines that intersect a projective bundles Konik. The Lemma allows us to prove a theorem on the constancy of the ratio of the radii of curvature at the touching point of conic sections in an arbitrary projective transformation. Examples of practical application of the theorems in descriptive geometry problems. It is shown that to determine the relationship of the radii of curvature of two conic sections at their point of contact need not find the circle of curvature data conics, and complex enough to calculate the ratio of four points on an arbitrary secant line passing through the point of tangency of curves.
the projective transformation, the beam conic sections, the circle of curvature, involution on the conic section, theorem Tissot.
Введение. На плоскости Π начерчены соприкасающиеся кривые a, b (рис. 1). В точке касания T обе кривые дважды дифференцируемы, радиусы их кривизны Ra и Rb отличны от нуля и не равны между собой. Кривые a, b находятся в двухточечном соприкосновении (первого порядка гладкости), поскольку имеют две общие бесконечно близкие точки A и B, в пределе совпадающие с точкой T, через которые проходит их общая касательная t [2; 3].
Пусть дано произвольное (нелинейное, непрерывное) взаимно однозначное отображение y = ϕ(x) точек xi плоскости П в точки yi плоскости П′. Образы a′, b′ кривых a, b в этом отображении сохраняют двухточечное соприкосновение. Радиусы кривизны Ra, Rb образов в точке их соприкосновения отличаются от радиусов кривизны Ra, Rb прообразов a, b. Как при таком отображении изменяется отношение q = Ra/Rb? Можно предположить, что q = q′, где q′= Ra, /Rb. Например, если кривые a, b имеют в точке соприкосновения равную кривизну (q = 1), то они находятся в трехточечном соприкосновении (второго порядка гладкости) [4].
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