DOI: 10.14489/vkit.2014.11.pp.020-024

Короткий В. А.
СИНТЕТИЧЕСКИЕ АЛГОРИТМЫ ПОСТРОЕНИЯ КРИВОЙ ВТОРОГО ПОРЯДКА
(с. 20-24)

Аннотация. Рассмотрены алгоритмы геометрически точного построения главных осей кривой второго порядка, проходящей через за-данные n точек и касающейся m прямых (n + m = 5) для всех возможных сочетаний заданных инцидентностей, основанные на известных проективных свойствах конических сечений. Составлена Лисп-программа, реализующая построение главных осей и вычерчивание кривой второго порядка для трех специализаций – по пяти точкам, пяти касательным, трем точкам и двум каса-тельным.

Ключевые слова: коническое сечение; инволюция сопряженных диаметров; инволюция на окружности; главные оси кривой второго порядка; пучки конических сечений.

Korotky V. A.
SYNTHETIC ALGORITHMS OF CONSTRUCTION OF THE CURVE OF THE SECOND ORDER
(pp. 20-24)

Abstract. While solving some geometric problems you want to build a curve of the second order, satisfying the specified conditions pits. As conditions usually indicate points must pass through the desired curve, and tangent to it. If the plane are n points, and m direct (n + m = 5), there is a conic curve passing through these points and concerning this direct. To build it is proposed to use known by the projective properties of conic sections. Compiled projective construction algorithms main axes and asymptotes curve of the second order, given a set of five conditions. There are six different combinations of points n and m tangent, identified abused curve of the second order: five points, five tangent, four points and tangent, four tangent and point, three point and two tangent, three tangent and two points. In these combinations possible existence of infinitely distant points and different variants of incidence points and tangent. Considering all possible variants, we obtain 53 cases, for each of which an algorithm of construction of the curve of the second order. Algorithm of construction of the main axes of the curve of the second order, passing on five arbitrarily given points, consists of three actions: construction of involution conjugate diameters, construction of the main dia meter and asymptotes, building tops (main axes) integral curve. Drawing a continuous curve of the second order is means-you three-dimensional computer modeling. Compiled LISP-program (AutoCad) the volume of 40 KB performing geometrically exact structure of the main axes of the conic curve passing through these points and on direct. The Program is used only three graphics operations – construction of the direct line through two points, to construct a circle and building of points of intersection of the straight line with a circle. Examples are given of application compiled program for designing and manufacturing of smooth composite contours and build beams of the second order with real or imaginary, of basic points.

Keywords: Conic section; Involution conjugate diameters; Involution on a circle; Main axis curve of the second order; Bundles of conic sections.