Habarovsk, Khabarovsk, Russian Federation
Habarovsk, Khabarovsk, Russian Federation
Habarovsk, Khabarovsk, Russian Federation
When studying the theory of contour construction in “Affine and Projective Geometry” course on educational program specializations “Computer-Aided Design Systems” and “Applied Informatics in Design” a unit of computational and graphic task "Contour Construction" is carrying out in structural design. In this computational and graphic task the contour constructions are carrying out by second-order curves (a circle — by the radius and graphical method; a hyperbola, an ellipse, a parabola — by means of Pascal curves, taking into account positions of engineering discriminant). The constructions of an arc of ellipse, hyperbola, and parabola are carried out based on Pascal theorem: in any hexagon, which vertices belong to a second-order series, three points of the opposite sides’ intersection lie on one straight line — the Pascal line. However, in construction of a conic (a second-order curve), it is necessary to draw students’ attention to the fact that the points belonging to a second-order series (a second-order curve, or a conic) make a geometrical locus of intersection of Pascal hexagon’s adjacent opposite sides. By this method students successfully construct conjugate arcs of an ellipse and a hyperbola with other conics. The construction of a parabola arc, conjugated with other conics, is carried out by the method of engineering discriminant (it is more convenient to divide line segments in halves: a median and a triangle side, which is opposite to its vertex lying on a parabola arc). It should be noted that theoretical and practical material on this subject corresponds to the assimilation of Study Plan’s necessary competences (in accordance with each educational program), however, some aspects of this subject are accepted by students simply by trust. The aim of this paper is research of construction methods for parabola, applied to contour simulation.
parabola, Pascal line, engineering discriminant, Hermite curve’s segment, Bezout, B-spline.
Плоская алгебраическая кривая n-го порядка имеет параметрическое число, равное
Для коники в соответствии с выражением (1) параметрическое число равно 5. Это объясняется и общим положением с позиции аналитической геометрии: исходя из уравнения кривой 2-го порядка в декартовых координатах
Ax2 + By2 + Cxy + Dx + Ey + F = 0 (2) имеем 6 коэффициентов (A, B, C, D, E, F). Если поделить каждый член уравнения (2) на коэффициент F, то уравнение (2) примет вид
A x2 + B y2 + C xy + D x + E y +1 = 0, в котором 5 параметров. Поэтому заключаем, что для ее задания необходимо 5 параметров — количество коэффициентов: A′, B′, C′, D′, E′, т.е. это ∞5 множество точек, или любые другие условия, сохраняющие именно 5 параметров: 5 точек; три точки и две касательные и т. д. [6; 12; 19]. Однако в случае, если коника проходит через начало координат, то в выражении (2) коэффициент F = 0 [3; 9; 14; 23]. Тогда уравнение такой кривой следует записать как
Ax2 + By2 + Cxy + Dx + Ey = 0, (3) т.е. 5 коэффициентов и, таким образом, 5 параметров. Коэффициенты A, B, C, D, E выражения (3) можно определить, подставив координаты точек в уравнение кривой. Получаем 5 уравнений первой степени. Решая систему пяти уравнений, узнаем искомые коэффициенты [5; 11; 14; 15; 19]. Приведем несколько примеров, применяемых как при конструктивном моделировании параболы (только при помощи прямых Паскаля и свойства инженерного дискриминанта, не затрагивая другие известные способы построения), так и с использованием средств компьютерной визуализации. Пример 1. Принимая «на веру», что при определении параболы и ее касательной t в точке A отрезки OAy и OAy равны, т.е. OA OA y y = (рис. 1), следует доказать это положение известной теоремы. Попутно приведем цитату: «Заметим синтез и анализ, не в математическом, а в общелогическом смысле слова совершенно равноправны, и во всяком исследовании они постоянно переплетаются друг с другом; поэтому едва ли может быть речь о предоставлении господства одному из этих орудий чело-
веческой мысли» [21]. В связи с этим проводимые исследования следует рассматривать и с аналитических позиций [18; 24; 25], которые обеспечивают моделирование с применением информационных технологий [10].
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