Conics As Sections of Quadrics by Plane (Generalized Dandelin Theorem)

Aleksandr L. Kheyfets

doi:doi:10.12737/article_5953f32172a8d8.94863595

UDC 51

Conics As Sections of Quadrics by Plane (Generalized Dandelin Theorem)

Published в Geometry & Graphics · Volume 5, Issue 2, 2017 · Pages 45–58 · Rubrics: Scientific problems of geometry

DOI 10.12737/article_5953f32172a8d8.94863595

Received: 28.06.2017 Accepted: 04.07.2017 Published: 04.07.2017 Language of publication: RUS

Authors

Aleksandr L. Kheyfets ¹

1 Yuzhno-Ural'skiy gosudarstvennyy universitet
Russian Federation

ABSTRACT

Has been presented a geometrical proof of a theorem stating that when a plane section crosses second-order revolution surfaces (rotation quadrics, RQ), such types of conics as ellipse, hyperbola or parabola are formed. The theorem amplifies historically famous Dandelin theorem, which provides geometric proof only for the circular cone, and extends the proof to all RQ: ellipsoid, hyperboloid, paraboloid and cylinder. That is why the theorem described below has been called as Generalized Dandelin theorem (GDT). The GDT proof has been constructed on a little-known generalized definition (GDD) of the conic. This GDD defines the conic as a line, that is a geometrical locus of points (GLP) P, for which ratio q = PT / PD = const, where PT is tangential distance from the point to the circle inscribed in the line, and PD is distance from the point to the straight line passing through the tangency points of the circle and the line. Has been presented a proof of GDD for all types of conics as their necessary and sufficient condition. The proof is in the construction of a circular cone and inscribed in sphere which is tangent to a cutting plane line at two points. For this construction is defined the position of a cutting plane, giving in section the specified conic. On the GDD basis has been proved the GDT for all the RQ with the arbitrary position of the cutting plane. For the proving a tangent sphere is placed in the quadric. An auxiliary cutting plane passing through the quadric axis is introduced. It is proved that in a section by axial plane the GDD is performed as a necessary condition for the conic. The relationship between the axial section and the given one is established. This permits to make a conclusion that in the given section the GDD is performed as the conic’s sufficient condition. Visual stereometrical constructions that are necessary for the theorem proof have been presented. The implementation of constructions using 3D computer methods has been considered. The examples of constructions in AutoCAD package have been demonstrated. Some constructions have been carried out with implementation of 2D parameterization. With regard to affine transformations the possibility for application of Generalized Dandelin theorem to all elliptic quadrics has been demonstrated. This paper is meant for including the GDT in a new training course on theoretical basis for 3D engineering computer graphics as a part of students’ geometrical-graphic training.

KEYWORDS

Dandelin theorem Dandelin structure Dandelin spheres conic sections quadrics second-order surfaces degree of point parameterization AutoCAD.

Information Text References

Authors:

1. Yuzhno-Ural'skiy gosudarstvennyy universitet

Russian Federation

Type:

Article

Pages:

from 45 to 58

Status:

Published

Subject area:

UDC 51

Language:

Russian, English

Keywords:

Dandelin theorem, Dandelin structure, Dandelin spheres, conic sections, quadrics, second-order surfaces, degree of point, parameterization, AutoCAD.

Text

Введение

Конические сечения, их построение и исследование являются разделом учебного курса начертательной геометрии (НГ) [6; 9; 13; 23]. Основу этого раздела составляет теорема Данделена, доказывающая возникновение кривых второго порядка — коник:
эллипса, гиперболы, параболы — при сечении прямого кругового конуса плоскостью.

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