The objective of this talk was to present a result on conic sections that has been recently obtained by Prof. G. Heimbeck. The target audience included students, mathematics teachers, subject specialists at the Ministry of Education, physics teachers, research physicists, portfolio managers, mathematics lecturers and research mathematicians. Conic sections (circle, ellipse, parabola and hyperbola) find applications in physics as the trajectories of the motions of planetary bodies, in optics and portfolio theory. In the talk, Prof. G. Heimbeck provided an explanation as to when a conic section has interior points. From a pure mathematics point of view, the talk presented a contribution to the projective theory of conic sections. The talk was kept as understandable as possible and the presenter refreshed the audience’s memory on the necessary concepts.
1. Contents
1. History: How did the theory of conic sections develop?
2. Basic concepts from projective geometry.
3. Interior points of conic sections.
2. From the internet
Menaechmus introduced conic sections in 375 BC in order to study
the three problems ‘doubling a cube’, ‘trisecting an angle’ and
‘squaring a circle’.
3. Contributions of the Greeks
1. Pappus (400 BC) did something about conic sections one can
still find in school books.
2. Apollonius wrote eight books on conic sections. He
introduced the names parabola, hyperbola and ellipse and he
gave a description of conic sections which was then used as a
definition of conic sections.
7. Analytic Geometry
In the 17th century, Descartes invented analytic geometry. Then
conic sections were investigated using the methods of analytic
geometry and then results we all know from high school were
derived.
8. Projective geometry
The 17th century also marks the beginning of the modern theory
of conic sections. Desargues introduced projective geometry and
since then, conic sections have been investigated using the
methods of projective geometry.
9. Steiner
Around 1850, Steiner gave a purely geometric definition of conic
sections which is known under the keyword ‘Steiner’s generation of
conic sections’.
10. Application
Kepler’s Laws.
I. Each planet moves a round the sun in an ellipse, with the sun
at one focus.
II. The radius vector from the sun to the planet sweeps out equal
areas in equal intervals of time.
III. The square of the period of a planet is proportional to the
cube of the semimajor axis of its orbit.
(From the Feynman Lectures)
11. Projective plane – affine plane
If one removes from a projective plane a line and all points which
lie on this line one obtains an affine plane.
12. Affine plane – projective plane
If one adds to an affine plane a line whose points are the parallel
classes one obtains a projective plane.
13. Projective plane of a field F
The points of this plane are the subspaces of dimension 1 of F 3 .
The lines are the subspaces of dimension 2 of F 3 . The incidence
relation is inclusion. C := {x ∈ F 3 | x1 x2 − x3 = 0} is a conic
2
section on this plane.
14. Passants, tangents and secants
Let F be a field and C a conic section on F 3 . Every line of F 3
contains at most two points of C . A line which contains no point
of C is called a passant. A line which contains exactly one point of
C is called a tangent. A line which contains two points of C is
called a secant.
15. Exterior and interior points
A point which is not a point of the conic section but lies on a
tangent is called an exterior point. A point such that each line
which passes through this point is a secant is called an interior
point.
16. Pythagorean fields
A pythagorean field is a field F such that
1. the sum of two squares is a square;
2. −1 is not a square.
17. Existence of interior points
Let F be a field and C a conic section on F 3 . Then there exist
interior points if and only if F is pythagorean.