Contents    1. History: How did the theory of conic sections develop?    2. Basic concepts from projective geometry.    3....
From the internet   Menaechmus introduced conic sections in 375 BC in order to study   the three problems ‘doubling a cube...
Contributions of the Greeks    1. Pappus (400 BC) did something about conic sections one can       still find in school boo...
Ellipse as a conic section
Hyperbola as a conic section
Parabola as a conic section
Analytic Geometry   In the 17th century, Descartes invented analytic geometry. Then   conic sections were investigated usi...
Projective geometry   The 17th century also marks the beginning of the modern theory   of conic sections. Desargues introd...
Steiner   Around 1850, Steiner gave a purely geometric definition of conic   sections which is known under the keyword ‘Ste...
Application   Kepler’s Laws.     I. Each planet moves a round the sun in an ellipse, with the sun        at one focus.    ...
Projective plane – affine plane   If one removes from a projective plane a line and all points which   lie on this line one ...
Affine plane – projective plane   If one adds to an affine plane a line whose points are the parallel   classes one obtains a ...
Projective plane of a field F   The points of this plane are the subspaces of dimension 1 of F 3 .   The lines are the subs...
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 p...
Exterior and interior points   A point which is not a point of the conic section but lies on a   tangent is called an exte...
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 s...
Existence of interior points   Let F be a field and C a conic section on F 3 . Then there exist   interior points if and on...
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Interior Points of Conic Sections

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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.

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Interior Points of Conic Sections

  1. 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. 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. 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.
  4. 4. Ellipse as a conic section
  5. 5. Hyperbola as a conic section
  6. 6. Parabola as a conic section
  7. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

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