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Contents




    1. History: How did the theory of conic sections develop?
    2. Basic concepts from projective geometry.
    3. Interior points of conic sections.
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’.
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.
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 using the methods of analytic
   geometry and then results we all know from high school were
   derived.
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.
Steiner




   Around 1850, Steiner gave a purely geometric definition of conic
   sections which is known under the keyword ‘Steiner’s generation of
   conic sections’.
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)
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.
Affine plane – projective plane




   If one adds to an affine plane a line whose points are the parallel
   classes one obtains a projective plane.
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.
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.
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.
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.
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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Interior Points of Conic Sections

  • 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.
  • 4. Ellipse as a conic section
  • 5. Hyperbola as a conic section
  • 6. Parabola as a conic section
  • 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.