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 studythe 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 canstill find in school books.

2. Apollonius wrote eight books on conic sections. Heintroduced the names parabola, hyperbola and ellipse and hegave a description of conic sections which was then used as adefinition 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. Thenconic sections were investigated using the methods of analyticgeometry and then results we all know from high school werederived.

Projective geometry

The 17th century also marks the beginning of the modern theoryof conic sections. Desargues introduced projective geometry andsince then, conic sections have been investigated using themethods of projective geometry.

Steiner

Around 1850, Steiner gave a purely geometric definition of conicsections which is known under the keyword ‘Steiner’s generation ofconic sections’.

Application

Kepler’s Laws.

I. Each planet moves a round the sun in an ellipse, with the sunat one focus.

II. The radius vector from the sun to the planet sweeps out equalareas in equal intervals of time.

III. The square of the period of a planet is proportional to thecube 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 whichlie 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 parallelclasses 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 incidencerelation is inclusion. C := {x ∈ F 3| x1x2 − x2

3 = 0} is a conicsection 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 pointof C is called a passant. A line which contains exactly one point ofC is called a tangent. A line which contains two points of C iscalled a secant.

Exterior and interior points

A point which is not a point of the conic section but lies on atangent is called an exterior point. A point such that each linewhich passes through this point is a secant is called an interiorpoint.

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 existinterior points if and only if F is pythagorean.