INSTRUCTIONAL UNIT: CONIC SECTIONS

by

A. KURSAT ERBAS & GOOYEON KIM  Click Here for a movie demo

A conic (or conic section) is a plane curve that can be obtained by intersecting a cone with a plane that does not go
through the vertex of the cone. There are three possibilities, depending on the relative position of the cone and the plane (Figure 1).

If no line of the cone is parallel to the plane, the intersection is a closed curve, called an ellipse.

If one line of the cone is parallel to the plane, the intersection is an open curve whose two ends are asymptotically parallel; this is called a parabola.

Finally, there may be two lines in the cone parallel to the plane; the curve in this case has two open pieces, and is called a hyperbola. History of Conic Sections

Conic sections are among the oldest curves, and is an old mathematics topic studied systematically and thoroughly. The conics seem to have been discovered by Menaechmus (a Greek, c.375-325 BC), tutor to Alexander the Great. They were conceived in an attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle. The conics were first defined as the intersection of: a right circular cone of varying vertex angle; a plane perpendicular to an element of the cone. (An element of a cone is any line that makes up the cone) Depending on whether the angle is less than, equal to, or greater than 90 degrees, we get ellipse, parabola, or hyperbola respectively. Appollonius (c. 262-190 BC) (known as The Great Geometer) consolidated and extended previous results of conics into a monograph Conic Sections, consisting of eight books with 487 propositions. Quote from Morris Kline: "As an achievement it [Appollonius' Conic Sections] is so monumental that it practically closed the subject to later thinkers, at least from the purely geometrical standpoint." Book VIII of Conic Sections is lost to us. Appollonius' Conic Sections and Euclid's Elements may represent the quintessence of Greek mathematics.

Appollonius was the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola.

In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, especially in the development of projective geometry where conics are as fundamental objects as circles are in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic sections is a rich classic topic that has spurred many developments in the history of mathematics.

(from http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html)

For specific history of each conic section fallow the "History" link in each below!

Notes for Teachers

Objectives for Students

CIRCLE

History

Lesson I: Definition and geometric construction of a circle

Lesson II:

Lesson III:

Lesson IV:

Lesson V:

Lesson VI:

In-class Worksheet:

Exploration:

PARABOLA (by A. Kursat ERBAS)

History

Lesson I: Definition and geometric construction of a parabola

Lesson II: Introduction to the algebraic representation of a parabola

Lesson III: Analytic Equation of a Parabola

Lesson IV: Understanding the simplest parabolas of the form Lesson V: Polar and Parametric Equations of a Parabola

Lesson VI: In class worksheet

Lesson VII: Reflective (Optical) Property of Parabola

Lesson VIII: Properties of tangents to a parabola

Lesson IX: Area property of a parabola

Student Projects

ELLIPSE (by A. Kursat ERBAS)

History

Lesson I: Definition and Basic Properties of Ellipse

Lesson II: Introduction to the algebraic representation of an ellipse

Lesson III: Polar and Parametric Equations of an Ellipse

Lesson IV: Circumference and Area of an Ellipse

Lesson V: Reflective (Optical) Property of an Ellipse

Investigation 1

Investigation 2

Investigation 3

HYPERBOLA

History

Lesson I: Introduction

Lesson II:

Lesson III:

Lesson IV:

Lesson V:

In-Class Worksheet:

Lesson VI:

References:

http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html

http://www.astro.virginia.edu/%7Eeww6n/math/ConicSection.html

http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/

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A. Kursat ERBAS & GooYeon Kim © 2000

mail: aerbas@coe.uga.edu

gkim@coe.uga.edu 