Conic curves
Construction / Reflection / Equations
General equation of 2nd degree / Eccentricity and Directrix

Conic curves is the name given to those which are obtained by cutting a cone with a plane.  Given their origin, conic curves are sometimes calles conic sections.

The Greek mathematician Apolonius (262-190 A.C.) from Perga (an old city in Asia Minor) was the first person to study conic curves in detail. Apolonius discovered that the conic curves can be classified in three types: ellipses, hyperbolas y parabolas.

Ellipses are curves which are formed cutting a cone with a plane which only touches one of the sides of the cone and it is not parallel to one of its edges.

Hyperbolas are curves which are formed on cutting a cone with a pane that touches the two sides of the cone.

Parabolas are curves which are formed on cutting a cone with a plane parallel to one of its edges.

Apolonius showed that conic curves have many interesting properties.   Some of these properties are those which are used nowadays to define them.  In the page construction the definitions of conic curves as loci are studied, they are the most commonly used in modern mathematics.   

Perhaps the most interesting and useful properties that Apolonius discovered about conic sections are the so called reflection properties.  If you build mirrors with the form of a conic curve which rotates about its axis, you obtain the so called eliptic, parabolic or hyperbolic mirrors, according to the curve that rotates.   Apolonius demonstrated that if a source of light is put in the focal point of an elliptic mirror, then the light reflected in the mirror is concentrated in the other focal point.  If light is received from a far away source with a parabolic mirror in a way that the incident rays are parallel to the axis of the mirror, then the light reflected by the mirror is concentrated on the focal point.  Thus it is possible to burn a piece of paper if it is put in the focal point of a parabolic mirror and the axis of the mirror is pointed towards the sun.  The legend exists that Arquímedes (287-212 A.C.) managed to set fire to the roman ships during the defense of Siracuse using the properties of parabolic mirrors.  In the case of hyperbolic mirrors, the light coming from one of the sources is reflected as if it were coming from another souce.  On the page reflection the reflection properties of conic sections are studied.

In the XVI century, the philosopher and mathematician René Descartes (1596-1650) developed a method for relating curves with equations.  This method is called Analytic Geometry. In Analytic Geometry conic curves can be represented by second degree equations in the variables x and y.  In the page equations of conics  simple equations of conic curves are shown which correspond to those which have their centre or, in the case of the parabola, their vertex, at the origin, and its axis of symmetry coincides with one of the coordinate axes.  Perhaps the most surprising result of Analytic Geometry is that all the second degree equations in two variables represent conic sections.  This fact is the subject of the page about general second degree equations in two variables.        

In eccentricity and directrix general equations of conics are shown, expressed in terms of eccentricity, the distance to the directrix and the angle of the axis of symmetry.

Without doubt, conic sections are the most important curves that Geometry offers Physics.  For example, their reflection properties are highly useful in Optics.  However, without doubt what makes them most important in Physics is the fact that the orbits of the planets around the sun are ellipses and that, even more, the course of any body subject to a gravitational force is a conic curve.  The German astronomer Johannes Kepler (1570-1630) discovered that the orbits of the planets around the sun are ellipses which have the sun as one of their focuses.  There is a nice page on the internet about the three Laws of Kepler (it is in spanish).  Later on, the famous well-known English mathematician and physicist Isaac Newton (1642-1727) showed that the orbit of an object around a gravitational force field is always a conic curve.  

A simple way of summarizing the importance of conic sections is to mention some names of  people whose work had something to do ith conic curves: Apolonius, Arquimedes, Kepler, Descartes and Newton.

Author: José Luis Abru León

	Ministerio de Educación, Cultura y Deporte. Año 2000