Conics and the Cone: Part 1

Introduction

While studying co-ordinate geometry (aka analytic geometry) in intermediate classes we normally arrive at the study of conic sections or in short "conics". Three new curves namely "ellipse", "parabola", and "hyperbola" come into picture and their theory is quite unlike those of the elementary geometrical objects (line, triangle, circle etc) studied in secondary classes. In case of the elementary geometrical objects like points, lines, triangles, circles we have two approaches: 1) using the axioms of Euclid and then deducing the properties of these objects logically from Euclid's axioms and, 2) using the language of coordinate geometry which transforms the subject of geometry into algebra. Unfortunately the beautiful approach of using Euclid's axioms is discarded in higher secondary classes in favor of the approach using coordinate geometry which makes the subject dull with huge amount of laborious algebraical manipulations. In fact most students think that the only way to study new curves like ellipses, parabola and hyperbola is through coordinate geometry.

Even more unfortunate aspect of this approach to conics is that the student is first taught that these particular curves can be obtained as sections of a right circular cone by various planes and then he is given definitions based on the focus directrix properties. Nowhere in the higher secondary curriculum do we find the link between these two definitions. Thus the "conics" get detached from the "cone" forever. In this post I will demonstrate the connection between these definitions via a very simple geometrical proof based on the axioms of Euclid.

Circular Cone

Before we proceed further it is better to precisely define the term "circular cone".  A circular cone is surface generated by set of all possible lines passing through the circumference of a circle and also passing through a fixed point not lying in the plane of the circle. The circle mentioned in this definition is called the base of the cone and the fixed point is called the vertex (apex) of the cone.  The adjective circular is used because of the  base being a circle. A line joining the vertex and some point on the circumference of the base is called a generator. Line joining the vertex and center of the base is called the axis of the cone. If the axis of a cone is perpendicular to the base of the cone then the cone is said to be right otherwise it called oblique. The figure below shows a right circular cone as well as an oblique circular cone. In both the cases A is the vertex, AO the axis, and BPC the base of the cone.

Right Circular Cone and an Oblique Circular Cone

Here in the above figure AB, AP, AC are all generators. Any plane passing through the axis is called an axial plane. Portion of an axial plane between the cone and the base is called an axial triangle. In the above figure ABC is an axial triangle, but ABP is not. In a right circular cone all axial planes are perpendicular to the plane of the base, but in an oblique cone only one axial plane is perpendicular to the base and this plane is the one containing the axis as well as the perpendicular from vertex to base. Here we must understand that we are dealing with three dimensional figures represented on a plane and hence some level of imagination of is needed to fully grasp what is being presented.

One more point worth mentioning is that since the cone is made by lines passing through vertex and circumference, the cone actually extends beyond the base as well beyond the vertex in opposite directions. So the cone ideally consists of two identical parts (each having infinite expanse) which meet at the vertex.

Sections of a Cone

Let a plane intersect the cone. The points of intersection between the cone and the plane give rise to a variety of curves and we have the following options:

1. The plane passes through the vertex. In this case the section obtained is a pair of straight lines with vertex as their point of intersection. In the degenerate case it may happen that the section consists only of a generator or only the vertex.
2. The plane is parallel to one of the generators. In this case we get a curve of infinite expanse (meaning it cannot be contained in a finite portion of a plane) which is called parabola.
3. The plane is parallel to the base and does not pass through vertex. In this case the section is a circle.
4. The plane cuts only one part of the cone and is not parallel to the base. In this case the section is an ellipse which may be a circle (in case of oblique cone).
5. The plane cuts both parts of the cone, but does not pass through the vertex. In this case the section is a hyperbola.

The above definitions are presented in almost every book on coordinate geometry,  but with reference to a right circular cone. The cases 1 and 3 are simple and obvious to prove. The rest of cases namely 2, 4 and 5 which give rise to new kind of curves are the most interesting and these are normally called conics.

We will now mention another set of definitions given for these conics. A conic section can be defined as a set of points whose distance from a fixed point varies in a constant ratio with their distance from a fixed line not passing through the fixed point. The fixed point is called the focus, the fixed line the directrix, and the constant ratio the eccentricity of the conic section. A conic section with eccentricity equal to unity is called parabola. If the eccentricity is less than unity it is called an ellipse and if the eccentricity is greater than unity it is called a hyperbola.

We now establish the equivalence between these two definitions. To make things simpler we first restrict ourselves to a right circular cone.

Focus Directrix property of Conics

Conic Sections

In the above figure we have a right circular cone with vertex O, with OV, OQ as generators. VQ is a diameter of the circular base. UAP is a plane cutting the cone, so that the curve AP is a conic section. The plane UAP is perpendicular to the axial triangle OVQ and line AU is the intersection of plane UAP with axial plane OVQ. We have a sphere which is touching the cone as well as the plane. It touches the cone in a circle EFR and the plane in point S. Line EF is parallel to line QV and it meets UA extended in X. XK is a line in plane UAP and is perpendicular to the line UX. PN is perpendicular AU in plane UAP and K is such that PNXK is a rectangle. To visualize we need to think OVQ is plane of webpage and UAP is perpendicular to it.

Since SP and PR and tangents to the sphere, we have SP = PR = EQ. Also Since AS and AE are tangents we have AS = AE.

Since the triangles AEX and AQN are similar (because EX is parallel to NQ) we have
$ \displaystyle \frac{AE}{AX} = \frac{AQ}{AN} \,\, \Rightarrow \frac{AQ}{AE} = \frac{AN}{AX} = k$
$ \displaystyle \Rightarrow \frac{EQ}{NX} = \frac{AE + AQ}{AX + AN} = \frac{AE + k\cdot AE}{AX + k\cdot AX} = \frac{AE}{AX} = \frac{SA}{AX}$

Since NX = PK it now follows that
$ \displaystyle \frac{SP}{PK} = \frac{SA}{AX}$

It is now quite obvious that the point S is the focus, line KX is the directrix and the distances PS and PK bear a constant ratio for any point P on the conic section, so that SA/AX is the eccentricity of the conic. Also note that the eccentricity is less than unity (equal to unity, greater than unity) according as angle ANQ  is less than angle OVQ (equal to angle OVQ, greater than angle OVQ). In other words if the intersecting plane is parallel to generator OV then the eccentricity of the conic is unity. If this plane cuts both generators OV and OQ towards same side of O and is not parallel to the PQV then the eccentricity is less than unity. If the plane cuts generators OV and OQ on opposite sides of Q, then eccentricity of the conic is greater than unity.

Thus with the above simple geometrical proof we obtain the focus-directrix property of the conics. The same proof for the case of oblique cone is bit difficult and will be discussed in the next post. Almost entire theory of the conics was set by Apollonius (sometime during 200 B.C.) in his famous 7 books titled "Conics" using the cone to derive some planar properties of the conics and was written in the style of Euclid's Elements. It is rather unfortunate that this masterpiece of Apollonius is totally left out of the current mathematical curriculum making students believe that there is no Euclidean geometry (i.e. geometry in the style of Euclid) after Elements.

P.S.: The proof above and the accompanying figure is taken from "Conic Sections Treated Geometrically" by W. H. Besant.

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