by Jongsuk Keum
Historical introduction
Conic sections are among the oldest curves, and is one of the oldest
mathematics subjects studied rigorously. The conics were discovered
by Menaechmus (a Greek, c. 375-325 BC) who was a pupil of Plato and Exodus
in an attempt to solve the famous problem duplicating the cube. (See appendix
for his solution), Euclid (c. 325-265 BC) studied about them, and Appollonius
(c. 262-190 BC) consolidated and extended previous results of conics
into a monograph Conic Sections, consisting of eight books with 487 propositions!
He applied his work to his study of planetary motion and used this to
aid in the development of Greek astronomy. 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, espcially in
the development of projective geometry where conics are fundamental objects
as circles in Greek geometry. Conic sections is a rich classic topic
that has spurred many developments in the history of mathematics.
Description of Conic Sections
The conic sections are the nondegenerate curves generated by the
intersections of a plane with a cone. For a plane perpendicular to the
axis of the cone, a circle is produced. For a plane which is not perpendicular
to the axis and which intersects only a single nappe which is one piece
of the cone, the curve produced is either an ellipse or a parabola. The
curve produced by a plane intersecting both nappes is a hyperbola.
Formal Definition of Conic
Sections
Conics may more formally be defined as follows. Given a line D
called the directrix and a point F called the focus not on D, a conic
is the locus of points P that moves in the plane such that the distance
from P to F divided by the distance from P to D is a constant called the
eccentricity. That is, the eccentricity is
e = distance[P,F]/distance[P,D].
If 0 < e < 1, the conic is an ellipse, if e = 1, it is a
parabola, and if e > 1, it is a hyperbola and has two branches. The
figure shows conic sections with eccentricity {.2, .4, .6, .8, .9 1,
1.5, 2, 2.5, 3, 4, 8} with their focus at the origin and the directrix
the dotted line x=-1.
The intersection of a cone and a plane is really a conic section
We will prove more precisely that any such section of a right circular
cone has the property that distance[P,F]/distance[P,D] = e, constant,
so it is a conic section in the sense of our formal definition.
Proof: We first inscribe a sphere that is tangent to the cone along
a circle C and the cutting plane at a point F. Let P be any point on the
conic, i.e., the intersection of the cone and the cutting plane. We shall
see that F is the focus of the conic and that the corresponding directrix
is the line D in which the cutting plane intersects the plane of the circle
C.
Let Q be the intersection point of the line through P parallel
to the axis of the cone and the plane of C, A be the point where the
line joining P to the vertex of the cone intersects C, and R be the point
on the line D and the perpendicular form P to the line D.
Then since the lines PA and PF are tangent to the sphere at A and
F, respectively,
distance[P,A] = distance[P,F].
Let us denote now alpha to be the cone angle and beta the angle
between the cutting plane and the plane of C. Then from the right triangles
PQA abd PQR we have
cos(alpha) = distance[P,A]/distance[P,Q] = distance[P,F]/distance[P,Q]
sin(beta) = distance[P,R]/distance[P,Q]
Hence,
distance[P,F]/distance[P,R] = sin(beta)/cos(alpha) =: e.
a constant. This proves our claim.
Alternative characterization
of conic sections
* Circle. A circle is the set of points P in a plane that are a
fixed distance r called the radius from a specified point O called the
center. Using GSP and identifying the center and the radius, create the
circle:
* Ellipse. An ellipse is the locus of points the sum of whose distance
from two fixed points is constant. The two fixed points are called the
foci of the ellipse, and the midpoint of the line segment joining the
foci is called the center. The points where the ellipse crosses the line
through the foci are called vertices of the ellipse. The chord of the
ellipse joining the vertices is called the major axis, and the chord through
the center perpendicular to the major axis is called the minor axis. Identify
the fixed points as A and B (these are the foci) and use GSP to create an
ellipse.
* Parabola. A parabola is the set of all points in a plane such
that each point in the set is equidistant from a line called the directrix
and a fixed point called the focus. The line through the focus perpendicular
to the directrix is called the axis, and the point where the parabola
crosses its axis is called the vertex. The directed distance from the vertex
to the focus is called the focal length, and is usually denoted p. If the
focus is either to the right of or above the vertex, then the value of
p is positive; if the focus is either to the left of or below the vertex,
then the value of p is negative.
* Hyperbola. A hyperbola is the set of points in a plane for which
the absolute value of the difference in their distances from two fixed
points in the plane is a constant. Just as with the ellipse, we called
the fixed points the foci, and the midpoint of the line segment joining
the foci is called the center. The line through the foci is called the
transverse axis. The vertices are the points where the hyperbola crosses
its transverse axis. The distance from the center to either of the vertices
is a. Let c be the distance from each focus to the center. Then c is called
the focal length.
Algebraic view of conic sections
We now move from a geometric point of view to an algebraic approach.
Note that all of the conics that we will see below will satisfy a second-degree
equation of the form
Ax^2+Bxy+Cx^2+Dx+Ey+F=0
where A, B, C, D, E, and F are constants. As we change the values
of some of these constants, the shape of the corresponding conic will
also change. Hence, it will be very important to focus on these differences
in the algebraic equations as we study the individual conic sections.
Knowing the subtle differences in the equations will help us to quickly
identify the type of conic that is represented by a given equation.
1. The type of section can be found from the sign of: B^2 - 4AC:
If B^2 - 4AC is... then the curve is a(an)...
< 0 ellipse, circle, point or no curve.
= 0 parabola, 2 parallel lines, 1 line or no curve.
> 0 hyperbola or 2 intersecting lines.
2. Examples of each case:
Possible Cases Example
Ellipse 4x^2 + 9y^2 = 36
Circle x^2 + y^2 = 4
Point x^2 + y^2 = 0
No Curve x^2 = -1
Parabola y^2 = 9x
Two Parallel Lines (x - 1)(x - 2) = 0
One Line x^2 = 0
Hyperbola x^2 - y^2 = 1
Two Intersecting Lines (x - 1)(y + 1) = 0
3. The Conic Sections.
conics Circle Ellipse Parabola Hyperbola
Equation x^2 + y^2 = r^2 x^2 / a^2 + y^2 / b^2 = 1 4px = y^2 x^2
/ a^2 - y^2 / b^2 = 1
Asymptotes y = ± (b/a)x
Eccentricity e 0 0<e<1 1 e>1
Definition distance to the origin is constant sum of distances
to each focus is constant distance to focus = distance to directrix difference
between distances to each focus is constant
Applications
Astronomical application: Why do the planets
move in an ellipse?
This is a very deep question to answer. Kepler, in 1602, said he
believed that the orbit of Mars was oval, then later showed empirically
that it was an ellipse with the sun at one focus. The eccentricity of
the planetary orbits is small (i.e., they are close to circles). In
fact, the eccentricity of Mars is 1/11 and of the Earth is 1/60. In 1705,
Halley showed that the comet, which is now called after him, moved in
an elliptical orbit round the sun. The eccentricity of Halley's comet
is 0.9675 so it is close to a parabola (eccentricity 1).
More precisely, Kepler showed that the planets are obeying his
three laws of planetary motion (see appendix 3) but he had no real physical
understanding of why they did so (he believed it was a magnetic phenomenon).
It took Newton's genius to explain why. Newton was able to show mathematically
that the inverse square law of force (his universal law of gravitation)
leads to motion of one body about another in the path of a conic section
(i.e. a circle, ellipse, hyperbola or a parabola). The initial values of
the position and velocity determine the type of motion. In the case of
the planets the motion is that of an ellipse. Of course the planets also
perturb one another and so these ellipse gradually rotate and their geometries
change, but the motion is always elliptical. In contrast some of the long
period comets may be on parabolic orbits that are not bounded
Satellite trajectory.
The orbits of satellites about a central massive body can be described
as either circular or elliptical.
Conic section mirrors.
Most of the mirrors used in telescopes have geometrical figures
that one can generate by rotation of two-dimensional conic sections about
their axes of symmetry. Why are conic section mirrors? Because
1. Paraboloidal mirrors reflect all on-axis parallel light to their
focus.
2. Ellipsoidal or hyperboloidal mirrors reflect light that would
converge on one focus and make it instead converge on the other focus.
Examples.
* The big dish at Arecibo. See the following website
http://www.math.iupui.edu/m261vis/LGM/LGM.htlm
* Cassegrain telescope.
Projectile motion:
The trajectory of a projectile becomes, neglecting the influence
of air resistance, a parabola where the horizontal velocity component
is constant and the vertical component is subject to gravity. Galileo
was the first who accurately described projectile motion. He showed that
it could be understood by analyzing the horizontal and vertical components
separately.
Appendix
Menaechmus's cube duplication.
Etymological study of the name of conic sections.
It is believed that Aristotle who took over the mathematical words
that have now become "hyperbola", "ellipse", "parabola" into rhetoric,
where they have become ""hyperbole", "elliptic speech" or "ellipsis", and
"parable".
The Greek originals mean
[hyperbola] "thrown beyond"
[ellipse] "falling short"
[parabola] "thrown beside"
and for the curves probably refer to the fact that the distance
to the focus exceeds, falls short of, or equals, that to the
directrix. [It may however, instead refer to the fact that the plane
we take to cut the cone, either "goes beyond" so as to hit the other portion
of the cone, or "falls short of " hitting that part, or "runs parallel
beside" a generator of the cone, as some dictionaries think.]
Kepler's three laws of planetary motion and Newton's modification.
First Law (Elliptic Orbits): The orbits of the planets around the
sun are ellipses with the Sun at one focus.
Newton's revision: The orbit of any pair of objects are conic section
with the center of mass as one focus.
Second Law (Areas): The line joining the
planet to the Sun sweeps out equal areas in equal times as the planet
travels around the ellipse.
Newton's revision: Angular momentum is conserved.
Third Law (Harmonic Law): The ratio of
the squares of the revolutionary periods P for two planets is equal to
the ratio of the cubes of their semimajor axis a. That is, P^2 = a^3.
Newton's revision: P^2 = {4pi^2/[G(m1+m2)]}a^3. Here G =
gravitational constant and m1 and m2 are mass of orbiting bodies.
Conic Section in the Bible.
There is even Biblical reference to conic sections. I King 7:23
discusses Solomon's "sea", which was to be a circular basin used for
ceremonial cleansing.
Now he made the sea of cast metal ten cubits from brim to brim,
circular in form, and its height was five cubits, and thirty in circumference.
(NASB)
Observe that the value of pi here is being estimated in some sense
by 3. Thus,
C = pi*d or pi = C/d,
where C is the circumference and d is the diameter
of a circle.
