D. N. Seppala-HoltzmanSt. Joseph’s College

A Canonical Conical FunctionTo appear in The College Mathematics

JournalIntended for a general audienceThis presentation can be downloaded from

the “downloads” page of: faculty.sjcny.edu/~holtzman

EccentricityAs the angle of tilt increases, the ellipses

become flatter and more elongatedMathematicians say the that the eccentricity

is increasingThis will be defined later

Eccentricity IIAs the angle of tilt increases, the hyperbolas

will open up more and moreAgain, the eccentricity is increasingI still owe you a definition

The Conic SectionsThe circle, ellipse, parabola and hyperbola

make up the family of conic sectionsThese were studied by the ancient Greeks

ApolloniusApollonius (262 – 190 B.C.) wrote a treatise

on them

Euclid Apollonius √ Pythagoras

Foci and VerticesConics have important points called foci and

verticesWe will need these to define eccentricity

Let us start with the ellipseHammer two nails into a boardTake a piece of string whose length is greater

than the distance between the stringsTie each end to one of the nailsPull the string taut with a pencil and draw a

curve that keeps the string taut at all timesThis will produce an ellipse

The foci of an ellipseThe two nails represent the foci of the ellipseAn ellipse is defined to be the set of points in

the plane the sum of whose distances to two fixed points (the foci) is a constant

Note that the length of the string is this constant distance

Foci and vertices of an ellipseThe foci of an ellipse are equidistant from the

center, lying on its central axisThe vertices of an ellipse are those two points

where the ellipse intersects its central axisTraditionally, we call the distance from the

center to either focus “c” and the distance from the center to either vertex “a”

Eccentricity of an ellipseThe eccentricity, e, of an ellipse is defined to be

e = c/aAs c < a, we have 0 < e < 1 The closer e gets to 1, the more elongated the ellipse

becomesThe closer e gets to 0, the more circular it becomesThe limiting case occurs when the foci coincide with

the center and the result is an actual circle. Circles have eccentricity, e = 0

The focus and vertex of a parabolaA parabola has a single focusThis is the unique point on the central axis

with the property that, if the parabola were a mirror, every light ray emitted from the point would reflect off the curve and travel parallel to the axis

Conversely, all in-coming rays parallel the axis would pass through the focus

The vertex is the point where the parabola crosses its axis

The focus of a parabola IIThis is why car headlights have parabolic

reflectors around the light source which lies at the focus

This is also why radio telescopes and dish antennae are parabolic bowls with the receiver at the focus

Eccentricity of the parabolaThe eccentricity of any parabola is equal to 1

The foci and vertices of a hyperbolaA hyperbola is defined to be the set of points

in the plane the difference of the distances to two fixed points is a constant

Recall that in the elliptical case, the sum of the distances was held constant

The two fixed points are the foci of the hyperbola

The points where the hyperbola intersect its central axis are the vertices

Foci and vertices of a hyperbolaThe foci and vertices of a hyperbola are

equidistant from the center, lying on its central axis

Traditionally, we call the distance from the center to either focus “c” and the distance from the center to either vertex “a” just as in the elliptical case

The eccentricity of a hyperbolaThe eccentricity, e, of a hyperbola is defined

to the quotient e = c/a just as it is in the elliptical case

As c > a, we have e > 1 for all hyperbolas

Conic eccentricities summarizedCircle: e = 0Ellipse: 0 < e < 1Parabola: e = 1 Hyperbola: e > 1

Why all the fuss about eccentricity?Any two conics with the same

eccentricity are similarThus, any two circles are similar as they all

have e = 0Likewise, any two parabolas are similar since

they all have e = 1For ellipses and hyperbolas, similarity

classes vary with e

What is similarity, anyway?Two shapes are similar if one can be scaled

up or shrunk down so that it can be placed over the other, matching it identically

Similarity of circlesClearly, given two circles, one could increase

or decrease the radius of one of them, making the two identical

Here, the radius is the scaling factor

Similarity of parabolasLikewise, one could increase or decrease the

distance from the vertex to the focus of one parabola to make it identical to any other parabola

Here, the distance from focus to vertex is the scaling factor

Similarity leads to constantsAny geometric construct on a similarity class

that is independent of the scaling factor, leads to a constant for that class

For example, consider the circleTake any circleCompute the ratio of the circumference divided

by the diameter. Note that the scaling factor, R, cancels. The result is a very famous constant:

2

2

C R

D R

Two ConstantsPursuing this pattern Sylvester Reese and

Jonathan Sondow made a pair of geometric constructs, one for all parabolas and one for a special hyperbola

These gave rise to two constants:The Universal Parabolic ConstantThe Equilateral Hyperbolic Constant

Two Constants IITheir respective values

were:

2 ln(1 2)

2 ln(1 2)

Holy Cow!The similarity of these two constants was

either an indicator of a profound mysterious truth or a mere coincidence

No one knows which

The ProblemTrying to get to the bottom of this question

one faces a big problem:The two constructions yielding the two

constants are incompatibleThe one carried out on the parabola could not

be done on the hyperbola and vice versa

A Unifying Construction is NeededA unifying construction that can be carried

out on all conics yielding a value that depends only upon the eccentricity is called for

One would want this construction to yield a smooth, continuous function of e

A Canonical Conical FunctionMotivated by this need, I created what I call

(with a nod to Dr. Seuss) a Canonical Conical Function

This function has the desired properties just discussed

Latus RectumTo define the function, I must first define a

line segment that all conics have: the latus rectum

Latin for “straight side,” the latus rectum is chord passing through a focus and orthogonal to the axis

Canonical Conical Function IIFor any conic, let A denote the area of the

region bounded by the curve and its latus rectum

Let L denote the length of its latus rectum We define our Canonical Conical Function by:

2

A

L

Canonical Conical Function IIIThis construction generates a smooth,

continuous function depending only on eIt is calculated for each class of conic and is

defined piecewiseWe denote this function by C(e)

The Circle (e = 0)We compute the value of A/L2 for the circle

and we get the value:

As the eccentricity of the circle is 0, we assign:

8

(0)8

C

The Ellipse (0 < e < 1)Doing the same construction for an ellipse,

we get the value of our function solely in terms of e:

2 3 2

2 2

1 2 2 2arcsin( ) 1( )

8(1 )

e e e e eC e

e

The Parabola (e=1) We compute the value of A/L2 for the

parabola and we get the value: 1/6As the eccentricity of a parabola is 1, we

assign:

1(1)

6C

The Hyperbola (e > 1)We compute the value of A/L2 for each

hyperbola and we get the value:

3 2 2

2 2

[ln( 1) 1]( )

4( 1)

e e e e eC e

e

Canonical Conical Function IVThus, C(e) is defined by the exact same

construction for each conic but has a different algebraic expression in each of the four categories

Amazingly, this function is smooth and continuous

Behold its graph on the next slide

ConclusionAnd so we have achieved what we set out to

doWe have found a single construction that can

be carried out on every conic which yields a smooth, continuous function, uniting the entire family of conics

Thank you for listening