D. N. Seppala-Holtzman St. Josephs College faculty.sjcny.edu/~holtzman The Canonical Conical Function Submitted to The College Mathematics Journal This presentation can be downloaded from the downloads page of:

faculty.sjcny.edu/~holtzman Circles All circles are similar Increasing or decreasing the radius of any circle can make it congruent to any other Parabolas

Likewise, all parabolas are similar Here, adjusting the focal parameter (the distance from the focus to the directrix) can make any two parabolas congruent Ellipses and Hyperbolas Alas, this property of similarity does not hold

for the other two families of conics: ellipses and hyperbolas However, the following does hold: Any two conics with the same eccentricity are similar. Scaling Factor Let us define a scaling factor to be a

parameter whose adjustment can make any member of a similarity class congruent to any other Scaling factors are, in general, not unique The radius of a circle and the focal parameter of a parabola are examples Similarity leads to constants

Any geometric construct on a similarity class that is independent of the scaling factor, leads to a constant for that class For example, consider the circle Take any circle Compute the ratio of the circumference divided

by the diameter. Note that the scaling factor, R, cancels. The result is a very famous constant: C 2 R D 2R

Two Constants Pursuing 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 Constant The Equilateral Hyperbolic Constant

The Universal Parabolic Constant This is defined as the ratio of the arc length of the portion of the parabola delineated by the latus rectum to the focal parameter The Equilateral Hyperbolic Constant This is defined to be the ratio of the area

bounded by the curve and its latus rectum to the square of the semi-transverse axis Two Constants II Their 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 Problem Trying to get to the bottom of this question one faces a big problem: The two constructions yielding the two constants are incompatible The one carried out on the parabola could not be done on the hyperbola and vice versa

A Unifying Construction is Needed A 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 continuous, differentiable function of e

The Canonical Conical Function Motivated by this need, I created what I call (with a nod to Dr. Seuss) The Canonical Conical Function This function has the desired properties just discussed Canonical Conical Function II

For 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: A 2

L Canonical Conical Function III This construction generates a continuous, differentiable function depending only on e It is calculated for each class of conic and is defined piecewise We denote this function by C(e)

The Circle (e = 0) We compute the value of A/L2 for the circle value: and we get the 8 As the eccentricity of the circle is 0, we

assign: C (0) 8 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 1 e 2e 2e 2 arcsin(e) 1 e C ( e) 2 2

8(1 e ) 2 The Parabola (e=1) We compute the value of A/L2 for the parabola and we get the value: 1/6 As the eccentricity of a parabola is 1, we

assign: 1 C (1) 6 The Hyperbola (e > 1) We compute the value of A/L2 for each

hyperbola and we get the value: e3 e [ln(e e 2 1) e 2 1] C ( e) 4(e 2 1) 2 Canonical Conical Function IV Thus, 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 continuous and differentiable everywhere To prove this, we must verify several things Continuity at e = 0 The limit of C(e) in the elliptical range as e

approaches zero from above yields C (0) 8 Continuity at e = 1 The limit of C(e) in the elliptical range as e

approaches 1 from below yields C(1) = 1/6 The limit of C(e) in the hyperbolic range as e approaches 1 from above yields C(1) = 1/6 Differentiability at e = 1 The limits of the derivatives of C(e) in the elliptical and hyperbolic ranges as e approaches 1 from below and above yield the

same result C(e) Properties and Observations I The function is monotonically decreasing It asymptotically approaches zero as e approaches C(e) provides a bijection from [0, ) to [/8, 0)

C(e) Properties and Observations II In the elliptical range, both AL2and are decreasing as e grows As C(e) is decreasing, we conclude that A is 2

shrinking Lfaster than In the hyperbolic range, both AL2and are increasing as e grows Thus, in this range, L2 is growing faster than A

C(e) Properties and Observations III The eccentricity of the equilateral hyperbola 2 is 2 C( ) = (The Equilateral Hyperbolic Constant) As L is twice the length of the semitransverse axis in this case, this result does

not surprise From Discontinuous to Continuous Consider an epsilon neighborhood about (1 ,1 1: ) Over this interval, at the geometric level,

the conic transitions from a closed curve with two foci to an open curve with one focus to a pair of open curves with two foci This is violent, dramatic discontinuous change At the algebraic level, over this same interval, all is tranquil and continuous As Dr. Seuss might have said:

Oh, the turbulent lives of the conics Marked by fits of rambunction But calming them down And uniting them all: The Canonical Conical Function Conclusion And so we have achieved what we set out to

do We have found a single construction that can be carried out on every conic which yields a continuous, differentiable function, uniting the entire family of conics Thank you for listening