A Canonical Conical Function D. N. Seppala-Holtzman St. Joseph’s College

A Canonical Conical Function To appear in The College Mathematics Journal Intended for a general audience This presentation can be downloaded from the “downloads” page of: faculty.sjcny.edu/~holtzman

Start with a cone

Slice the cone with a horizontal plane

The intersection will be a circle

Tilt the cutting plane slightly

The intersection will be an ellipse

Tilt the plane a bit more

The result will be a wider, flatter ellipse

Eccentricity As the angle of tilt increases, the ellipses become flatter and more elongated Mathematicians say the that the eccentricity is increasing This will be defined later

When the tilt of the plane matches that of the side of the cone, we get a parabola

A parabola

Tilting more will yield a hyperbola

A hyperbola

Eccentricity II As the angle of tilt increases, the hyperbolas will open up more and more Again, the eccentricity is increasing I still owe you a definition

The Conic Sections The circle, ellipse, parabola and hyperbola make up the family of conic sections These were studied by the ancient Greeks

Apollonius Apollonius (262 – 190 B.C.) wrote a treatise on them Euclid Apollonius √ Pythagoras

Foci and Vertices Conics have important points called foci and vertices We will need these to define eccentricity

Let us start with the ellipse Hammer two nails into a board Take a piece of string whose length is greater than the distance between the strings Tie each end to one of the nails Pull the string taut with a pencil and draw a curve that keeps the string taut at all times This will produce an ellipse

The ellipse

The foci of an ellipse The two nails represent the foci of the ellipse An 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 ellipse The foci of an ellipse are equidistant from the center, lying on its central axis The vertices of an ellipse are those two points where the ellipse intersects 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”

Foci and vertices of an ellipse

Eccentricity of an ellipse The eccentricity, e, of an ellipse is defined to be e = c/a As c < a, we have 0 < e < 1 The closer e gets to 1, the more elongated the ellipse becomes The closer e gets to 0, the more circular it becomes The 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 parabola A parabola has a single focus This 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 II This 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

The parabola with focus and vertex

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

The foci and vertices of a hyperbola A 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 hyperbola The 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

Foci and vertices of a hyperbola

The eccentricity of a hyperbola The 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 summarized Circle: e = 0 Ellipse: 0 < e < 1 Parabola: e = 1 Hyperbola: e > 1

Why all the fuss about eccentricity? Any two conics with the same eccentricity are similar Thus, any two circles are similar as they all have e = 0 Likewise, any two parabolas are similar since they all have e = 1 For 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 circles Clearly, 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 parabolas Likewise, 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 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:

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

Two Constants II Their respective values were:

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 smooth, continuous function of e

A Canonical Conical Function Motivated 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 Rectum To 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

Circle with Latus Rectum

Ellipse with Latus Rectum

Parabola with Latus Rectum

Hyperbola with Latus Rectum

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:

Canonical Conical Function III This construction generates a smooth, continuous 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 and we get the value: As the eccentricity of the circle is 0, we assign:

The Ellipse (0 < e < 1) Doing the same construction for an ellipse, we get the value of our function solely in terms of e:

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:

The Hyperbola (e > 1) We compute the value of A/L2 for each hyperbola and we get the value:

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 smooth and continuous Behold its graph on the next slide

The Graph of C(e)

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 smooth, continuous function, uniting the entire family of conics Thank you for listening