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

2. 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

3. Start with a cone

4. Slice the cone with a horizontal plane

5. The intersection will be a circle

6. Tilt the cutting plane slightly

7. The intersection will be an ellipse

8. Tilt the plane a bit more

9. The result will be a wider, flatter ellipse

10. 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

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

12. A parabola

13. Tilting more will yield a hyperbola

14. A hyperbola

15. 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

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

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

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

19. 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

20. The ellipse

21. 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

22. 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”

23. Foci and vertices of an ellipse

24. 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

25. 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

26. 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

27. The parabola with focus and vertex

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

29. 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

30. 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

31. Foci and vertices of a hyperbola

32. 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

33. Conic eccentricities summarized
Circle: e = 0
Ellipse: 0 < e < 1
Parabola: e = 1
Hyperbola: e > 1

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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:

40. 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

41. Two Constants II
Their respective values were:

42. 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

43. 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

44. 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

45. 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

46. 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

47. Circle with Latus Rectum

48. Ellipse with Latus Rectum

49. Parabola with Latus Rectum

50. Hyperbola with Latus Rectum

51. 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:

52. 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)

53. 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:

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

55. 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:

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

57. 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

58. The Graph of C(e)

59. 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