Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

The Parabola

March 2004

Menaechmus (c. 375-325 BC), a pupil of Eudoxus, tutor to Alexander the Great, and a friend of Plato [Smith, p. 92], is credited with the discovery of the conics. A more revealing term is conic sections, on account of their being found as the intersections of circular cones by planes. If the planes pass through the vertex of the cone, the conics are said to be degenerate, otherwise they are not. There are three non-degenerate conics: the ellipse, the parabola, and the hyperbola.

The parabola results when the plane is parallel to a generating line of the cone. This is an unbounded curve some of whose properties will be discussed and illustrated below. The plane can be thought of as being hinged on a straight line perpendicular to the axis of the cone. Rotate the plane even a little bit in one direction so that it still cuts only one nappe of the cone, and the intersection will become a closed curve — an ellipse; turn it in the other direction, and the plane will cut the second nappe, so that the curve will acquire a second infinite branch. This 2-piece curve is known as a hyperbola.

Following Pappus (c. 290-350 AD), the common belief is that it was Apollonius of Perga (c, 262-190 BC) who gave the curves their names.

The term parabola comes [Schwartzman, p. 158] from Greek para "alongside, nearby, right up to," and -bola, from the verb ballein "to cast, to throw." Understandably, parallel and many of its derivatives start with the same root. The word parabola may thus mean "thrown parallel" in accordance with the definition.

The story is interesting, and a short etymological regression won't be entirely out of place. As I. Kant once appropriately (see below) remarked, "... even clever persons occasionally chatter" [Kant, p. 55].

Mathematics is known to heavily borrow its terminology from the common language. Sometimes the process goes in reverse, and the conics serve one such example. As J. Conway has noted years ago in an online discussion,'... it was 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"... In rhetoric, "hyperbolic" speech is the kind that goes beyond the facts, "elliptic speech" falls short of them, while a "parable" is a story that exactly fits the facts.' Since parables were originally spoken stories, the Latin verb parabolare came to mean "to speak." Via French we also got parley, parlance, parlor, parliament, parole [Schwartzman, p. 158] and from the Portuguese metathesis offshoot palavra the word palaver [J. Conway]. (pa·lav·er n. 1.a. Idle chatter. b. Talk intended to charm or beguile.)

The etymology of the conics is bolstered by the form of the equations that describe the curves. In modern notations, conics are the curves generally given by a second degree equation:

(1) Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0.

The discriminant D = B2 - AC of the equation serves to differentiate between various curves: for ellipse, parabola, and hyperbola D is respectively less, equal, and greater than 0. In a special system of coordinates, (1) could be transformed into the canonical
(2) y2 = 2px + qx2,

which represents an ellipse, a parabola, or a hyperbola depending on whether q is less than, equal to, or greater than 0. The number p is known as the focal parameter for reasons that will be partially explained below. (Note that the word parameter has the same Greek origin as parabola and signifies an attribute of a mathematical object given either alongside other attributes or the one that exactly determines the object.)

Further nomenclature is as follows: p/2 is called the focal distance, the point F(p/2, 0) is a(the) focus, and the straight line x = - p/2 is (a)the directrix of the conic. The ellipse and the hyperbola each have two foci and two directrices, whilst the parabola has only one of each. (The opinions of course vary. Some say that parabola has two foci, as do all other non-degenerate conics, with the second focus being a point at infinity.) The chord through the focus parallel to the directrix is known as the focal chord, or latus rectum. The latus rectum measures exactly 2p.

In polar coordinates (r, θ) centered at a focus [Brannan, Coxeter, Salmon], the conics are given by

(3) r = p / (1 + e·cos θ),

where the parameter e — the conic's eccentricity — is less than, equal to, or greater than 1 for the ellipse, parabola, and hyperbola, respectively. (It is thought to be non-negative, although for hyperbola, (3) describes only one of its branches. The second branch is obtained with the plus sign "+" in (3) replaced by "-".)

The focus, directrix and eccentricity combine in the definition of the conics as loci of points: a conic section is the locus of points the ratio of whose distances to a fixed point (the focus) and a fixed line (the directrix) is a fixed quantity (the eccentricity.) Thus, a distinguishing property of parabola is that its points are equidistant from the focus and the directrix. In addition, "... the shape of a conic is determined by its eccentricity. That is, all conics of the same eccentricity are similar, differing from one another merely in size. Since all parabolas have eccentricity 1, all parabolas are similar to one another, differing from one another only in size" [Eves, p. 5]. The similarity of the curves with the same eccentricity follows easily from (3). By having such a rare property, the parabola is analogous to circle (which has eccentricity 0.)

All this (and more) was known to Apollonius, but two revolutionary discoveries involving conics had to wait until the 17th century. Johannes Kepler (1571-1630) published his three laws of planetary motion in 1609 and 1619. In a major revision of Copernicus' heliocentric theory, Kepler placed the planets on elliptical orbits with the sun at their common focus. In a momentous departure from the Aristotelian theory, Galileo Galilei (1564-1642) realized that the motion of material objects could be split into independent components, say horizontal and vertical for a projectile. This understanding led him to revise even his own beliefs. As he himself wrote, "... where the senses fail us reason must step in." By 1604 he concluded that projectiles travel along parabolic trajectories. However, the results were not published until 35 years later.

It can be observed (and of course proven from the focal definition or, as easily, from the canonical or polar equation) that parabola has an axis of symmetry which passes through the focus and is perpendicular to the directrix. The point midway between the focus and the directrix is an extremity of the parabola, known as its apex.

The applets below illustrate several purely geometrical properties of the parabola. For entirely idiosyncratic reasons, the parabola has been rotated 90o such that wherever a parabola had to be drawn, I used the equation y = x2/2p instead of (2). In the following, the feet of perpendiculars dropped from points A, B, etc. on the parabola to the directrix will be denoted A', B', etc. F will always denote the focus of the parabola at hand.

Theorem 1

Let A lie on a parabola. Then the tangent to the parabola at A makes equal angles with AF and AA'.

Proof

By the definition, FAA' is isosceles. Let T be the midpoint of FA'. Then the perpendicular bisector AT divides the plane into two parts: one consists of points that are nearer to F than they are to A'; the other consists of points that are nearer to A'. Except for A, all points of the parabola lie in the former half. Indeed, let B be a point on the parabola. Then, since BB' is the shortest segment from B to the directrix, FB = BB' < BA'. In particular, B does not belong to AT. We conclude that A is the only point of intersection of that line with the parabola. Therefore, AT is tangent to the parabola at A.

Corollary (Parabolic mirror)

If a light source is placed at the focus of parabola and the light is reflected from its inner surface, the reflected rays are all parallel to the axis. Radio telescopes are built on a reversed principle. Incoming signals parallel to the axes all pass through the focus.

Corollary (The Parabola as Envelope)

The x-axis lies midway between the focus and the directrix. Therefore, the midpoint T of FA' lies on the x-axis. In addition, FT is perpendicular to AT, the tangent at A. In a reversed process, assume a point and a line are given. (We'll take F as a given point and the x-axis as the given line.) Connect the point F to an arbitrary point T on the x-axis and construct a line t perpendicular to FT at T. The family of those straight lines drawn for many points T clearly delineates a parabola. This is the parabola with focus F and the apex on the x-axis. The parabola is said to be the envelope of the family of straight lines {t}. The process of obtaining parabola in this manner is known as the pedal construction [Yates, p. 50]. The x-axis is said to be the pedal of the parabola with respect to its focus.

Corollary (Parabola by paperfolding)

Now imagine a parabola, its focus and directrix drawn on a sheet of paper. Since FAA' is isosceles, and AT is its perpendicular bisector, one may fold the paper along AT. The crease thus obtained, which is none other than AT, is a particular case of a mathematical object known as paper line. A paper point is a result of intersection of two paper lines [Martin, Ch. 10]. The whole business of drawing with paper points and lines (there are also paper circles) was introduced in 1983 by T. Sundra Row from India. The first rigorous treatment of paperfolding is attributed to R. C. Yates (1949). The best known axiomatics of paperfolding is due to Humiaki Huzita (1991.)

If, at the outset, the parabola is not given, but only a point and a line, we may produce any number of creases by folding the paper so that the given point falls onto the given line. In time, a parabola will emerge as the envelope of paper lines. Note that it does not matter whether the goal of a particular folding is to place a point on a line, or make the line pass through the point. As a practical matter, if the given line coincides with a paper edge then it is much easier to pursue the latter goal.

Archimedes triangle

The triangles formed by two tangents to a parabola and the chord connecting the points of tangency were used by Archimedes in his study of the area of parabolic segments and bear his name. The chord is usually considered the base of the triangle.

Archimedes' Lemma

Let, in an Archimedes triangle ABS, M be the midpoint of the base. Assume that SM crosses the parabola in O, and let the tangent to the parabola at O crosses the sides of ABS in A1 and B1, as shown. Then

  1. The median MS is parallel to the axis of the parabola,

  2. A1 and B1 are the midpoints of the sides AS and BS of ABS,

  3. O is the midpoint of MS.

Proof

  1. In A'B'F, lines AS and BS serve as perpendicular bisectors of sides A'F and B'F. They intersect at S, so that the perpendicular from S to the third side A'B' is bound to bisect the latter. In trapezoid ABB'A', this line is parallel to the bases AA' and BB' and passes through the midpoint of one side A'B'. It therefore passes through the midpoint M of other side as well.

  2. The triangles AA1O and BB1O are also Archemedean. So the first part applies: the medians from A1 and B1 are parallel to the axes of the parabola. But those medians serve as midlines in triangles AOS and BOS.

  3. As we just showed A1B1 is a midline in ABS. It therefore cuts in half any cevian from S, MS in particular.

Theorem 2 (Squaring of parabola)

(See [Dörrie, pp. 239-242, Stein, pp.51-62].)

The parabola divides the area of an Archimedes triangle in the ratio 2:1. In other words, the area of the parabolic segment AB equals 2/3 of the area of the Archimedes triangle ABS.

Proof

Let area(ABS) = 1. Two thousand years before the invention of Calculus, Archimedes filled the parabolic segment with triangles, whose areas are easily arranged into a geometric series whose sum he already knew.

The first triangle in the series is the "inner triangle" ABO. From Lemma, area(ABO) = 1/2 — the first term of the series. Area(A1B1S) = 1/4. Therefore, area(AA1O) + area(BB1O) = 1/4.

AA1O and BB1O are Archimedes triangles with inner triangles (filled triangles in the above applet) inside the parabola segment. The combined area of the filled triangles is half that of their Archimedes progenitors: 1/2·1/4. This is the second term of the progression. The next term comes from four smaller Archimedes triangles with the total area of 1/4 of the two preceding Archimedes triangles (which was 1/4.) The combined area of their inner triangles is therefore 1/2·(1/4)2, etc. Continuing this process, the total, which is the area of the parabola segment, is

  1/2 (1 + 1/4 + (1/4)2 + ...) = 2/3.

Theorem 3 (Equal angles and similar triangles)

Assume there are two points A and B on a parabola, with tangents AS and BS meeting in S. Then

  1. S is the circumcenter of A'B'F,

  2. FAS = FA'B' = FSB,

  3. FBS = FB'A' = FSA,

  4. The triangles BFS, SFA, and B'FA' are similar,

  5. F lies on AB iff S lies on A'B' and, in this case, ASB = 90o.

Proof

  1. By Theorem 1, SA is the perpendicular bisector of FA'. Therefore, SA' = SF. Similarly, SB' = SF.

  2. An inscribed angle FA'B' and a central angle FSB' subtend the same chord. Hence,

      FSB = FSB'/2 = FA'B'.

    On the other hand, angles FA'B' and SAA' have pairwise perpendicular legs and are thus equal. Since, SAA' = FAS,

      FA'B' = FAS.

  3. Similar to 2.

  4. Follows from 2-3. Triangles BFS, SFA, and B'FA' have the same orientation and are directly similar. Triangles SA'A and BB'S are also similar the above three, but have a different orientation.

  5. To shorten the expressions, let's introduce a = FAS and b = FBS. If S lies on the directrix, 2(a + b) = 180o. Then AFS = (90o - a) + (90o - b) = 90o, and similarly BFS = 90o. So AFB is a straight line. The argument is obviously reversible.

Directrix is the Polar of Focus

Rephrasing #5, we can say that the tangents at the extremities of a chord that passes through focus F, meet on the directrix. This exactly means that the directrix is the polar of the focus, while the focus is the pole of the directrix with respect to the parabola.

Lambert's Theorem

(See [Dörrie, pp. 206-208].)

The circumcircle of a triangle formed by three tangents to a parabola, passes through the focus of the parabola.

Proof

Let the tangent at C intersect tangents AS and BS in points U and V, respectively. Theorem 3, applied twice, gives

  FSU = FBS = FVU,

which tells us that the quadrilateral SUFS is cyclic.

x-Axis is simson of the focus

As we saw earlier, x-axis is the pedal curve of the parabola with respect to its focus. In other words, x-axis consists of the feet of the perpendiculars from the focus to the tangents to the parabola. This means that x-axis is the simson of the focus with respect to the circumcircle of any three tangents to the parabola [Honsberger, p. 48].

Parabola from four tangents

Lambert's theorem suggests a construction of parabola from four tangents. Any three tangents determine a circle that passes through F. Two such circles determine F uniquely. Reflections of F in any two tangents produce two points on the directrix.

Apollonius' Theorem

Two tangents of a parabola are divided into segments of like proportion by a third and this third is divided in the same proportion by its point of tangency [Dörrie, p. 220]. More accurately,

  AU/US = UC/CV = SV/VB.

Proof

The three triangles AFS, SFB and UFV are similar. Furthermore, their corresponding vertices A, U, S are collinear, as are the vertices S, V, B. Vertex F is shared by all three triangles. The configuration is reminiscent of the Theorem of Directly Similar Figures. Its converse immediately implies that

(4) AU/US = SV/VB.

The same reasoning applied to tangents BV and CV crossed by the tangent AUS gives

  CU/UV = VS/SB,

or which is the same

  UV/CU - 1 = SB/VS - 1,

from where

(5) CV/UC = UV/CU - 1 = SB/VS - 1 = VB/SV.

(4)-(5) prove the theorem.

Parabola as Envelope II

Assume a parabola with two points A and B and their tangents AS and BS are given. Pick a number n and divide AS and BS into n equal intervals. Label division points on AS with numbers 1, 2, 3, ... counting from S, and mark those on BS counting from B. Connect the points with the same labels. From Apollonius theorem, the lines will envelope the parabola [Dörrie, pp. 220-222, Wells, p. 171].

If one starts with just two segments AS and BS, the emerging parabola will touch them at points A and B.

References

  1. D. A. Brannan et al, Geometry, Cambridge University Press, 2002
  2. H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, NY, 1961
  3. H. Dörrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965
  4. H. Eves, Mathematical Reminiscences, MAA, 2001
  5. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.
  6. I. Kant, Observations on the Feeling of the Beautiful and Sublime, U. of California Press, Berkeley, 1960
  7. G. E. Martin, Geometric Constructions, Springer, 1998
  8. G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960
  9. S. Schwartzman, The Words of Mathematics, MAA, 1994
  10. D. E. Smith, History of Mathematics, v 1, Dover, 1958
  11. S. Stein, Archimedes: What Did He Do Besides Cry Eureka?, MAA, 1999
  12. D. Wells, Curious and Interesting Geometry, Penguin Books, 1991 p26,Galileo-catenary
  13. R. C. Yates, Curves and Their Properties, NCTM, 1974 (J. W. Edwards, 1959)

Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. In November 2003, his site has welcomed its 8,000,000th visitor. The site is a recipient of the 2003 Sci/Tech Award from the editors of Scientific American.

Copyright © 1996-2004 Alexander Bogomolny