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three theorems on parabolas

(Topic)

In the Cartesian plane, pick a point with coordinates (subtle hint!) and construct (1) the set of segments joining with the points , and (2) the set of right-bisectors of the segments $s\in S$ .

Theorem 1 :

The envelope described by the lines of the set

is a parabola with

-axis as directrix and focal length $\vert f\vert$ .

Proof:

$\includegraphics[width = 6.5 cm, height = 4.5 cm, clip]{const.eps}$

We're lucky in that we don't need a fancy definition of envelope; considering a line to be a set of points it's just the boundary of the set $C=\cup_{b\in B} b$ . Strategy: fix an

coordinate and find the max/minimum of possible

's in C with that

. But first we'll pick an

from

by picking a point

on the

axis. The midpoint of the segment $s\in S$ through

is $M = (\frac{w}{2},f)$ . Also, the slope of this

is $-\frac{2f}{w}$ . The corresponding right-bisector will also pass through $(\frac{w}{2},f)$ and will have slope $\frac{w}{2f}$ . Its equation is therefore

$\displaystyle \frac{2y-2f}{2x-w} = \frac{w}{2f}.$

Equivalently,

$\displaystyle y = f + \frac{wx}{2f} - \frac{w^2}{4f}.$

By any of many very famous theorems (Euclid book II theorem twenty-something, Cauchy-Schwarz-Bunyakovski (overkill), differential calculus, what you will) for fixed

is an extremum for

only, and therefore the envelope has equation

$\displaystyle y = f + \frac{x^2}{4f}.$

I could say I'm done right now because we ``know'' that this is a parabola, with focal length

and

-axis as directrix. I don't want to, though. The most popular definition of parabola I know of is ``set of points equidistant from some line

and some point

." The line responsible for the point on the envelope with given ordinate

was found to bisect the segment $s\in S$ through

. So pick an extra point $Q\in b\in B$ where

is the perpendicular bisector of

. We then have $\angle FMQ = \angle QMH$ because they're both right angles, lengths

, and

is common to both triangles

and

. Therefore two sides and the angles they contain are respectively equal in the triangles

and

, and so respective angles and respective sides are all equal. In particular,

. Also, since

and

have the same

coordinate, the line

is the perpendicular to the

-axis, and so

, a general point on the envelope, is equidistant from

and the

-axis. Therefore etc.

QED.

Because of this construction, it is clear that the lines of are all tangent to the parabola in question.

We're not done yet. Pick a random point outside (``inside" the parabola), and call the parabola $\pi$ (just to be nasty). Here's a nice quicky:

Theorem 2 The Reflector Law:

For $R \in \pi$ , the length of the path

is minimal when

produced is perpendicular to the

-axis.

Proof:

$\includegraphics[width = 4.9cm, height = 6.9 cm, clip]{refl.eps}$

Quite simply, assume

produced is not necessarily perpendicular to the

-axis. Because $\pi$ is a parabola, the segment from

perpendicular to the

-axis has the same length as

. So let this perpendicular hit the

-axis at

. We then have that the length of

equals that of

. But

(and hence

) is minimal when it's a straight line; that is, when

produced is perpendicular to the

-axis.

QED

Hey! I called that theorem the ``reflector law''. Perhaps it didn't look like one. (It is in the Lagrangian formulation), but it's fairly easy to show (it's a similar argument) that the shortest path from a point to a line to a point makes ``incident" and ``reflected" angles equal.

One last marvelous tidbit. This will take more time, though. Let be tangent to $\pi$ at , and let be perpendicular to at . We will call the normal to $\pi$ at . Let meet the -axis at .

Theorem 3 :

The radius of the ``best-fit circle'' to $\pi$ at

is twice the length

Proof:

$\includegraphics[width = 6.4cm, height = 7.3cm, clip]{curve.eps}$

(Note: the $\approx$ 's need to be phrased in terms of upper and lower bounds, so I can use the sandwich theorem, but the proof schema is exactly what is required).

Take two points on $\pi$ some small distance $\epsilon$ from each other (we don't actually use $\epsilon$ , it's just a psychological trick). Construct the tangent and normal at R, normal at . Let intersect at , and intersect the -axis at . Join . Erect perpendiculars to the -axis through respectively. Join . Let intersect the -axis at . Let be points on not in . Construct perpendicular to with in . We now have

i): $\angle PRO = \angle ORF = \angle GRH \approx \angle P'R'O = \angle OR'F$
ii): $ER \approx FR \cdot \angle EFR$
iii): $\angle R'RE + \angle ERO \approx \frac{\pi}{2}$ (That's the number $\pi$ , not the parabola)
iv): $\angle ERO + \angle ORF = \frac{\pi}{2}$
v): $\angle R'ER \approx \frac{\pi}{2}$
vi): $\angle R'OR = \frac{1}{2} \angle R'FR$
vii): $R'R \approx OR \cdot \angle R'OR$
viii)

From (iii),(iv) and (i) we have $\angle R'RE \approx \angle GRH$ , and since

is close to

, and if we let

approach

, the approximations approach equality. Therefore, we have that triangle

approaches similarity with

. Therefore we have $RR':ER \approx RG:RH$ . Combining this with (ii),(vi),(vii), and (viii) it follows that $RO \approx 2 RG$ , and in the limit $R'\rightarrow R$ ,

QED

This last theorem is a very nice way of short-cutting all the messy calculus needed to derive the Schwarzschild ``Black-Hole" solution to Einstein's field equations, and that's why I enjoy it so.

"three theorems on parabolas" is owned by CWoo. [ full author list (3) | owner history (2) ]

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See Also: properties of parabola

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Cross-references: solution, Calculus, QED, limit, similarity, triangle, equality, approximations, number, perpendiculars, intersect, normal, tangent, distance, proof, lower bounds, terms, path, argument, similar, Lagrangian, right, extremum, differential calculus, theorems, equation, pass through, slope, midpoint, axis, fix, strategy, boundary, envelope, length, directrix, parabola, lines, segments, coordinates, point, plane

This is version 16 of three theorems on parabolas, born on 2002-05-28, modified 2007-08-11.
Object id is 2958, canonical name is ThreeTheoremsOnParabolas.
Accessed 5200 times total.

Classification:

AMS MSC:

51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

Pending Errata and Addenda

None.[ View all 5 ]

Discussion

forum policy

I like this by drini on 2002-05-28 00:42:48

I like your fresh writing style.
Have you considered changing this to "topic" or writing topic-kind entries? we're in need of them. There are lots of definitions and theorems, but we need some texts on general stuff.

also, if you write some stuff like an exposition or a paper we have special sections for posting them.
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

[ reply | up ]

Re: I like this by quincynoodles on 2002-05-28 00:51:45
- Re: I like this by drini on 2002-05-28 01:36:46

little advice by drini on 2002-05-28 00:08:58

when creating entries titles, it's wise to capitalize only proper nouns and its derivates. This is because the way system works.

If we don't do it this way, when automatically linking happens, text like
"The Hypotenuse of a Square is equal to the Sum of the Squares of its Legs".

f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

[ reply | up ]

Re: little advice by quincynoodles on 2002-05-28 00:15:00
- Re: little advice by vampyr on 2002-05-28 00:27:07

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