three theorems on parabolas
Theorem 1 :
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 . 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 through is . Also, the slope of this is . The corresponding right-bisector will also pass through and will have slope . Its equation is therefore
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
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 through . So pick an extra point where is the perpendicular bisector of . We then have 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.
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 (just to be nasty). Here’s a nice quicky:
Theorem 2 The Reflector Law:
For , the length of the path is minimal when produced is perpendicular to the -axis.
Quite simply, assume produced is not necessarily perpendicular to the -axis. Because 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.
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 and “reflected” angles equal.
One last marvelous tidbit. This will take more time, though. Let be tangent to at , and let be perpendicular to at . We will call the to at . Let meet the -axis at .
Theorem 3 :
The radius of the “best-fit circle” to at is twice the length .
(Note: the ’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 some small distance from each other (we don’t actually use , it’s just a psychological trick). Construct the tangent and normal at R, normal at . Let intersect at , and intersect the -axis at . . Erect perpendiculars to the -axis through respectively. . Let intersect the -axis at . Let be points on not in . Construct perpendicular to with in . We now have
(That’s the number , not the parabola)
From (iii),(iv) and (i) we have , 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 . Combining this with (ii),(vi),(vii), and (viii) it follows that , and in the limit , .
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 equations, and that’s why I enjoy it so.
|Title||three theorems on parabolas|
|Date of creation||2013-03-22 12:40:51|
|Last modified on||2013-03-22 12:40:51|
|Last modified by||CWoo (3771)|