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Re: [Phys-l] elliptical thinking



Ludwig Kowalski wrote:

"That would be a convincing argument if an ellipse were defined as a
shadow (due to a parallel beam of light) of a tilted circle. That is
good enough for me. But a mathematician would probably begin be
convincing students that the shape of the boundary of a tilted circle
satisfies the ellipse definition--the sum of distances from two focal
points being constant everywhere on the boundary."

I seem to remember a proof in our high school trigonometry book that the
cross section of a plane surface passing through a right circular
cylinder, and which did not pass through the ends of the cylinder,
produced, of course, a circular cross section for a 90 deg angle and an
ellipse for a non-90 deg angle.

I was reminded of this result when installing a new electrical outlet in
our kitchen. I extended the wiring from the old outlet by first
drilling a hole in the side of the old outlet box at about a 60 deg
angle with respect to the side of the box. I got a nice elliptical hole
(well not quite because my drill bit did wobble a little as I drilled
the hole producing a less than perfect ellipse). I have to admit I was
a little surprised by the shape of the hole until I thought about it.

Don Polvani
Northrop Grumman Corp.
Undersea Systems
Annapolis, MD

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of ludwik
kowalski
Sent: Monday, July 21, 2008 5:32 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] elliptical thinking

On Jul 21, 2008, at 3:53 PM, LaMontagne, Bob wrote:

Area of a circle is pi*r^2.

Tilt circle away from you by angle theta and projected area is
pi*r^2 cos(theta), but r cos(theta) is same as the a for the observed
projected ellipse - therefore pi*a*r = pi*a*b.


That would be a convincing argument if an ellipse were defined as a
shadow (due to a parallel beam of light) of a tilted circle. That is
good enough for me. But a mathematician would probably begin be
convincing students that the shape of the boundary of a tilted tilted
circle satisfies the ellipse definition--the sum of distances from two
focal points being constant everywhere on the boundary.

Ludwik Kowalski, a retired physics teacher
5 Horizon Road, Apt. 2702, Fort Lee, NJ, 07024, USA Also an amateur
journalist at http://csam.montclair.edu/~kowalski/cf/







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