Re: [Phys-l] elliptical thinking

["LaMontagne, Bob" <RLAMONT@providence.edu>]

Ellipses and circles are all part of what are called "conic sections". If one places a basic right circular cone base first on a table, any horizontal cross-sectional cut through the cone produces a circle. If one imagines a tiny light bulb at the apex of the cone, the shadow of any one of these horizontal cuts is the circle that forms the base of the cone. Likewise, a non-horizontal cut defines the conic section called the ellipse. It's shadow is also the base circle that forms the cone. A tilted cut parallel to the sides of the cone produces a parabola. Its shadow is a portion of the circle that forms the base. An hyperbola comes from a vertical cut - it's shadow also forms a portion of the base circle. One of the science supply companies sells a lovely wooden model of all this that disassembles so the conic sections can be seen.
 
Bob at PC

________________________________

From: phys-l-bounces@carnot.physics.buffalo.edu on behalf of Polvani, Donald G.
Sent: Tue 7/22/2008 9:18 AM
To: Forum for Physics Educators
Subject: 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/







_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l