hyperbolic slot

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From: Monya Baker (monyab@eastside.org)
Date: Fri Aug 20 1999 - 16:09:26 PDT 

Message-Id: <3.0.6.32.19990820160926.007f07d0@pop.walltech.com>
Date: Fri, 20 Aug 1999 16:09:26 -0700
From: Monya Baker <monyab@eastside.org>
Subject: hyperbolic slot

Hi folks,

I just made a snack version of the hyperbolic slot, in which a straight
stick can be rotated through a curved slot. Now, I am driving myself
slightly crazy trying to come up with a mathematical connection between the
angle that the stick is placed at (45 degrees) and the curvature of the
hyperbola (you only see one side of it in the snack version.

What I want to know is, is there a way that I can use the slope of the
stick (which is tilted at a 45 degree angle) to come up with an equation
for the hyyperbola? Also, since only one half of thhe hyperbola is shown,
can I come up with an equation for the parabola?

Here's how my mathematical thoughts are running so far: A
side-to-side-opening hyperbola on the origin has the formula negative(x
over x-radius)squared plus (y over y-radius)squared = one. To figure out
the curvature of the hyperbola, you use the asymptotes - a line that goes
through the center and has a slope of y-radius over the x-radius. For a
side-to-side opening hyperbolas, steeper asymptotes will mean a shallower
curve and less steep asymptotes mean a more elongated curve. But, the
slope of the asymptotes here will be 1 and -1. If you have a line at a 45
degree angle, you have y = x, Which you can rearrange to y-x = 0. So
somehow, I have to have a reason to square both the terms but still keep a
negative sign. Help, my algebra isn't meshing with my geometry!

Here's just my observations: If you have the stick at a 45 degree angle,
you have a hyperbola. If you have the stick at a 0 degree angle
(perpendicular (sp?) to the floor), it's as if the asymptotes got
infinitely big, so you have no hyperboal at all - just a straight up and
down line as your slot. If you have the stick at a 180 degree angle
(parallel to the floor), you have a straight, horizontal line as your slot
- or for your hyperbola - the asymptotes go to zero. Can you get a
straight stick to make a hyperbola at anything but a 45 degree angle?

If anyone can think through this clearly let me know! It's been puzzling
me personally - and I'd love my class to grapple with it.

Happy beginning of the school year.

Monya

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Monya Baker, Science Teacher
Eastside College Preparatory School
2101 Pulgas Avenue
East Palo Alto, CA 94303
Phone: (650) 323-5898
Fax: (650) 688-0859

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ACADEMIC EXCELLENCE IN EAST PALO ALTO
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