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From: (no name) (no email)
Date: Mon Mar 03 2003 - 13:17:17 EST
Norm,
IF the curved line has the same amount of curvature all along its length,
THEN we can approximate it as an arc of a circle.
Let the length of the straight line be 2*b, and the separation between
that straight line and the arc be "h". If the curved line is an arc of
a circle, call the radius of that circle R. The straight line, of length
2*b, is a chord of that circle of radius R. Drawing a little sketch,
construct a straight line, perpendicular to the original straight line
segment, extending to the center of the circle of which the curved line is
an arc. Call the length of the line segment, from the middle of the chord
of length 2*b to the center of the circle, a length "a". The radius of the
circle is R = a+h, and there's a right triangle, with hypotenuse (the long
side)
of length R extending from the center of the circle to the end of the
circular arc,
and two legs, of lengths "b" (half the length of the chord) and "a" (from
middle
of the chord to the center of the circle).
(I would draw a picture, but I'm trying to confine this message to plain
text.)
Pythagoras's theorem on the right triangle gives R*R = (a+h)*(a+h) = a*a +
b*b
Expanding (a+h)*(a+h) gives
a*a +2*a*h + h*h = a*a + b*b
Subtracting a*a from both sides gives
2*a*h * h*h = b*b
2*a*h = b*b - h*h
a = sqrt( (b*b - h*h)/2 )
Plug in the number b (half the length of the straight line you began with)
and the number h (the separation between the arc and the straight line) into
that formula, and you get "a" the distance from the center of the chord
to the center of the circle. Then b/a is the tangent of the smaller angle
in the right triangle. Find that angle. (Use a calculator's inverse tangent
function.) If you express that angle in radians (2 pi radians = 360
degrees),
then that angle in radians, multiplied by the radius of the circle (a + h),
is half the length of the arc. So the length of the arc is
arc length = 2* [inverse tangent in radians of (b/a)]*(a+h)
An ordinary scientific calculator, such as the one provided with MS windows
on your computer, should let you perform these calculations.
Jim
Salem, Oregon
-----Original Message-----
From:
[mailto:] On Behalf Of
Sent: 02 March 2003 19:02
To:
Subject: lv-ab: Question
I have a geometry problem.
Imagine two horizontal lines, one straight and one curved, a few inches
long, sorta parallel with the ends of each line touching the other. Like
you sliced a piece of the crust of a pizza off and the sliced-off piece is
lying on the table.
I know the length of the straight line that the knife cut, and I know the
distance apart the lines are in the middle of the sliced off pizza crust,
how do I find the length of the curved line?
Norm
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