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Hop
Posts:
420
Registered:
12/4/04
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Re: length of ellipse projection?
Posted:
Jul 24, 2003 3:24 PM
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Rouben Rostamian wrote:
> Hop David <hopspage@tabletoptelephone.com> wrote:
>
>
>>If you project an ellipse onto a plane perpendicular to
>>the major axis you get a line segment of length 2b
>>
>>Projecting this ellipse onto a plane perpendicular to the
>>minor axis and you get a line segment of length 2a
>>
>>Now take an arbitrary line coplanar with the ellipse and
>>forming angle alpha with the major axis. When you project
>>the ellipse onto a plane perpendicular to this line
>>segment, what is the length of the resulting line segment?
>
>
> So you are asking for a formula for the width of an ellipse.
> Let me write t for what you call alpha.
>
> Here is the width:
>
> w = 2 sqrt[ (a sin t)^2 + (b cos t)^2 ]
I see this:
A circle with radius a centered on the origin.
A radial line has end point (a cos t, a sin t)
Clone the radial line and rotate it 90 degrees.
New endpoint is (-a sin t, a cos t)
Scale the whole shebang vertically by b/a.
The endpoint of rotated radius is now
(-a sin t, b cos t) and the width of the ellipse
is indeed
2 sqrt[ (a sin t)^2 + (b cos t)^2 ]
Here's a pic:
http://clowder.net/hop/etc./ellanim.gif
However the vertical scaling changes the angle t to t'..
tan t' = b/a tan t.
Hmmm. With a few aspirins I can whomp up a spreadsheet to do what I want.
Thanks,
Hop
http://clowder.net/hop/index.html
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