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Topic: length of ellipse projection?
Replies: 2   Last Post: Jul 24, 2003 3:24 PM

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 Hop Posts: 420 Registered: 12/4/04
Re: length of ellipse projection?
Posted: Jul 24, 2003 3:24 PM

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

Date Subject Author
7/24/03 Hop
7/24/03 Rouben Rostamian
7/24/03 Hop