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Topic: Re: Locus for ratio of cosine of angles to be constant
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Narasimham

Posts: 307
Registered: 9/16/06
Re: Locus for ratio of cosine of angles to be constant
Posted: Nov 18, 2013 7:18 AM
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> cos(s) = cos(p) ecc where ecc is a constant.
>
> Use differential relations between r,th (polar) and
> d x,y (cartesian) coordinates with respect to arc
> length s.
>
> dr/ds - dx/ds ecc = 0 which integrates to
>
> r - x ecc = r - r cos(th) ecc = p, an arbitrary
> y constant.
>
> 1 - ecc cos(th) = p/r the polar equation of all
> l conics.
>
> Given constant is eccentricity and arbitray constant
> t is semi-latus rectum.
>
> This rotational definition imho is much better than
> n the standard distance definition of conics with
> respect to the meaning and significance of
> eccentricity.


Graphic showing equivalence of these two definitions is given:

http://i43.tinypic.com/2med8x4.jpg

It considers the right angle subtended by tangent point to directrix line segment at focus.

Regards
Narasimham



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