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Topic: Cross Ratio of Four Concurrent Lines
Replies: 6   Last Post: Jul 9, 2013 11:29 AM

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Avni Pllana

Posts: 513
Registered: 12/6/04
Re: Cross Ratio of Four Concurrent Lines
Posted: Jun 30, 2013 4:54 PM
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Xratio.jpg (67.1 K)

> Cross Ratio can be defined in projective geometry
> either for 4 collinear points or 4 concurrent lines.
>
> Point P connects 4 fixed points in x-y plane. Find
> locus of P so that cross ratio of the four ray pencil
> is a positive constant.
>
> Regards
> Narasimham


Hi Narasimham,

the problem is very interesting. I studied the special case A=[1 0], B=[0 1], C=[-1 0], D=[0 -1], where point P is element of the box [-2 2]x[-2 2], and tet=angle(APB), phi=angle(BPC), psi=angle(CPD). For the Cross Ratio we have

CR=sin(tet+phi)*sin(phi+psi)/(sin(tet+phi+psi)*sin(phi)).

The result is shown in the attachment.

Best regards,
Avni



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