fisheye perimeter & projective measures ?

Higher-dimensional geometry (previously "Polyshapes").

fisheye perimeter & projective measures ?

Postby thigle » Sat Mar 18, 2006 1:20 am

i don't know much about elliptic geometries, but a fisheye lens seems to be an exemple of it.

in halfsphereVision (180deg around), an edge contained in vertical frontal plane that contains the point-of-view, wherefrom the observer looks forward, is a horizon of vision, its boundary, perimeter.

now is this vision-edge in 180degFOV an ideal line at infinity ?
what about ideal lines ? do they twist ? is that wherefrom the 'strangeness'(nonorientability) of projective plane comes from ?

anyway, my question is this: can we measure in projective geometry at all, and if so, what is the diameter of the space represented through halfspherical fisheye lens ?

half infinity ? or one infinity ? what kind of ?

or what is the diameter(extent) of RP3 ?
is diameter of RP4 'bigger', like 2 infinities ?
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Postby wendy » Wed Apr 05, 2006 9:51 am

The projective plane preserves straight lines.

It derives from the artistic projection, where the eye is not on either the flat plane or the picture. A ray from the eye intersects both the picture and the flat plane.

The horizon projects as the "line at infinity".

Because the projective plane is so constructed, it is possible for things like antipodal points to appear as one (this happens in the spherical model, where the eye is at the centre of the sphere).

There is no direct realisation of the projective plane in its entirity, because the thing is non-orientable, while any fragment of real space is orientable. The best one can do is "lines passing through a point".

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