[ptx] difference between rectangular and cylindrycal pano

[Rik Littlefield]

JD Smith wrote:

>And astronomers call the rectilinear projection "gnomic" or, more
>typically, "tangent-plane".
>

>...  The often stated property of "straight lines are mapped
>to straight line" is more correctly stated as "great circles are mapped
>to straight lines".  
>  
>
Well, sort of.

I tried to be careful about this issue in my post, when I wrote

"It's not immediately obvious that gnomonic projection maps lines that 
are straight in 3D to lines that are straight in the map, but it does."

The point is kind of subtle.  In fact the tangent-plane projection does 
map lines that are straight in Euclidean 3-space onto lines that are 
straight in the 2D map.  It also maps great circles of the sphere onto 
straight lines in the map.  Of course this must mean that it maps lines 
that are straight in Euclidean 3-space onto great circles of the sphere, 
and that also is true, as you can see by noticing that the plane 
determined by the center of the projection and any straight line must 
bisect the sphere.

The issue is not quite academic from my standpoint.  Panorama Tools 
essentially models the world as being painted on the inside of a sphere 
with the entrance pupil of the lens at its center.  Straight lines in 
the 3D world that was actually photographed, map to great circles of 
that imaginary sphere, and are then mapped back to straight lines when 
you select a rectilinear projection.  I have to keep this straight while 
working on the code and when trying to analyze unusual situations like 
"flat stitching".  The concept often makes my head hurt a little.  
Fortunately most people get to ignore it.

Anyway, if you want to talk about mapping a sphere to a plane, then it 
is more correct to talk about great circles.  If you want to talk about 
about normal photography, it is equally correct and probably more 
natural to talk about projecting straight lines to straight lines.

--Rik