[ptx] difference between rectangular and cylindrycal pano

[JD Smith]

On Thu, 2004-10-28 at 20:12, Rik Littlefield wrote:
> Marek,
> 
> PTviewer assumes that its input image is equirectangular projection, 
> possibly less than full spherical 360x180 degrees.  For partial panos 
> with small fov, there is not much difference between equirectangular and 
> rectilinear or cylindrical, so visually it works OK to use those with 
> PTviewer also.
> 
> Whether you should use rectilinear or cylindrical for website and 
> printing depends on what you are trying to accomplish.  With 
> rectilinear, all lines that are straight in the world are also straight 
> in your picture, but you get severe distortion if you try to go beyond 
> roughly 120 degrees fov.  With cylindrical, the horizon and verticals 
> are the only straight lines in the world that end up straight in your 
> picture, but you can go up to full 360 degrees horizontal.
> 
> Cylindrical and equirectangular have many of the same characteristics -- 
> up to a full circle around, vertical lines in the world are also 
> vertical in your picture, things near the poles get badly distorted.  
> The difference is that in equirectangular, you can go clear to the poles 
> and things near the poles get "squashed" vertically, where in 
> cylindrical they get "stretched" and you can go only about +-60 degrees 
> vertically before distortion gets too much.  (PTViewer remaps from 
> equirectangular input to rectilinear on the screen, so when you look up, 
> you see an undistorted view of the stuff above you.)
> 
> For technical discussion and pictures using maps of the Earth, see 
> http://mathworld.wolfram.com/EquirectangularProjection.html , 
> http://mathworld.wolfram.com/CylindricalProjection.html, and 
> http://mathworld.wolfram.com/RectilinearProjection.html , which links to 
> http://mathworld.wolfram.com/GnomonicProjection.html .  (What 
> photographers call "rectilinear" projection, geographers call 
> "gnomonic".  

And astronomers call the rectilinear projection "gnomic" or, more
typically, "tangent-plane".  The last is the best, I think, since it
describes exactly what the projection does.  It's also a fundamental
projection in that it is the projected coordinate system most cameras
and imaging instruments with flat focal planes provide in their output
images, at least approximately.  Pin-hole cameras, in fact, provide
exactly a tangent-plane mapping of the sphere onto their detector
planes, and most simple imaging systems (consumer cameras with
non-fisheye lenses among them) approximate this quite well.  

There are deviations, of course, but typically the tangent-plane
projection is used as a convenient mathematical base, from which
distortions are expressed in terms of additional radial polynomial
distortion corrections (this is what the a,b,c parameters of PanoTools
are all about).  The often stated property of "straight lines are mapped
to straight line" is more correctly stated as "great circles are mapped
to straight lines".  For panoramic imaging, this distinction is largely
academic, since the center of the coordinate system moves with the
camera.

JD