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 By Bill Claff
Lenses project images, usually
onto an image plane, sometimes onto a curved surface. I will assume only image
planes in this article.
The way the image is projected is classified according to where rays of an
incoming angle are projected onto the image.
The angle of the incoming ray ranges from zero on the optical axis to the
halfangle of view of the lens.
I will use Θ to
indicate that angle. Since optical prescriptions generally use ω to indicate the
halfangle of view this means that 0
<= Θ <= ω.
And I will use y to
indicate how far from the optical axis the ray arrives at the image plane
although you may see the letter r (for radius) elsewhere.
Optical prescriptions specify image height as Y
so my choice reminds us that 0
<= y <= Y.
Naturally, in any of the following formulas f
indicates focal length.
The projections follow
trigonometric functions that I will list in the following table.
These formulas are often shown in terms of y
(or r) but I prefer to
restate them in terms of (y / f).

y = 
(y / f) = 
Rectilinear 
f tan( Θ ) 
tan( Θ ) 
Stereographic 
2 f tan( Θ / 2 ) 
2 tan( Θ / 2 ) 
Equidistant 
f Θ 
Θ 
Equisolid angle 
2 f sin ( Θ / 2 ) 
2 sin ( Θ / 2 ) 
Orthographic 
f sin ( Θ ) 
sin ( Θ ) 
You might protest that the
Equidistant formula is not trigonometric.
However, note that both k * tan (
Θ / k ) and k *
sin ( Θ / k ) approach Θ
as k approaches
infinity.
It's useful to look at these
projection formulas graphically:
I limited the yaxis of this chart so I could better maintain the proper aspect
ratio. Notice that when the yaxis is y
/ f the Equidistant line has a slope of 1.
The xaxis is limited to an angle of view of 270 degrees which is the largest
angle of view for which I have ever seen an optical prescription.
And the Rectilinear line is limited to an angle of view of 140 degrees which
exceeds the angle of view of the widest rectilinear lens that has been
produced.
I don't think of these
projections as distinct but rather as members of two families of curves that
meet at the Equidistant (also Equiangular) line.
Above the Equidistant line we have k
* tan ( Θ / k ) and below that line k * sin ( Θ / k ).
To give the reader a sense for how quickly these families converge on the Equidistant line I added curves for 4 * tan ( Θ / 4 ) and for 4 * sin ( Θ / 4 ).
No lens follows a projection
formula perfectly. In a separate article we'll learn more about where the
boundary between Rectilinear and the other projections actually lies.
For now simply note that below the Rectilinear line is barrel distortion
and above it is pincushion distortion.
Also, consider this; when the angle of view is small the projections are
indistinguishable.
So we might label a telephoto lens as Rectilinear but it might be
indistinguishable from even Orthographic!
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