tangential & sagittal planes


Sagittal image plane

In an object plane perpendicular to the optic axis, construct a 
line intersecting the optic axis.  From any point on this object 
line, send out a fan of rays all of which lie in the plane which 
contains the object line and the optic axis.  Since there is 
nothing about the lens which can refract a ray which originally 
lies in this plane out of this plane, all rays will strike any 
arbitrarily-placed screen in image space someplace on the image 
line, regardless of whether the line is in focus or not.  That is, 
although the rays do "triangulate in" to a common point in image 
space, we can't find that point because even when they do not 
"triangulate in" (intersect), they still land someplace along the 
line and so don't generate any fuzziness.  Therefore, the correct 
location of the screen or image plane is indeterminate if we try 
to use only this fan of rays to find it.  

Now construct a new fan of rays from any off-axis object point on 
the object line.  This fan's plane will be perpendicular to the 
fan of the previous paragraph, and the central ray of this fan 
will start at the object point and pass through the point which is 
the middle of the lens.  Since these rays do not lie in the common 
plane of the object and image lines, there is a determinate 
distance from the lens to the image screen at which the rays will 
"triangulate in" to a point (intersect) and not broaden the width 
of the image line.  With the screen at this distance from the 
lens, the line on the image screen will be at its sharpest.  The 
screen now lies in the "sagittal image plane".  Sagittal means 
"radial" and it is a radial line which is sharpest at this image 
plane location.


Tangential image plane

In an object plane perpendicular to the optic axis, construct a 
circle centered on the optic axis.  Select a point on that circle 
and then construct a plane which is tangent to the circle at that 
point and which contains the point which is the middle of the 
lens.  Send out a fan of rays from the selected object point but 
constrain them to lie in the plane just defined.  To the first 
order, it does not matter what distance the image screen lies from 
the lens: the circle will still be sharp because any out-of-focus 
condition will broaden the image point only along the image of the 
circle.

Now construct a new fan of rays from the same object point on the 
circle described above, but this time constrain the rays to lie in 
a plane containing the object point and the optic axis.  (Note 
that this plane is perpendicular to the plane described in the 
previous paragraph.)  This fan will "triangulate in" on a definite 
point in image space and if the rays are not triangulated in, they 
will broaden the width of the line which is the image of the 
circle.  A screen located at the place where the circle is 
sharpest is said to lie in the "tangential image plane" because 
any small line tangent to a circle centered on the optic axis will 
be in focus in this plane.


Observations

It is somewhat counterintuitive that to find the tangential image 
plane, one uses a "tangential ray fan" and the tangential ray fan 
lies in a radial plane.  Similarly, to find the sagittal image 
plane, one uses a "sagittal ray fan" and the sagittal ray fan lies 
in a tangential plane.  So live with it.  

The sagittal ray fan generates skew rays and skew rays can 
intersect the image plane slightly exterior to the plane of the 
fan.  This is usually not a large enough error to worry about.

These two image planes are actually curved.  That is, they are 
curved surfaces which tend to be somewhat axisymmetric.  For 
instance, if several concentric circles of different diameters are 
drawn centered on the optic axis and in an object plane which is 
perpendicular to the optic axis, each circle's sharp image will 
lie at a slightly different distance from the lens in object 
space.  Similarly, with the sagittal plane, different parts of a 
radial line will be sharpest in image space at different distances 
from the lens.