Symmetry orbits and curvature maps

The cornea is the main refracting surface of the human eye. Its front (anterior) surface is approximately 1.3 cm^2 with an average radius of curvature of about 7.8mm.  Typical computer algorithms for corneal curvature measurement (known as keratometry) are based on projecting a pattern of concentric rings of light onto the anterior surface of the cornea and numerically determining the relative separation between the images of these reflected rings of light. This models the  anterior surface of the cornea as a highly polished spherical mirror.  By sampling the curvature at specific circularly equidistant points, a numerical model for the surface curvature may be obtained.

The corneal curvature, (k), and its refractive index, (n), contribute to determining the surface's ptical refractive power k(n-n'), where n' is the reference refractive index of the air. Most of the light refraction takes place at the surface of the cornea, which has refractive index n=1.3376.

Light then passes through the aqueous humor (n=1.336, close to the refractive index of  water) to the lens (n=1.386 to 1.406, where it is refracted further) and through the vitreous humor to the retina at the back of the eye.

The standard unit of refractive optical power is the diopter (D) and is defined to the inverse of the radius of curvature, or approximately equal to the inverse of the focal length of the refracting element. One diopter equals one inverse meter (m^{-1}). For example, using the standard keratometric index n=1.3375, a cornea with a curvature of 7.50mm at 0 degrees has  power (1/0.0075)x0.3375=45D, whereas if the curvature at 90 degrees is 7.25mm, the power is 46.50D.

The difference between these steep (maximum) and the flat (minimum) curvatures, as illustrated in the diagram above (From W.F. Long), is the amount of regular astigmatism present in the optics of the eye, which interferes with a sharp formation of the image on the retina. In the present example, the regular astigmatism is 1.50D. 

The graph below shows a typical curvature contour near the apex of the cornea:

In analogy with the D_2 symmetry orbits derived  for the  visual field data we can evaluate the D_2 orbit

C, vC, hC, vhC

for the curvature contours by applying the corresponding planar reflections {v, h} and double reflection {vh} to the support of the contour. The following graph shows the contour for the difference

C-hC 

shifted by 7D (red contour) to allow its polar plotting relative to that reference value (black contour).

Recalling the multiplication table for D_2 we now know that

h(C-hC) = hC - h^2 C = hC - C = - (C - hC)

so that the contour is horizontally anti-symmetric. 

Here is the 7D-shifted contour for C - vC:

showing its vertical anti-symmetry, and here

is the contour for C - hvC, showing its central (double-reflection) anti-symmetry.

Again as introduced earlier in the study of the  visual field , we seek to describe the orbits invariants for D_2, namely:

I1 = C + vC + hC + hvC,

Iv = C + vC - hC - hvC, 

Ih = C - vC + hC - hvC,

Ihv = C - vC - hC + hvC. 

Here is the contour for the full symmetric invariant I1,

which is vertically, horizontally and centrally (double-reflection) symmetric. Here is the contour for the invariant Ih,

which is horizontally symmetric, vertically and centrally anti-symmetric.  Here

is the contour for Iv, which is vertically symmetric, horizontally and centrally anti-symmetric. And here

is the contour for Ihv, which is centrally symmetric, and vertically and horizontally anti-symmetric. These are then the invariant summaries of a single contour under the D_2 orbit applied to its support. Clearly, its support allows for many more refined groups of symmetries. We will, in the future, revisit these contours after learning the general method for determining the orbit invariants. 

First revised 06/20/2011
These  postings are based on "Symmetry Studies" An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)