A schema of a 3D eyeball sphere model is shown in
Figure 2, where
O and
N are the center and the north pole of the sphere, respectively. In
Figure 2, angle θ is formed by lines
ON and
OA where
A is an arbitrary point on the sphere. The south pole of this sphere model corresponds to the optic nerve head; the top dome of the sphere represents the cornea. In this coordinate system, the length of the shortest arc (i.e., the geodesic distance) from point
A to south pole
S on the sphere is
L =
R × (π − θ), where
R is the sphere radius. To clarify terminology, a line of latitude for point
A is “latitude
A” and its corresponding concentric circle after projection on the south pole plane is “circle
A”. The change in arc length during the flatmount process is assumed to be negligible. Therefore arc length
L is identical to the radius of “circle
A” on the 2D flatmount plane. With
Figure 2, the length of “latitude
A” can be computed as
R × sin θ × 2π. The perimeter of “circle
A” on the south pole plane is
L × 2π. If the sphere is evenly cut into four lobes unfolded on the south pole plane, some gaps would occur between the neighboring lobes. This is illustrated in
Figure 3(a). The sum of the lengths of all the gap arcs (
G) on “circle
A” is
G =
L × 2π −
R × sin θ × 2π. From the previous discussion,
L =
R × (π − θ). Therefore the following equation governs the relationship across the gap arc length (
G) in the 2D south pole plane, the geodesic distance from point
A to the south pole on the sphere, and the sphere radius:
\begin{equation}G = 2\pi \times \left( {L - R \times \sin \left( {\pi - \frac{L}{R}} \right)} \right)\end{equation}