Archimedean Metronomy II

Archimedean Area Metronomy by Descent

The area of a  spherical dome is equal to the area of a disk, the radius of which is the length of the segment joining the top vertex of the dome to its boundary. Equivalently, if $S$ is this vertex of the spherical dome and $A$ any point on the boundary circle, then the area of the spherical dome is equal to the area of a disk with radius $SA$. If the spherical dome is the whole sphere, then $S$ and $A$ are antipodes, and thus the area of the sphere is $\pi (SA)^2=4 \pi \rho^2$, where $\rho=SA/2$.



\[S C=h, O C=R-h, O A=R\]
Hence, we have:
\[A C^{2}=R^{2}-(R-h)^{2}=2 R h-h^{2}\]
\[S A^{2}=A C^{2}+C S^{2}=2 R h\]

Thus, we identify $SA$ with the geometric mean of $2SO$ and $SC$:
\[S A=\sqrt{2 S O \cdot S C}\]

We conclude that the area of a spherical dome is measured by descending to the area of the corresponding disk in a measure-preserving manner.

Let \( A_{1}, A_{2} \) be two points on the half-sphere; we consider the domes delimited by the planes passing through these points. The dome with \( A_{1} \) on its boundary, \( S A_{1} \), has area \( E\left(S A_{1}\right)=\pi\left(S A_{1}\right)^{2} \), and similarly \( E\left(S A_{2}\right)=\pi\left(S A_{2}\right)^{2} . \) 

Concomitantly, we may consider the morphism $\phi: A \rightarrow B$ from the half-sphere onto the unit disk, such that $B$ lies on the same vertical plane as $A$, and $2 OB^2=SA^2$.

The images are respectively the disks of radius \( O B_{1} \) and \( O B_{2}, \) with areas \( \pi\left(O B_{1}\right)^{2}=\frac{\pi}{2}\left(S A_{1}\right)^{2} \) and \( \frac{\pi}{2}\left(S A_{2}\right)^{2} . \) The area of the spherical zone between these disks is the following:
\[E(\text {zone})=\frac{\pi}{2}\left(S A_{2}\right)^{2}-\frac{\pi}{2}\left(S A_{1}\right)^{2}=\frac{1}{2}\left(E\left(\operatorname{dome}\left(S A_{2}\right)\right)-E\left(\operatorname{dome}\left(S A_{1}\right)\right)\right)\]

Consequently, the morphism $\phi$ is a measure-preserving bijection from the half-sphere onto the unit disk.

This measure-preserving property is actually precisely what underlines the Archimedean equi-areal projection of the sphere onto the plane.

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