Budapest $(B)$ has a geographic latitude of $\varphi_{1}=47.5^{\circ}$ and an eastern longitude of $\lambda_{1}=19.1^{\circ}$. The earthquake-stricken Agadir $(A)$ is located at $\varphi_{2}=$ $30.5^{\circ}$ north latitude and $\lambda_{2}=9.6^{\circ}$ west longitude. How far are $A$ and $B$ from the Equatorial plane and the Earth's axis of rotation? How far are they from the $0^{\circ}$ and $90^{\circ}$ meridian (or half-meridian) planes? Based on this, what is the straight-line distance (i.e., the chord of the sphere) between $A$ and $B$? What is the distance on the Earth's surface, measured along the shortest arc? - How far is the deepest point of the $A B$ spherical chord from the Earth's surface? (Assume the Earth is a sphere with a radius of $6370 \mathrm{~km}$, and ignore the altitude above sea level.) - Provide a general formula for determining the distance between any two points on the Earth's surface based on their geographic coordinates.
Solution: I. 1. The distance of a point $P$ on the Earth's surface from the equatorial plane $E$ and from the axis of rotation, in the plane of the meridian circle of $P$, is $z=R \sin \varphi$, and $d=R \cos \varphi$ (upper part of the 1st figure).

![](https://cdn.mathpix.com/cropped/2024_05_02_f68fa3fdfac5ca8a2cdbg-1.jpg?height=961&width=610&top_left_y=265&top_left_x=736)

When drawing the Earth in its usual position and considering southern latitudes as negative, $z$ should also be considered with its sign. For $B$, $z_{1}=4696, d_{1}=4304$; for $A$, $z_{2}=3233, d_{2}=5488$ (all distances are given in km, so we do not exceed the accuracy of four-digit tables anywhere).

2. The distance from the plane of the $0^{\circ}$ meridian $H_{1}$, in the plane of the parallel circle of $P$, from the previously mentioned $d$ and the longitude $\lambda$ is $y=d \sin \lambda$. (lower part of the 1st figure). Considering western longitudes as negative, i.e., if $-180^{\circ}<\lambda \leq 180^{\circ}$, $y$ also has a sign. Similarly, the distance from the planes of the $\pm 90^{\circ}$ meridians $H_{2}$ is $x=d \cos \lambda$. For $-B$, $y_{1}=1408, x_{1}=4066$; for $A$, $y_{2}=-915, x_{2}=5411$.

(Thus, we essentially gave the points in a spatial rectangular coordinate system known from descriptive geometry, with the origin at the Earth's center $O$, the $X$- and $Y$-axes intersecting the equator at $\lambda=0^{\circ}$ and $\lambda=+90^{\circ}$ (south of Accra and Calcutta), and the $Z$-axis at the North Pole.

3. From our results, we can obtain the edges $a, b, c$ of the rectangular parallelepiped $T$ whose two vertices are $A$ and $B$, and whose faces are parallel to $E, H_{1}, H_{2}$. From the edges, we can then find the distance $A B=h$ measured along the chord, as $A B$ is the space diagonal of $T$. Specifically, $a=x_{2}-x_{1}=1345, b=y_{1}-y_{2}=2323, c=z_{1}-z_{2}=1463$, and thus $h=\sqrt{a^{2}+b^{2}+c^{2}}=3057 \mathrm{~km}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_f68fa3fdfac5ca8a2cdbg-1.jpg?height=464&width=309&top_left_y=1893&top_left_x=884)

Figure 2
4. The shortest arc $A B$ on the Earth's surface is cut by the plane $A B O$. The intersection is a great circle (Figure 2), so its radius is $R$. Let $\angle A O B=\vartheta$, then from the cosine rule in $\triangle A O B$,

$$
\cos \vartheta=1-h^{2} / 2 R^{2}
$$

From this, $\vartheta=27.78^{\circ}$, in radians 0.4848, and thus the distance measured on the surface $A B=R \vartheta=3088 \mathrm{~km}$.

5. The deepest point of the chord $A B$, i.e., the point closest to $O$, is the midpoint $F$, and its depth is $m=R(1-\cos \vartheta / 2)=187 \mathrm{~km}$.

II. Based on the above, we provide the general formula for calculating the angle $\vartheta$. The arc can then be easily obtained. The lengths of the edges of $T$ - possibly with opposite signs, but this is not significant due to the subsequent squaring:

$$
\begin{aligned}
& x_{1}-x_{2}=d_{1} \cos \lambda_{1}-d_{2} \cos \lambda=R\left(\cos \varphi_{1} \cos \lambda_{1}-\cos \varphi_{2} \cos \lambda_{2}\right) \\
& y_{1}-y_{2}=d_{1} \sin \lambda_{1}-d_{2} \sin \lambda_{2}=R\left(\cos \varphi_{1} \sin \lambda_{1}-\cos \varphi_{2} \sin \lambda_{2}\right) \\
& z_{1}-z_{2}=R\left(\sin \varphi_{1}-\sin \varphi_{2}\right)
\end{aligned}
$$

With these, rearranging and using known trigonometric identities,

$$
\begin{gathered}
\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=R^{2}\left[\cos ^{2} \varphi_{1}\left(\cos ^{2} \lambda_{1}+\sin ^{2} \lambda_{1}\right)+\cos ^{2} \varphi_{2}\left(\cos ^{2} \lambda_{2}+\right.\right. \\
\left.\left.+\sin ^{2} \lambda_{2}\right)-2 \cos \varphi_{1} \cos \varphi_{2}\left(\cos \lambda_{1} \cos \lambda_{2}+\sin \lambda_{1} \sin \lambda_{2}\right)\right]= \\
=R^{2}\left[\cos ^{2} \varphi_{1}+\cos ^{2} \varphi_{2}-2 \cos \varphi_{1} \cos \varphi_{2} \cos \left(\lambda_{1}-\lambda_{2}\right)\right]
\end{gathered}
$$

Thus,

$$
\begin{aligned}
& h^{2}=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}=R^{2}\left[\left(\cos ^{2} \varphi_{1}+\sin ^{2} \varphi_{1}\right)+\right. \\
& \left.+\left(\cos ^{2} \varphi_{2}+\sin ^{2} \varphi_{2}\right)-2 \sin \varphi_{1} \sin \varphi_{2}-2 \cos \varphi_{1} \cos \varphi_{2} \cos \left(\lambda_{1}-\lambda_{2}\right)\right]= \\
& =2 R^{2}\left[1-\sin \varphi_{1} \sin \varphi_{2}-\cos \varphi_{1} \cos \varphi_{2} \cos \left(\lambda_{1}-\lambda_{2}\right)\right]
\end{aligned}
$$

Finally, from (1),

$$
\cos \vartheta=\sin \varphi_{1} \sin \varphi_{2}+\cos \varphi_{1} \cos \varphi_{2} \cos \left(\lambda_{1}-\lambda_{2}\right)
$$

From this, $\vartheta$ can be uniquely determined - clearly $0<\vartheta \leq 180^{\circ}$.

Jenő Székely (Pécs, Nagy Lajos g. III. o. t.)

Note. Our general result can also be obtained by applying the cosine rule of spherical trigonometry to the spherical triangle ÉAB, where É is the North Pole. Several solutions, aiming only at the final result, immediately wrote this down and considered it a solution, or a second solution. Here, the goal was to show that the formula can be derived without systematically developing the theorems of spherical trigonometry, beyond the numerical example.