We can model the surface of the earth as a sphere of radius $R$. Even if in reality, the earth is more a kind of ellipsoid with an irregular surface, where each point of its surface owns an altitude in relation to the sea level.

Coordinates on earth with latitude and longitude
Coordinates on earth with latitude and longitude

Let $p1 = (\text{lat1}, \text{long1})$, $p2 = (\text{lat2}, \text{long2})$ be 2 points on the surface of the earth with a latitude and a longitude given in radian respectively. The latitude has a value between $[-\frac{\pi}{2}, \frac{\pi}{2}]$, and the longitude between $]-\pi, \pi]$. So each point is identified by unique way with this tuple: latitude and longitude. There is at least a great circle of the sphere passing both through $p1$ and $p2$, either an infinity if $p1$, $p2$ and the center of the sphere are aligned in the ambient space, or an unique one. The smallest length of a path joining $p1$ and $p2$ on the earth surface is the size of the minor circular arc of a great circle. A such arc is called a geodesic, that we note $d(p1, p2)$ its size. We put $\alpha \in [0, \pi]$ is the smallest central angle, to avoid all ambiguities, because $p1$ and $p2$ belong to a circle, and define 2 circular arcs: the minor one and the major one, respectively with a size $d(p1, p2)$ and $2\pi R - d(p1, p2)$. If $\alpha = 0$, then $p1$ and $p2$ are confused, and if $\alpha = \pi$, then $p1$ is the antipode of $p2$ (diametrically opposite, and reciprocally).

Smallest distance between 2 points on the earth surface
Smallest distance between 2 points on the earth surface

So:

\[d(p1, p2) = R\;\alpha\]

With the formula of scalar product, we can find $\alpha$, because:

\[p1\,.\,p2 = ||p1||\,||p2||\,\cos(\alpha) = R^2 \cos(\alpha)\]

But also, in the cartesian coordinates system, with $( 0, 0, 0 )$ the center of the earth, we have that a point $p = (\text{lat}, \text{long})$ on the surface of the earth has for cartesian coordinates:

\[p = \begin{pmatrix} R\cos(\text{lat})\cos(\text{long}) \\\ R\cos(\text{lat})\sin(\text{long}) \\\ R\sin(\text{lat}) \end{pmatrix}\]

Indeed, even if it means changing reference points, we can always by symmetry come back to this diagram:

Cartesian coordinates on earth
Cartesian coordinates on earth

Then, we have also:

\[\begin{align*} p1\,.\,p2 &= R^2\left(\sin(\text{lat1})\sin(\text{lat2}) + \cos(\text{lat1})\cos(\text{lat2})(\cos(\text{long1})\cos(\text{long2}) + \sin(\text{long1})\sin(\text{long2})\right) \\\ &= R^2\left(\sin(\text{lat1})\sin(\text{lat2}) + \cos(\text{lat1})\cos(\text{lat2})\cos(\text{long1} - \text{long2})\right) \end{align*}\]

So:

\[\cos(\alpha) = \sin(\text{lat1})\sin(\text{lat2}) + \cos(\text{lat1})\cos(\text{lat2})\cos(\text{long1} - \text{long2})\]

By applying the function $\text{arccos}: [-1, 1] \rightarrow [0, \pi]$ the reciprocal bijection of $\cos$ on $[0, \pi]$, we get the smallest center angle $\alpha$, and:

\[\boxed{ d(p1, p2) = R\,\text{arccos}\big(\sin(\text{lat1})\sin(\text{lat2}) + \cos(\text{lat1})\cos(\text{lat2})\cos(\text{long1} - \text{long2})\big) }\]

For a historical reason and numerical computation often the haversine fonction $\text{hav}: \mathbb{R} \rightarrow [0, 1]\quad x \mapsto \sin^2\left(\frac{x}{2}\right)$.

We have $\cos(\alpha) = 1 - 2\text{hav}(\alpha)$, so:

\[1 - 2\text{hav}(\alpha) = \sin(\text{lat1})\sin(\text{lat2}) + \cos(\text{lat1})\cos(\text{lat2}) - 2\text{hav}(\text{long1} - \text{long2})\cos(\text{lat1})\cos(\text{lat2})\]

That is to say:

\[\begin{align*} \text{hav}(\alpha) &= \dfrac{1 - (\sin(\text{lat1})\sin(\text{lat2}) + \cos(\text{lat1})\cos(\text{lat2}))}{2} + \text{hav}(\text{long1} - \text{long2})\cos(\text{lat1})\cos(\text{lat2}) \\\ &= \dfrac{1 - \cos(\text{lat1 - lat2})}{2} + \text{hav}(\text{long1} - \text{long2})\cos(\text{lat1})\cos(\text{lat2}) \end{align*}\]

Thus, we get the harvesine relation:

\[\boxed{ hav(p1, p2) = \text{hav}(\text{lat1} - \text{lat2}) + \text{hav}(\text{long1} - \text{long2})\cos(\text{lat1})\cos(\text{lat2}) }\]

Where $hav: \left(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \times ]-\pi, \pi]\right)^2 \rightarrow [0, 1]$ is defined by overloading of $hav$ on $[0, \pi]$ by a change of variables $hav(p1, p2) = hav(\alpha)$, because the couple $(p1, p2)$ defines a unique $\alpha$ (it is not reciprocal).

By using $\text{archav}: [0, 1] \rightarrow [0, \pi]$ the reciprocal bijection of $\text{hav}$ on $[0, \pi]$, we have for $\alpha \in [0, \pi]$:

\[d(p1, p2) = R\,\text{archav}(\text{hav}(\alpha))\]

In addition $\forall x \in [0, 1]$:

\[\text{archav}(x) = 2\,\text{arcsin}(\sqrt{x})\]

And by putting $y = \text{arcsin}(x)$ and $x = \sin(y)$ with $\tan(y) = \sqrt{\dfrac{\sin^2(y)}{1 - \sin^2(y)}}$ we have for $x \neq 1$:

\[\text{arcsin}(x) = \text{arctan}\left(\sqrt{\dfrac{x^2}{1 - x^2}}\right)\]

So:

\[d(p1, p2) = 2\,R\,\text{arctan}\left(\sqrt{\dfrac{\text{hav}(p1, p2)}{1 - \text{hav}(p1, p2)}}\right)\]

In order to have a valid formula for the case where $\text{hav}(p1, p2) = \text{hav}(\alpha) = 1$, that is to say for $\alpha = \pi$, we introduce the function:

\[\text{atan2}: \mathbb{R}^+ \times \mathbb{R}^+ \backslash \left\{0, 0\right\} \quad (x,y) \mapsto \left\{ \begin{array}{ll} \text{arctan}\left(\frac{x}{y}\right) & \text{if } y > 0 \\\ \frac{\pi}{2} & \text{if } y = 0 \end{array} \right.\]

So:

\[\boxed{ d(p1,p2) = 2\,R\,\text{atan2}\left( \sqrt{\text{hav}(p1, p2)}, \sqrt{1 - \text{hav}(p1, p2)} \right) }\]

We can check that the coherence of the formula where $d$ is well a metric distance, it is a positive function, symetric $d(p1, p2) = d(p2, p1)$, and homogeneous $d(p, p) = 0$ respecting the triangle inegality.

In practice, the radius of the earth is $R = 6371\,km$, and the latitude and longitude are given often in decimal degree and not in radian.


Creative Commons Attribution 4.0 International License:
Also read
Comments