JonathanSturm and Bo Leuf keep excellent online journals. During some mutual correspondence a few months ago it occurred to me to ask, given our diverse locations --- Tasmania, Sweden, and the Washington DC area of the USA --- the question, "What's the farthest point on Earth from all three of us?"

My spherical geometry is next to nonexistent, and so it took me a while to figure out how to attack this poser. If our locations were on the same parallel of latitude then things would be easy: the answer would obviously be either the North Pole or the South Pole. But the coordinate conversions required to rotate the actual, arbitrary situation to that simple case were too tough for me to compute.

Finally, I realized that in the universe of vector algebra, the puzzle is (relatively) straightforward. In brief:

- Take the latitudes and longitudes of Jonathan, Bo, and
*^z* - Convert them to (x, y, z) coordinates, assuming the Earth is a perfect sphere
- Get the vectors from Bo to Jonathan and from Bo to
by subtracting their respective coordinates*^z* - Take the cross product of those two difference vectors
- Turn that cross product back into a latitude and longitude

The answer is either that location or the point on the Earth opposite to it. A cross product is a standard way of combining two vectors to create a third, perpendicular to the first two ... and that's precisely what is needed in order to find the point equidistant (and maximally or minimally far) from three places. I implemented the calculation on a simple spreadsheet.

And the answer, please? For Jonathan, Bo, and me, the site as far as possible from us is *(drum roll)* at ~37.3 South and ~11.6 West --- fortuitously near the tiny island **Tristan da Cunha** in the southern Atlantic Ocean. It's ~6,750 miles from every one of us. A few hundred people live on Tristan da Cunha. They were all evacuated to England in 1961 when a volcano erupted there; most of them returned in 1963. A couple of years ago a major hurricane struck, destroying the only pub on the island and washing many cows out to sea. Probably none of the folks on Tristan da Cunha are much interested in online journalling ....

On the opposite side of the globe, however, if Jonathan and Bo and I wish to travel an equal and minimal distance to meet together then we have to go to the antipodal solution to the equations: an empty area in the northern Pacific Ocean northwest of Midway. It's ~5,750 miles for each of us to voyage. Not within my budget, alas ....

TopicScience - TopicZhurnal - 2002-07-11

*(correlates: SturmUndLeuf, SillyAnniversaries, SirJonathan, ...)*