Destination Point from Bearing
Find where a bearing and a distance take you, plus why holding that heading lands you 3,000 km from where the great circle would.
Two destinations are shown because they are genuinely different places: one follows the great circle that starts on your bearing, the other holds that bearing the whole way. Nothing is uploaded.
Holding a heading is not following a great circle
The great circle from London to New York starts on 288°. Follow that circle for 5570 km and you arrive in New York — to within a millimetre, because that is what a great circle is. Hold the needle on 288° for the same distance and you land 3232 km away.
| Method | Lands at | Miss |
|---|---|---|
| Following the great circle | New York | 0.000 m |
| Holding 288° throughout | 67.26°, -95.56° | 3232 km |
The miss is more than half the length of the journey. To reach New York on a single fixed heading you would steer 258° instead — a different number entirely — and travel 224 km further. Both are legitimate routes; they are simply not the same route, and the initial bearing belongs to only one of them.
A great circle climbs above both its ends
The shortest route between two points at the same latitude does not follow that latitude. It bulges toward the nearer pole, and Clairaut's relation fixes exactly how far.
| Both points at | Longitude apart | Route reaches | Climb |
|---|---|---|---|
| 45° | 60° | 49.1° | +4.1° |
| 45° | 120° | 63.4° | +18.4° |
| 60° | 120° | 73.9° | +13.9° |
| 20° | 120° | 36.1° | +16.1° |
The wider the separation, the higher it climbs. Two points at 45° separated by 120° of longitude are joined by a route that reaches past 63°. Two exceptions prove the rule: an east-west route along the equator stays on the equator, because the equator is itself a great circle, and a due-north route reaches the pole exactly.
The arithmetic checks out
Travel a distance on a bearing and measure the result back, and you get the distance you asked for — at every leg length from 100 km to most of the way round the planet.
| Asked for | Landed at | Measured back |
|---|---|---|
| 100 km | 52.139°, 0.908° | 100.000 km |
| 1,000 km | 57.337°, 11.690° | 1000.000 km |
| 5,000 km | 59.859°, 84.242° | 5000.000 km |
| 10,000 km | 26.170°, 127.885° | 10000.000 km |
| 15,000 km | -13.928°, 148.804° | 15000.000 km |
Some sanity checks that also happen to be the interesting cases. Due north from the equator keeps the same meridian. Due east from the equator keeps the same latitude, because there the two coincide. Due east from 45° does not — it drifts south, which is the same fact as the bulge above, seen from the other side.
How to use
- Enter a starting coordinate.
- Set a bearing and a distance.
- Two destinations are shown — they are different places.
- One follows the great circle; one holds the heading.
Frequently asked questions
Why are there two destinations?
Because setting off on a bearing and holding a bearing are different things. Following the great circle that starts on your heading means turning continuously all the way; holding the needle on one number traces a rhumb line instead. Over a transatlantic distance the two land about 3,200 kilometres apart.
Which one is right?
Both, for different questions. The great circle is the shortest route and is what an aircraft flies. The constant heading is what you get if you actually steer one number, and it is what dead reckoning gives you. The mistake is assuming the initial bearing describes the whole journey.
So how do I reach a place on one fixed heading?
Use the rhumb bearing rather than the great-circle one — a different number. For London to New York it is about 258 degrees rather than 288, and the route is around 220 kilometres longer. That is the trade for never having to turn.
Why does a great circle climb above both its endpoints?
Because the shortest route between two points at the same latitude does not follow that latitude — it bulges toward the nearer pole. Clairaut’s relation fixes exactly how far: two points at 45 degrees separated by 120 degrees of longitude are joined by a route reaching past 63.
Are there routes where it does not climb?
Two, and they are instructive. An east-west route along the equator stays on the equator, because the equator is itself a great circle. And a due-north route reaches the pole exactly, which is the extreme case of the same rule.
Does travelling east keep my latitude?
Only on the equator. Set off due east from 45 degrees north on a great circle and you drift south, which is the same fact as the poleward bulge seen from the other side. Holding a due-east compass heading does keep the latitude, and is the longer path.
How accurate is this?
The round trip closes to under a metre at every leg length from 1 kilometre to 9,000, and the bearing you set off on is reproduced to within a millionth of a degree. The spherical model is used throughout, so distances carry the same few tenths of a per cent that any sphere does.
Does this send anything anywhere?
No. Everything is computed in your browser, and nothing is uploaded.
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