# Latitude and longitude of points on a line

How could you get the latitude and longitude of four points (equal distance apart) on a line from $(27,-82)$ to $(28,-81)$? The four points should split the line into 5 parts.

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Given the latitudes of two points, $\beta_1$ and $\beta_2$, and their difference in longitude, $\Delta\lambda=\lambda_2-\lambda_1$, compute $\delta\in[0,\pi]$ with $$\cos(\delta)=\sin(\beta_1)\sin(\beta_2)+\cos(\beta_1)\cos(\beta_2)\cos(\Delta\lambda)\tag{1}$$ Then compute $\alpha$ using $$\tan(\alpha/2)=\frac{\sin(\Delta\lambda)\cos(\beta_1)\cos(\beta_2)}{\sin{\beta_2}-\sin(\delta-\beta_1)}\tag{2}$$

After computing $\delta$ and $\alpha$, compute $\beta'\in[-\frac{\pi}{2},\frac{\pi}{2}]$ with $$\sin(\beta')=\sin(\beta_1)\cos(k\delta/5)+\cos(\beta_1)\sin(k\delta/5)\cos(\alpha)\tag{3}$$ Then compute $\Delta\lambda'=\lambda'-\lambda_1$ using $$\tan(\Delta\lambda'/2)=\frac{\sin(k\delta/5)\sin(\alpha)\cos(\beta_1)}{\cos(k\delta/5)+\cos(\beta_1+\beta')}\tag{4}$$ Where $k$ in $(3)$ and $(4)$ ranges in $\{1,2,3,4\}$.

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The simplest is to plot $(27,-82)$ and $(28,-81)$ on your map using whatever projection you have, split the difference in pixel count into five, and plot the points that produces. That will produce a straight line on your display, which may be what you want. If you want something else, you have to define what route you want. It could be a great circle, which is the shortest distance. It could be the rhumb line, which maintains a constant bearing. Or it could be some others.

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I'm going to be using this to calculate points for the navigation of a Remote Control UAV. So it needs to be very accurate. –  AndrewFerrara Oct 21 '11 at 20:14
If you are using it for waypoints, any points will work. If they are not on the great circle, you will follow a longer path, but will get there. If not, how are you using it? –  Ross Millikan Oct 21 '11 at 20:16

Is the answer not simply found by changing each coordinate each time by a fifth of the difference in each of the lat and long elements.

(27, -82) (27.2, -81.8) (27.4, -81.6) (27.6, -81.4) (27.8, -81.2) (28, -81)

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That is a simple approximation and at this scale is probably good enough. If the start and finish points were much further apart it would be wrong as it does not take into account the curvature of the Earth: you would need to look at a great circle. –  Henry Oct 22 '11 at 9:19

You could proceed as follows:

1. Compute the distance $D$ between the two lat/lng points $P_1, P_2$ along the great circle.

2. Compute the heading (bearing) $h$ of the great circle through $P_1, P_2$.

3. Compute any point $P$ on the great circle between the two lat/lng points $P_1, P_2$, given the heading $h$ and the distance $d$ between $P_1$ and $P$, e.g. $d(n)=n D/5, n = 1..4$, in your case.

You will find formulae for all these computations on the Web, e.g. in the very good page Calculate distance, bearing and more between Latitude/Longitude points by Chris Veness.

ADDITION. I see that you included a Google Maps example. If you use Google Maps API v3, then everything is very easy:

(0) Include the Geometry library into your Web page for Spherical Computations:

<script type="text/javascript"

(1) Use var D = g.computeDistanceBetween(P1, P2); to compute the distance D between the two points.
(2) Use var heading = g.computeHeading(P1, P2); to compute the heading.
(3) Use var point = g.computeOffset(P1, D/5, heading); to compute the point at the distance D/5 from P1 along the great circle line P1 -> P2. Similarly for the other dividing points.