# Determining the distance between two points on the surface of the earth.

Question is simple:

1. Does the elliptical shape of the earth affect its radius? (Yes!!?)

2. If it is true: How?

3. How can I determine the exact distance between two points on the earth with this influence?

Notice: When I measure the distance between two points (the arc length) on any circle the length radius is unique at any point. So what about the earth? The equatorial radius of earth (from the center of the earth to the equator) is larger than the polar radius.

Also:

Can I use $d= r$ $\Delta \theta$ to determine distance between two points on the surface of the earth ?

In answer to question 3. Computing distance on an ellipsoid of revolution (oblate or prolate) is addressed in "Algorithms for geodesics". The algorithms given there are available in several different languages using GeographicLib. The methods are essentially exact for $|f|<1/50$ where $f$ is the flattening. (There's a C++ version of the algorithms which works for arbitrary values of $f$.)

Edit: Thanks to Christian Blatter I correct a misunderstanding in the "oblate ellipsoid" definition I previously used in this answer.

With "radius" we mean the distance between any given point on the Earth surface and the center of the Earth itself.

If we approximate the Earth to an oblate ellipsoid, then, cutting it by any plane parallel to the one containing the equatorial circle, we obtain a circle as section. This is because an oblate ellipsoid is obtained by revolution of a given ellipse along the minor axis. In the case of the Earth the minor axis lies on the pole-pole line.

The distance between a point on the section and the center of the section itself (i.e. the center of the circle) does not depends on the longitude of the point itself.

In other words, for points with the same latitude the radius does not depend on the longitude.

If we consider points with the same longitude, the radius depends on the latitude, instead. This is because the section obtained by considering any plane containing the the pole-pole axis is an ellipse. Moving along such ellipse the radius varies.

In summary, for points with the same longitude the radius depends on the latitude.

• I'd say that cutting the earth by a plane which is parallel to the equator we get a circle. But when we cut it with a plane containing the earth axis we get an ellipse. – Christian Blatter Jun 29 '13 at 17:54
• @Christian Blatter this is 100% true if the earth is a sphere. The OP considers an "oval shaped" earth, though. So I considered it as an oblated ellipsoid. – Avitus Jun 29 '13 at 18:01
• That's what I'm saying. – Christian Blatter Jun 29 '13 at 19:05
• @ChristianBlatter I am a bit confused now: aren't you referring to the "prolate" ellipsoid, instead? en.wikipedia.org/wiki/Prolate_spheroid – Avitus Jun 29 '13 at 19:24
• No. My earth is an oblate ellipsoid with a large equatorial circle, smaller circles in planes parallel to the equatorial plane, and a shorter axis from pole to pole. That's my last word on this matter. – Christian Blatter Jun 29 '13 at 19:32