How to calculate rhumb line distance “due East” between two points

First off, I am very new to GIS, so an explanation to the given solution/formula would be great.

How do I calculate a constant bearing distance (rhumb line) of "due East"/"due West" from point1 to point2?

p1: Lat: 40 N Long: 110 W

p2: Lat: 40 N Long: 75 E

I know there should be a formula for this, but everything I have found seems a too advance, since I have that lat1 = lat2.

Edit: I have earlier found the shortest great-circle distance for these points, by the formula (converted to degrees): arcos(sin(40) * sin(40) + cos(40) * cos(40) * cos(175)) * (180 / pi) * 111 = 11085.58km This result correspond to what I get from this site: http://www.movable-type.co.uk/scripts/latlong.html

According to the same site, the rhumb-line should give a result of 14910km for due West, but I want to calculate this myself.

• Welcome to GIS SE. Please take a look at the Tour. And please also edit some extra info into the body of the question for clarification: e.g. what do you mean exactly by "calculate a constant bearing"? You're given two locations and you're given a bearing. Do you wish to calculate the along-rhumb-line distance? Or maybe you actually wish to calculate the initial and final bearings of the great-circle route? – Martin F Sep 7 '15 at 20:02
• Yes, I wish to calculate the along-rhumb-line distance(km) from point 1 to point 2 – Giando Sep 7 '15 at 20:38
• I'm not matching your results on the movable type page. I'm getting 14910km on the rhumb line distance calculator (which is due West on your original coordinates. It is matching within 40km to the value I get in Esri's projection engine using WGS84 (40 75 to 40 -110). – mkennedy Sep 8 '15 at 20:58
• What value for Earth's radius are you using? – Martin F Sep 11 '15 at 4:16

Take a look at my answer to a related question Proof of part of haversine formula?

The first part is directly relevant:

The radius, r, of the small circle joining all points at latitude, φ is

r = R cos φ

where R is the radius of the sphere.

However, instead of the chord length of a straight line joining two points on the same latitude (a requirement of that question), you want the simpler arc length of part of the small circle. It is simply

AD = r dλ

where dλ is the difference in longitude of A and D.

As always in spherical trigonometry, things get more complicated in the case of an ellipsoid. The first equation above relates only to the globe being a simple sphere.

Assuming an Earth radius of 6,371km, the above formulas yield a result of 15,758km for the rhumb-line distance between your two points.