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Rhumb line by slide rule
From: Paul Hirose
Date: 2016 Nov 02, 14:41 -0700

```Formulas for rhumb line course from point 1 to point 2:

a = 45 + lat1 / 2
b = 45 + lat2 / 2
y = π * longitude difference
x = 180 * (ln tan b - ln tan a)
course = arc tan (y / x)

The longitude difference is 0 to 180 degrees.

A slide rule solution may be easier with some modifications to the x and
y formulas.

• Replace natural (base e) logarithms with common (base 10) logarithms.
We need only multiply the latter by 2.303 (the natural log of 10).

• Divide both equations by π to combine the constants in one place. That
has no effect on the computed course.

• Replace logarithm subtraction by division, since log a - log b = log
(a / b).

After those modifications, the equation for x contains the expression
180 / π * ln 10. To practical accuracy (better than 1 part in 1000),
that equals 132. Then the equations for rhumb line course are

a = 45 + lat1 / 2
b = 45 + lat2 / 2
y = longitude difference
x = 132 * log (tan b / tan a)
course = arc tan (y / x)

In the equation for x, the log is negative if the expression in
parentheses is less than 1. That's no problem. Read negative logs on the
L scale as if it's numbered in reverse, i.e., with 0 at the right

Negative natural logs are not so easy. In the original equation these
occur when a latitude is negative (south). In that case, 45 + lat / 2 is
less than 45, hence its tangent is less than 1 and its logarithm is
negative. There's a way to read it, however. If Ln is on the slide, set
a C index to the tangent on D, set the cursor to the D index, read the
negative natural log at the cursor on Ln.

For example, lat = -20. Then ln tan (45 + -20/2) = ln tan 35. Set a C
index to tan 35 on D in the obvious way. Set the cursor to the D index
and read -.357 on Ln. There's a similar trick if the Ln scale is on the
body of the slide rule.

If both points are south latitude, a simple dodge is to pretend the
latitudes are north. The resulting course is measured from the south.

But the following example is in north latitude, and I'll use the common
log version of the formulas. Initial latitude = +30, final latitude =
+50, longitude difference = 20°. Destination is to the east.

a = 45 + 30 / 2 = 60
b = 45 + 50 / 2 = 70
y = 20
x = 132 * log (tan 70 / tan 60) = 26.4
course = arc tan (20 / 26.4) = 37.2

Compare that to 37.21717 per
http://geographiclib.sourceforge.net/cgi-bin/RhumbSolve

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