[Math] Given latitude and longitude, how to find central angle and arc length of spherical triangle

algebra-precalculusspherical trigonometrytrianglestrigonometry

Lewis and Clark followed several rivers in their trek from what is now Great Falls, Montana, to the Pacific coast. First, they went down the Missouri and Jefferson rivers from Great Falls to Lemhi, Idaho. Because the two cities are on different longitudes and different latitudes, we must account for the curvature of the Earth when computing the distance they traveled. Assume the radius of Earth is 3960 miles.

Great Falls is at approximately 47.5 degrees N, and 111.3 degrees W. Lemhi is approximately 45.5 degrees N, and 113.5 degrees W. (We will assume the rivers flow straight from Great Falls to Lemhi on the surface of Earth).

  1. Apply the Law of Cosines for Sides of a spherical triangle to find the angle between Great Falls and Lemhi. (To find S2 and S3, subtract the latitude from 90 degrees. For example S2 = 90-47.5. To find a1, subtract the longitudes).

  2. Then find the length of the arc joining Great Falls and Lemhi. (Recall: s = θr, where θ is in radians. Assume the radius of Earth is 3960 miles ).

Best Answer

Let $R \in \mathbb{R}$ be the earth's radius : $R \leftarrow 3960$

Any of the points we are taking about here are non-zero, and therefore can be expressed in spherical coordinates as $\vec{v} = R.\vec{u}$ where $\vec{u}$ is a unit vector whose $\theta$ is the azimuth and $\varphi$ is the inclination.

Generalities

Phi

We now have the relation : $\varphi_p = 90 - \text{northenDegrees}(p)$

Because the degrees toward north are $0$ on the equator and 90 on the north pole, whereas phi is $0$ on the north pole and goes the other way (hence the *= -1)

Illustrations :

Latitude and Longitude

Spherical Coordinates

Theta

And $\theta$ seems to stay the same

In this instance

Conversion

Let's first convert the points to spherical coordinates :

  • Great Falls $(47.5~N, 111.3~W) = (R, 90 - 47.5, 111.3) = (R, 42.5 °, 111.3°) = (R, \varphi_F, \theta_F) = F$
  • Lemhi $(45.5~N, 113.5~W) = (R, 44.5°,113.5°) = (R, \varphi_L, \theta_L) = L$

Then we let some computer convert it in radians, but we're close to $\frac{\pi}{4}$ and $\frac{2.\pi}{3}$

Problem

Apply the Law of Cosines for Sides of a spherical triangle to find the angle between Great Falls and Lemhi.

You might be meaning the spherical law of cosines ?

It seems like what you are looking for is

$cos(\widehat{FL}) = F\bullet L = r_Fr_L(\sin \varphi_F \sin \varphi_L \cos (\theta_F - \theta_L) + \cos \varphi_F \cos \varphi_L)$ $= R^2\sin \varphi_F \sin \varphi_L \cos(\theta_F - \theta_L) + \cos \varphi_F \cos \varphi_L$

Here again a computer will help us out !

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