The PDF shown here was sent in by Seth Nelson, a student here in the Treasure Valley. 

Seth looked at the distance between two points on Earth (Boise and Accra). He found his distances using the Plate Carrée map projection and compared the answer he got with answers from an online distance calculator and with the formula provided by NASA. 

He took it a step further and analyzed how latitude and longitude values impacted the error (based off of the online distance calculator mentioned earlier) on pages 3 & 4 of his pdf. 

As well, check out this awesome Desmos visualization of calculations on the Plate-Carrée projection that he created!

Here is the distance calculator he used for distances between points given their latitude and longitude

Linked here is a model created by a local high school teacher upon receiving this problem, using the free online graphing tool, Desmos.

Check it out and see if you can make sense of the formulas! 

Using some skills I learned in Multivariate Calculus, I added to that Desmos visual to include the circle going through our two points. 

To get an idea of how I am decided on my goal for solving this problem, and how I am visualizing what's going on, I've described my thinking using an orange in the first video to the left. 

You can use the sliders in this Desmos visual to further visualize how latitude and longitude work.

The second video shows my actual calculations which rely on trigonometry and alludes to where NASA's formula likely came from. 

This Desmos visual shows our specific points on the Earth and lines connecting them and the center of the Earth to form a triangle.

This video doesn't directly generate the formula used by NASA, but I believe their formula is derived from similar thinking, and possibly the use of trigonometric identities.

The globe that I own is an inflatable one I purchased on Amazon several years ago (here is the link if you want your own). My only complaint is that the globe isn't nearly as round as it appears to be in the images provided by the seller, but it's been good enough for my purposes. 

To use a globe to approximate the distance, I used a wet erase marker to approximate the points in Oregon and South Caroline and then a flexible measuring tape that I usually use for crafts. I measured the distance to be 9.5cm on the globe. 

I then approximated the full distance two ways:

• Multiply by the provided scale factor of 46,500,000

• Measured the provided 1500km distance scale to be 3.1cm and then used that to convert

The two answers I got were

• 4,417.5km using the provided scale factor

• 4,596.77km using my measured ratio

See work shown to the left.

Both of those answers are a little bit off. Inspired by Seth's work, I found the percent error for each to be 10.13% and 14.60%, respectively. Not too shabby! 

For this approach, I knew that if I had a parametric equation, I could use Calculus to find the arc length of this curve. So the video here discusses the challenge of finding the parametric equation, and then applying the existing formula for arc length.

Here are links to some of the things I show in the video:

For calculation purposes I recreated the Desmos visualization linked earlier on this page using Radians and the radius of the Earth in km, and added a fun parameterization of a curve that goes through the two desired points. With some more time and google searching, I was able to finally get the actual parameterization of the circle (see in Desmos). 

• Read more about finding arc length with parametric curves here, and the extension of it to 3D here.

• Equation of a plane formula

• Here is an introduction to spherical coordinates

• Here is an introduction to cross products

• This spreadsheet was used to calculate cross products.

• PDF of Maple output for attempt 1 (very far off)

• PDF of Maple output for attempt 2 (close! but parameterization wasn't a circle)

• Math Stack Exchange thread on how to parameterize a circle

• PDF of Maple output for attempt 3, spot on!