Calculating the Shortest Distance from an Ionosonde to a QSO Path Using Sage, Cross Products, and Geometry

I've run into an interesting math problem. Project TouCans occasionally makes dx contacts. That's not a problem, that's actually really awesome! However, KO6BTY and I would like to be able to map the F2 skips our radio signal makes on its way to the receiving station. We have plenty of data about the F2 layer of the ionosphere that is captured by ionosondes around the world. The issue is how do we determine which set of ionosonde data to use? Our first guess is that we should use the ionosonde closest to the path of the signal at any point. OK... Now, how do we determine that?

First, we'll calculate the shortest distance from each ionosonde to the path of the QSO between ham radio stations. Then, we can sort the distances so that we can only include the ionosondes that are closest to the path. Armed with that data, we can perhpas used the distances to calculate a weighted, estimated height of the F2 layer at any point along the signals path as it skips from station to station.

Here's an example of the first problem we're trying to solve, namely what's the shortest distance from a given Ionosonde to the path between the two amateur radio stations involved in the QSO?

The yellow line is the path between our station and PG4I in the Netherlands. The red line is the perpendicular line from the ionosonde located in Alpena, Michigan to the path of our signal, (that's the shortest distance between the ionosonde and the signal path.)

But, how do we find the distance along the red line? That problem was handled and explained quite handily here, but I wanted to see the geometry, so we're going to model it using Sage cells on this page. To see the graphical demonstration of each step, click the 'Activate' buttons.

First, let's draw the two station locatiosn and the ionosonde location. Each location is denoted by a vector that extends from the origin (x=0, y=0, z=0) out to the point.

Taking the cross product of the vectors that point from the center of the Earth to the two radio staions defines a plane that contains both stations and the path between them.

The vector result of the cross product is shown a thicker green vector. We're going to use it again in just a moment. You might remember from geometry class that a line perpindicular to a plane defines a family of planes that will all be perpendicular to that original plane. Let's use that to find the perpendicular path from the ionosonde to the path between the radio stations. If we take a cross product of the green cross product vector we just calculated, (remember, it's perpendicular to the plane that contains the two radio stations and the path between them), and the vector to the ionosonde, we'll get back a vector that defines a plane parallel to and the green cross product vector, (and therefore perpendicular to the plane between the stations), and the ionosonde's position vector.

With the above result, you can see that a path from the ionosonde along the plane defined by the cross product in the last step is in fact a perpendicular line from the ionosonde to the path between the QSO stations. The only remaining issue is that even though we can see the point where the intersection occurs, we don't know where it is. We don't have a numeric latitude and longitude. One more cross product will provide the answer. If we take the cross product of the two vectors produced in the last two steps, we'll get a vector that has to be perpendicular to both of the original vectors. Since those vectors are perpendicular to the two planes they defined, the only place the result of the new cross product can exist is along the intersection of the two perpendicular planes! Here's what that looks like:

I've taken the liberty of converting the Cartesian coordinates back to Earth based spherical coordinates. From that point, it's a simple matter to use the haversine plugin in Datasette to determine the distance from the Ionosonde to the path of the QSO. If you're curious about the code that created these demonstrations, I've stashed it away at the repo for the ham radio and Reverse Beacon Netowrk logger KO6BTY and I have been working on.

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