Introduction

This post is the first in a series about "Time Difference of Arrival" methods and it covers just the facts about hyperbolas that you need to know in order to understand TDoA. If you're confident that you already know enough about hyperbolas (including how they work on the surface of a sphere), you can skip to the next post in the series, otherwise read on...

Note that I'll be using names like $R_A$, $R_B$, and $S_x$, in this post. This is a departure from names like $F_1$, $F_2$, and $P_x$ which are more commonly used in geometry discussions, but I am intentionally foreshadowing where (R)eceiving and (S)ending stations will appear in future posts.

What is a Hyperbola?

For our purposes, it will be more useful to work with "branches of hyperbolas" instead of "hyperbolas" and I'll be using the term "branch" as an abbreviation for "branch of a hyperbola". As for a "hyperbola", let's just say that it is a pair of branches whose respective constants differ only in sign. This will make more sense once we discuss "branches" in the next section.

What is a Branch?

When it comes to defining "branch", forget any of the algebraic equations involving $x$ and $y$ that you might have seen in the past because we'll be using a much simpler geometric definition:

That's it, and we can already see that branches are related to differences, which is why they come into play in TDoA.

In the next figure, the branch with $M={^{+}10}$ is highlighted along with example points on it and the distances that demonstrate that they satisfy definition. Its twin branch with $M={^{-}10}$ appears very faintly on the left and together these two branches make up the hyperbola with $M={^{\pm}10}$

Families of Branches

For any two foci there is actually an infinite family of branches, each member having it's own value of $M$ in the allowed range. The following figure shows 11 members of the family of $R_A$ and $R_B$ along with their $M$ values.

But a family actually covers the entire plane. The next figure gives you a sense of what that really looks like by shading the infinite continuum of branches with different colors.

No matter what point $S$ on the plane you pick, it will be on exactly o ne of the branch of that family.

Hyperbolas on a Sphere

Up until this point, we have been discussing hyperbolas and their branches on the flat, Euclidean plane. But when we perform TDoA analysis over substantial portions of Earth's surface, we'll be working on the surface of a sphere a "hyperbola" becomes a "spherical hyperbola". Luckily, everything we've said about regular hyperbolas and their branches still holds for spherical case, but there are some intereesting differences.

One such difference results from the fact that there are two straight lines between any two points on a sphere, one "short" and one "long". Since this post is primarily aimed at amateur radio operators, I'll assume that you are familiar with this having been exposed to the concept of short vs long path propagation. This short/long degree of freedom means that there are actually four different branch families for any two given foci on a sphere, depending on whether $\overline{S_x R_A}$ is long or short and $\overline{S_x R_B}$ is long or short. I call these the short/short, short/long, long/short, and long/long families.

Another difference is that branches on a sphere no longer have infinite length but curve back on themselves to become spherical ellipses. So a spherical hyperbola is composed of two branches each of which is a spherical ellipse, one branch having $M={^{+}10}$ the other having $M={^{-}10}$.

Now, recall that ellipses (like hyperbolas) have two foci so it's reasonable to wonder "What is the second focus for each of the branches?" It turns out that the additional points of interest in the spherical case are the antipodal points of the $R_A$ and $R_B$ foci.

Let's see if we can visualize all these changes that the spherical case introduces. The next figure is a lat/lon projection of Earth along with the SS, SL, LS, and LL hyperbola continua associated with foci at Hawaii and Argentina. The antipodes of Hawaii and Argentina are labeled in lower case. The shapes of the continua are distorted by the projection but you should still be able to see the elliptical nature of the branches and the differences between the various short/long combos.

Cycling through the four short/long combos, notice that SS and LL are the same but with signs flipped and that the same is true for SL and LS.

Conclusion

To get started with TDoA, that's about all you need to know about hyperbolas and their branches.

Note that I don't specifically say "difference in distances" in the second takeaway. That's because the difference can also be in time, without actually knowing the specific distances, and that's how we get to TDoA in the next post!

73s