If you notice CDs stumbling response, and his continued stumbling on the topic, he hasn't a clue as to what doppler shift is and how it could be used to track a spacecraft.

The notion that triangulation is the only means of precision tracking is simply absurd. I'm not surprised he mentioned it and professed understanding of it. A lot of people know about that and how it works.

Directional antennas are not all that difficult. You locate the direction of a signal from one point, and simultaneously the direction of that signal from another point, and those two vectors intersect at the source of the signal.

Sure, that would be necessary in the most general case, where the signal could be coming from literally anywhere. But the joy of truly understanding a problem according to its theory and not just on the basis of uncomprehended examples is knowing where you can cheat.

Suppose you constrain the motion of a signal source to a known path, say to a long straight stretch of highway. Your receiving station is some distance from the highway. A second station is not required because your direction-finding vector will intersect the highway at only one point.

In fact, you don't even need a directional antenna if you can assume the signal source moves at a constant rate. That tracking problem can be solved with an omnidirectional antenna and doppler shift alone. Remember that doppler gives you velocity toward or away from the receiver. That means at each point along the highway, the velocity toward or away from the station is unique.

If you consider the classical example of the train whistle heard by a stationary observer to the side of the track, the pitch of the whistle is different at each point along the track. If you know what the "natural" pitch is, and have a good ear, you can determine how far away the train is simply by comparing the pitches and knowing where you are relative to the track.

The notion of constraining the velocity state of the transmitter is paramount. The Apollo spacecraft was primarily a ballistic projectile, and ballistic projectiles obey very strict relationships between altitude and velocity.

Think of pitching a baseball. It moves very rapidly from the pitcher to the catcher along a very flat ballistic trajectory (excluding knuckle balls, curve balls, spit balls, and other aerodynamic and salivadynamic effects). Now imagine lobbing that ball in a very high trajectory from pitcher to catcher. It follows a specific path, and moves comparatively slowly -- extremely slowly at its apex.

You can't mix and match. You can't throw it slow and shallow. You can't lob it high and invariably fast. There are hard and fast rules that govern the velocity state for any point along any ballistic trajectory. Those apply to the Apollo spacecraft.

Now let's say you're on the ground and you have a precise dish antenna which allows you to determine the signal's direction precisely. You know how the earth rotates so you can convert altitude and azimuth to vectors in geocentric space. And let's say you have the ability to precisely measure doppler shift.

Now you take a series of observations -- directions and velocities over precise time intervals. That combination of time interval, direction, and ballistic constraint allows you to determine a trajectory, which in turn provides a unique mapping of velocity and displacement, which provides a basis for translating observations into actual positions.