# About the AltiCalc Calculator

## Altitude Tracking

/*Describe AltiTrak w/o using the word AlitTrak?*/ You can use this angle to estimate the height of the rocket by assuming that the rocket went straight up into the sky. This isn't a particularly bad assumption, but in windy weather, you could be off by a significant amount.

To correct for this inaccuracy, you need to get more information about the rocket's trajectory. One of the most common methods of tracking rockets is by using two theodolites. A theodolite not only measures the angle above the horizon (the altitude angle), but also the angle the rocket has moved horizontally (the azimuth angle). With two theodolites, you can determine the final altitude with very good accuracy.

The problem with theodolites is that they are more expensive than AltiTraks and it is much easier to make your own AltiTrak-like device that it is to make your own theodolite. Simply tape a weighted string to a protractor and you have a crude device to measure altitude angle. AltiCalc provides a way to track rockets accurately with only altitude angle measurements.

## The Theory Behind AltiCalc

Again consider the lone person with an AltiTrak. Imagine an infinite ray stretching out of the tracking device towards the rocket. This represents the locus of points that the rocket could occupy given only that the tracker has it in their sights. Since we don't know the azimuth angle in which the tracker is facing, this ray could point in any direction as long as it has the same altitude angle. This forms a right cone around the tracker. [show gif]

The AltiCalc program takes advantage of the fact that right cones aligned on parallel axes will intersect. More specifically, given three non-colinear right cones whose their apexes are on the xy-plane and whose axes are parallel to the z-axis, the only intersections of all three cones will be at a unique z-value. The equations involved in this method cannot be solved analytically, so a numerical solver is used instead.

The solver works as follows:

- A z-value guess is selected
^{1} - The cones are sliced at that height to yield three circles [add example]
- The x-value of the intersections between cones 1 & 2 and cones 1 & 3 are computed. (If the intersections don't exist, nothing is computed)
- If the intersection of 1 & 2 is to the left of cones 1 & 3, a bigger guess is selected. If it is to the right, a smaller guess is selected.
- This process continues until either
- The intersection of cones 1 & 2 is at the same
place
^{2}as cones 1 & 3 - The guesses get too close to the edges of the allowable range of guesses.

## Assumptions

- The elevation of each tracking device is at the same elevation as the launch pad.
- The x-y plane is defined as this elevation, with z directed upward
- The trackers are arranged along the x-axis.