Guide to telescience

So, I'm not going to dumb it down much for those who don't know math.  In order
to fully appreciate the guide, you will want to have experience with
trigonometry and polar coordinate systems.

So, you've come to telescience, and you're wondering how in the world it works.
You have bearing, elevation, power, and sector as variable controls and you
have send, receive, recalibrate crystals, and eject crystals.

What do all of these mean?  Well, you see, telescience is a lot like throwing a
ball.  When you press send, you are throwing a ball that can go through matter
and the object goes from your source to your destination.  It's the same thing
with receive, except the object you are teleporting goes from destination to
source.

So, if you're going to throw a ball somewhere, you want to face a
direction, figure out what angle you want to throw it at, and control how far
you throw it.

So, bearing is just the number of degrees that represent a compass or cardinal
direction.  Essentially, it's what your facing is.  Elevation is the angle you
throw the teleportation at.  If you know your basic ballistic trajectories, you
know the angle you can get the farthest distance with for a constant power is
45 degrees.  And your power is the initial velocity you throw the teleport at.

The sector is a special one.  Space around the station is divided up into
sectors.  Sector 1 is the station, sector 5 is the asteroid, and sector 2 is a
restricted sector that the Telepad Control Console won't let you teleport to.
I recommend using sector 3 for your calibrations.

Let's give each of these a symbol so we can refer to them in equations.

        B : bearing
        a : elevation
        P : power

        Px: power, x component, horizontal speed
        Py: power, y component, vertical speed

Please note that the x and y components used here are independent from the
often referred to x and y coordinates for the GPS on the station.  In other
words, x is the absolute distance the teleport travels, and y is the height.

Since we're given an angle of elevation, we can figure out Px and Py.  Given a
right triangle with hypotenuse P, angle a opposite from side Py and adjacent to
side Px, such as:

              /|
             / |
           P/  |
           /   |Py
          /    |
         /     |
        /a_____|
           Px

We can intuitively see the relationship between all of the variables.  Thus:

        Px = P*cos(a)
        Py = P*sin(a)

This describes only the initial conditions of the ballistic trajectory.  A
ballistic trajectory looks something like:

                                    Midpoint
                                       *
                                  *         *
                             *                   *

                        *                             *


                   *                                       *



              *                                                 *



         Source                                                  Destination
         *                                                           *

At the source, the initial conditions apply, and at the destination, the
teleport is done.  The midpoint is there for illustation in the time it takes
the teleport to rise, then fall.

And that brings us to time.

The teleport's time is completely dependent on how long it takes for the object
to go up and then down.  Now, the "gravity" for the teleport is 10, and the
teleport has to rise first, then fall, which doubles the time it takes for it
to arrive at its the destination.  Thus we have:

        t: time
        t = 2*Py/10
          = 1/5 * Py

And the horizontal distance travelled is just a product of the speed of the x
component, which stays the same through the trip, and time:

        d: distance
        d = t * Px = 1/5 * Px * Py
                   = 1/5 * P*cos(a) * P*sin(a)
                   = 1/5 * P^2*sin(a)cos(a)

If you know your trig, you should recognize something, but if not:

        d = 1/10 * P^2*2sin(a)cos(a)
          = 1/10 * P^2*sin(2*a)

The double-angle formula drastically simplifies what we need to know.  This
works on an intuitive basis, since the elevation has to be between 1 and 90
degrees.  At 90 degrees, you'd throw the teleport straight up and then it
should come back down to the same spot, and sin(2*90)=sin(180)=0.  This also
works when the angle of elevation is a 45, since sin(2*45)=sin(90)=1, which is
the maximum that the sine function can output.

So, now we know how to calculate distance.  Now to apply it to the actual
teleport.  To teleport, you specify a magnitude and a direction.  You control
the magnitude by setting the proper elevation and power for how far you want to
go, and you get the bearing by calculating what you know about the remote
coordinates.  Here is the relationship via graph:

        x : x distance from teleporter (can be negative)
        y : y distance from teleporter (can be negative)
        B : bearing (can be negative)
        B = arctan(x/y)

                                        0deg ^   x
                                             |--------/
                                             |       /
                                             |      /
                                             |     /
                                           y |    /d
                                             |   /
                                             |  /
                                             |B/
                                             |/                          90deg
        <-------------------------------------------------------------------->
        270deg                               |
                                             |
                                             |
                                             |
                                             |
                                             |
                                             |
                                             |
                                             |
                                             |
                                             v 180deg

Keep in mind your quadrant when taking inverse tangent.  Since distance cannot
be negative, you have to specify a correct bearing.  If the quadrant of the
angle is wrong, then you will just need to add or subtract 180.  For example,
let's say x = -2 and y = -20.  The angle given will be 5.71.  However, this
isn't what you wanted.  It's definitely not in the third quadrant.  For this,
you add 180, and get 185.71, which is the correct angle you need to have.

Now, all the above is what happens in an ideal situation.  There's one thing to
mentions: variance.  Variance, in short, is what you need to compensate for,
and it's what turns your job into something only math majors can understand.
If you're bad at finding variance properly, you could be using enough teleports
to justify another recalibration, and that's not good.  There's no really easy
way to find variance without breaking out the equations that would require a
bunch of tests (because we all love the infinite series for sine, right?).

The variances for each input are the following:

        Bearing: No more than 10 degrees off.
        Elevation: At most 25 degrees off.
        Power: No more than 4 units down from the setting (also read as at P-2,
               no more than 2 units off)

Additionally, you have between 30 and 40 uses of the teleporter before you need
to recalibrate.