Can someone check my math?

[flyboy67109]

Even though this question is related to RemoteTech, it's not really a question about RemoteTech. What I have is my first three satellites in orbit for my relay net v.1.0. When I put up my third satellite, I targeted one and then the other noting the distance to each satellite. I then found the difference and divided in half. I then took that number and added it to the distance to the satellite that had the shortest distance. My thought is, this should give close to the equal distance between each satellite. But, this is real caveman math and there is an equation to figure out the length one side of a equilateral triangle inside a circle. But, I can't seem to find it when I Google for it. It's been too long sense I last took geometry. Here's the orbit for the "mathpiles" out there:

Orbit = 1001km X 1006km

Right now I'm figuring 2550km between satellites. Is that right or even close?

Thanks

[kahlzun]

You have to remember that Kerbin is about 600km across, so those 1000km orbits are actually 1300km orbits.

If all 3 of them orbit the centre at the same distance, then you know the length of two sides of the triangle (1300km), and if the 3 of them are equidistant, you know the intended angles also (60 degrees).

if one angle is 60 degrees, and two 'arms' are the same length, then my logic states its an equilateral triangle, and all 3 sides are 1300km in length. Therefore, your satellites will need to be 1300km apart.

Edited March 20, 2014 by kahlzun

[toadicus]

That's not quite right. If we're circumscribing a circle of radius 'r' about an equilateral triangle 'ABC' with three sides of length 'l', we can define half the chord length by a triangle 'ADE' comprising the radial line bisecting one corner of the equilateral triangle, half of an adjacent chord, and the distance from the center of the circle out to the midpoint of the chord, giving us a 30/60/90 right triangle with hypotenuse 'r'. From this, it follows that:

l / 2 = r × cos(30°)
l = 2 × r × cos(30°)

Where l is the length of each chord defined by the equilateral triangle and r is the radius of the circle.

Kerbin's radius is actually 600km (not it's diameter), so when your altitude is 1000 km your orbital radius is actually 1600km. Thus, the distance between each satellite should be:

l = 2 × 1600 km × cos(30°)
l = 2771 km

[DocMoriarty]

You have to remember that Kerbin is about 600km across, so those 1000km orbits are actually 1300km orbits.

If all 3 of them orbit the centre at the same distance, then you know the length of two sides of the triangle (1300km), and if the 3 of them are equidistant, you know the intended angles also (60 degrees).

if one angle is 60 degrees, and two 'arms' are the same length, then my logic states its an equilateral triangle, and all 3 sides are 1300km in length. Therefore, your satellites will need to be 1300km apart.

Kerbin has 600km radius or 1200km across, according to that the 1000km orbits are actually 1600km away from center of Kerbin. The line from Kerbin center to satellite (in your case 1300km long) is NOT part of the equilateral triangle though, so your explanation is basically dead wrong, aside of the wrong radius for Kerbin.

The key to solve the problem is to derive a right angle triangle from the situation and use some math magic.

To do so imagine a triangle built from: a line from Kerbin center to satellite (1600 km), from satellite halfway to next satellite, and from "halfway point" back to Kerbin center. The inner angle at the "halfway point" is 90°, our right angle. The other angles are 30° (at the satellite) and 60° at the center of Kerbin.

Now we know that the line halfway to next satellite is cos(30°) and we know 1 = 1600km. So the distance between 2 satellites should be 2 * cos(30°) * 1600km = 2771 km. Sorry i'm too lazy to make a picture.

Edit: LOL i was a bit slower than the previous poster but we both had the same solution!

[capi3101]

Okay - this is basic trigonometry. You've got three satellites that you want equidistant from one another, each at an average orbital distance of 1003.5 kilometers (splitting the difference between 1001 and 1006). Kahlzun is correct in pointing out the 600 kilometer figure, but incorrect about its usage - if you check the wiki, the equatorial radius is 600 km. That figure is the radius, not the diameter, and so you have to add 600 km to the figures, not 300.

Okay. So you know that the distance from the center of the circle (the orbit) to the edge (where your sats are) is 1600 km. Each one is sixty degrees from the other. What you can do is draw the triangle from one satellite to another, and then draw a line from each satellite to the center of the circle. You can go ahead and then extend the lines headed through the center to the opposite side of the triangle. What does this do? It sets you up with a hell of a lot of 30-60-90 triangles is what it does, and the hypoteneuse of each (the one corresponding to the 90 degree angle) is your 1600 km figure.

So we look up wikipedia: http://en.wikipedia.org/wiki/30-60-90_triangle#30.E2.80.9360.E2.80.9390_triangle

And discover that the length of the sides are in a proportion of 1:2:sqrt3, with 2 corresponding to the hypoteneuse.

So - we half the 1600 figure and then multiply it by the square root of 3; half the distance from one satellite to the another should be 1385.641 km (you can verify that figure with the law of sines, incidentally). Multiply that by two -

2,771.281 kilometers. And that's your answer.

Here's a diagram, in case any of what I said above is unclear:

...and no, that's not supposed to be the Deathly Hollows.

EDIT: Double ninja'd. OP, look on the bright side - you've got three folks throwing you the same figure. Pretty sure that means it's correct.

[flyboy67109]

So, you think 2771 km is right, huh? Haha! Thanks for the fun math and the equation.

[Kryxal]

If you guys aren't averse to the more advanced trig equations, there's also the Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos©

a = b = 1603.5 km, C = 120 degrees

c^2 = 2571212.25 + 2571212.25 - 2*2571212.25*2571212.25*-.5 = 2571212.25*3 = 7713636.75

c = 2777.34 km (I used 1603.5 to everybody else's 1600, all the equations work just fine though