Great circle trajectories around a sphere

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What's the best way to calculate great circle trajectories, knowing the velocity (tangent to sphere) of an orbiting object?

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Answer by Fattie · Oct 27, 2012 at 08:50 AM

just draw a straight line (ie through the planet)

simply raise that to be a tangent to the surface

(go away from the center of the sphere until 90 away)

Hey Ina - do you need more than this? This is exactly how you do it. It's remarkably easy in Unity.

It was a lot harder for navigators in the 1800s! :)

You further wanted to know how to do this using built-in functions.

 direct line = B - A

(use the built-in subtraction of Vector3)

 AToCenter = planetCenter - A
 temp = direct line X AToCenter
 final result = AToCenter X temp

the large "X" represents Cross Product which is an operation between Vectors. Cross Product is built-in to Unity:

http://docs.unity3d.com/Documentation/ScriptReference/Vector3.html

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Bunny83 · Oct 27, 2012 at 09:15 PM 0

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+1 ;)

It's actually a quite off-topic question since it has nothing to do with Unity. As you said it's a pure mathematical problem.

Besides that the velocity of an object is never ever enough. The only reason why an object is orbitting another is the gravity. Therefore without the distance between the center object and the orbitting object and the mass of both, this isn't going anywhere.

Next thing is most the time an orbitting object doesn't really move on a great circle. It's usually an elliptic route unless the parameters (distance, mass, velocity) are very well choosen.

In this case when you really move on a great circle, simple trigonometry should be enough since it's just a circle.

I can just copy what Fattie said: We need more information!

ina · Oct 29, 2012 at 01:53 PM 0

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Well, it's a game question... Angry Birds but with planetary orbits. Continuing the analogy, the goal is to hit a pig on the "other side of the planet", after sending the bird into orbit...

I have several ideas on how to implement this but they're all very complicated. There's got to be an easier way.

DaveA · Oct 29, 2012 at 03:36 PM 0

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I loved Angry Birds Space. Are you remaking that?

ina · Nov 11, 2012 at 12:19 PM 0

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So basically, direction to move is always $$anonymous$$esh.tangent, or is there another way to get the tangent to the surface? @DaveA I'm just exploring for now, trying to figure out the best way to make these things in Unity

Fattie · Nov 11, 2012 at 12:30 PM 0

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the tangent to a sphere is a PLANE so your comment is meaningless.

the tangent of the mesh is largely irrelevant here, I don't see how it could help in any way.

to simply re-type what I typed above:

(1) make an imaginary straight line (ie, it will go through the planet) between the begin point and the end point. in other words it will look like an incredibly long, shallow, tunnel between the two points.

(2) now raise that line until it is on the tangent PLANE at the begin point. in other words, obviously, make the angle between center-begin and begin-end, a 90 degree right angle.

if you struggle with the math for this, I don't really know if it's the forum for a huge beginner geometry math intro you know. Also you never say thanks, so not many people want to do that.

to "make it a right angle" just use the usual trick of crossing twice. the first cross will give you a arrow, on the tangent-plane, but pointing 90 degrees away fromt your desired destination. just cross again to get your desired destination.

so (1) is nothing more than vector3 = B - A

and (2) is nothing more than temp = 1 X begin, result = temp X begin, result=-result

(I think I have the negative correct, recall unity is left handed in crosses)

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Great circle trajectories around a sphere

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