# (Geometry-Physics) Given a radius and velocity calculate position of an aircraft banking to make a turn in three dimensional space

I have a radius, R, for an aircraft traveling at velocity, V. If we start at point, (X,Y,Z), what is the position of the point at time, t in terms of coordinates(X1,Y1,Z1)?

For example: The aircraft is at point (0,0,0) and traveling at 250 knots and initiates a turn with a bank angle, phi, of 5 degrees. Assume that the aircraft can instantaneously rotate to the five degree bank. The equation for the turn radius, R where g is the acceleration due to gravity (9.81) is: R=V2gtanϕ

For this example, R = 10.4 nautical miles. Where is the aircraft at t = 2 if the aircraft is traveling at a heading of 90 degrees (straight along the y axis)in three dimensional space

Elaboration of the above question: I would like to elaborate the question a bit more. Suppose an aircraft is moving at a certain fixed altitude above the ground. It follows a path defined by latitude and longitude. Now if we want to define the position of an aircraft at any point in the air, three variable are required for example X for latitude, Y for longitude and Z for the altitude. Suppose an aircraft flies and reach a certain fixed altitude Z, it then follows a route defined by X and Y. Now suppose that at any stage during the flight the aircraft decides to take a turn. As long as Z remains constant to predict any future position of the aircraft during the turn, the answer you gave in " Given a radius and velocity calculate position of an aircraft banking to make a turn " works fine. But if the turn of the aircraft is on a sphere rather than a circle then in the case the new Z position also needs to be calculated. In other words if the aircraft does a maneuvre in such a way that it turns either to the left or right and increases or decreases it's altitude in the same time then a new equation for the Z needs to be found. Assuming knowing the speed and the current three Dimensional position of the aircraft, how can the future position of the aircraft after a known time t can be predicted? Also assume that other aircraft related constant parameters are also known like phi etc

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It seems that you still need to define in which plane the aircraft is going to travel. In principle, on this plane the movement equations are similar to the two dimensional case, in this question math.stackexchange.com/questions/12394/… – Américo Tavares Apr 19 '11 at 22:46
@Abdul: the reason you had difficulty editing this question is because you had two separate unregistered accounts. I've merged them. Please do consider registering: it will help the software keep track of who you are and which are your questions. – Willie Wong Apr 24 '11 at 18:23
"the turn of the aircraft is on a sphere rather than a circle". 1) Isn't the curve described by the aircraft still a circle with radius $R$, though with a non constant altitude? 2) If it is, you have to define it: give its equation, or other information to find it. – Américo Tavares Apr 24 '11 at 19:00
Thanks Americo, I thing i am getting your point. – Abdul May 13 '11 at 10:58
@Americo Tavares: I developed the equations for locating a point on the sphere, but the equations have two angles, one(phi) with respect to altitude(Z) and the other(theta) with respect to X or Y axis. The Phi can be found be calculating the arc length but how could one calculate theta with the given information. – Abdul Jun 7 '11 at 11:34