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I have a radius, R, for an aircraft traveling at velocity, V. If we start at a point, $(X,Y,Z)$, what is the position of the point at the time, t in terms of coordinates $(X_1,Y_1,Z_1)$?

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=V2gtan\phi $

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 on 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 is 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 maneuver 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 as $\phi$, etc

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    @Abdul: The equations you mention are the transformation from Cartesian into spherical coordinates, aren't they? If you don't get any answer here you might try posting your question in Physics Stack Exchange.2011-06-07

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