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