Movement on a Sphere: Find Lat & Long of Point P

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The discussion centers on determining the latitude and longitude of a point moving on a sphere along the shortest path between two points, P1 and P2, at a constant speed. Participants clarify that the goal is to express the angles as functions of time, using components of velocity derived from the initial and final points. The approach involves integrating the angular velocity for both latitude (theta) and longitude (phi), with adjustments for the radius of the sphere. It is noted that while the theta component can be approached similarly to a circle, the phi component requires additional trigonometric considerations. The conversation highlights the need for a clear setup of the velocity components to accurately represent the movement along a great circle.
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Hi all,

I have a question on the movement of a point on a sphere.

Consider a point P(\theta,\phi) (\theta and\phi are the latitude and the longitude, respectively)moving from a point P_1(\theta_1,\phi_1) to a point P_2(\theta_2,\phi_2) with a constant speed v.

How to find the latitude and the longitude of P(\theta,\phi) (the expression of \theta and \phi) if we suppose that the trajectory is the shortest distance between P_1 and P_2 ?

thanks.
 
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I'm not quite understanding the question. Do you want the angles as a function of time, or one angle as a function of the other?

To start working on the problem, you can find the theta and phi components of the velocity from the known P1, P2 and the radius of the sphere. Then it shouldn't be too bad to write down expressions for the two angles as a function of t , since the speed is constant. The phi equation will take a little more trig than the equation for theta. When you get this far, you should be able to answer your question.
 
thank you for the answer.

I'm not quite understanding the question. Do you want the angles as a function of time, or one angle as a function of the other?

angles as a function of time.

I see how to do it on a circle but on a sphere I don't arrive to visualize the movement.

Just to be sure, on a circle of radius R we have that d \theta=\frac{v}{R} dt, so by integrating we have \theta(t)=\frac{v}{R} t + \theta_1 ?
 
You have the circle approach correct. The same approach will work for the theta component on the sphere. For phi, R will have to be replaced by R cos(theta), but the approach is otherwise the same. You will still have to figure out how to set up the two components of v (theta and phi) to go along a great circle from the initial to the final point.
 
thanks. I still don't see how things work. Any other help ?

I have that x=R \cos(\theta) \cos(\phi), \\ y=R \sin(\theta) \cos(\phi), \\ z=R \sin(\phi)

and \frac{dr}{dt}=v

what am I missing ?. Is \tan(\theta)=\frac{r}{R} ?
 
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