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

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Discussion Overview

The discussion revolves around determining the latitude and longitude of a point P moving on the surface of a sphere along the shortest path (great circle) between two points P1 and P2, given a constant speed. The focus is on the mathematical expressions for latitude and longitude as functions of time.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant seeks to find expressions for latitude (\theta) and longitude (\phi) as functions of time while moving from point P1(\theta1, \phi1) to P2(\theta2, \phi2) at a constant speed v.
  • Another participant suggests finding the theta and phi components of velocity based on the known points and the sphere's radius, indicating that the phi equation will require more trigonometric considerations than the theta equation.
  • A participant confirms their understanding of the approach for a circle and seeks clarification on how to visualize the movement on a sphere, specifically asking if the relationship d\theta = v/R dt holds true.
  • Another participant agrees with the circle approach and notes that while the same method applies for the theta component on the sphere, the phi component will involve using R cos(\theta) instead of R.
  • A participant expresses confusion about the setup and shares their equations for the Cartesian coordinates of the sphere, questioning if \tan(\theta) = r/R is correct.

Areas of Agreement / Disagreement

Participants generally agree on the approach to use the great circle method for determining the angles, but there is no consensus on the specific expressions or the visualization of the movement on the sphere.

Contextual Notes

There are unresolved aspects regarding the integration of the velocity components and the specific relationships between the angles and time, as well as potential dependencies on the definitions of the angles in spherical coordinates.

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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} ?
 
Last edited:

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