# Point movement on a surface of a sphere

• Sobe118
In summary, the conversation discusses finding the resulting location of a point on a sphere in spherical or Cartesian coordinates, based on velocities from the perspective of an object on the sphere. Various methods are suggested, including rotating the coordinates, using the point's up direction and velocities, and creating a rotation matrix. The end goal is to compute the trajectory of the point on the surface of the sphere.
Sobe118
I'm trying to find the resulting location of a point on a sphere in spherical coordinates or Cartesian. Based on velocities from the perspective of an object on the sphere.

So given the:
location on the sphere (in spherical or Cartesian)
zy - plane rotation of the point
up direction of the point is always away from the origin
velocity from the perspective of the object

How do I find the resulting new location on the sphere. I am not sure what math to use, or what keywords to search in google.

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Is point moves along the surface of sphere ?

Yes, it is very commonly computed in video games. Self driven point on the surface of a sphere. I can't find any math on this topic. I've been looking for a week.

If velocity is upward then how can point move on surface of sphere ? It should leave the surface.

The point would normally only ever have velocities on its green and red arrows. The radius is fixed.

Think of it as if a car is driving on a very tiny earth

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Maybe we can try to rotate the coordinate? Make the orbit to the xy plane and do the motion and then rotate it again.
Is this a feasible way?

The point trajectory will be a circle If two velocities are equal then you can make resultant speed that speed direction will be circle trajectory and then you can use 3d circle equation

Thanks Arman, I will try to test that. Seems simple.

I found a post on another form related to this problem from a game programming perspective, I haven't got this to work yet.
1) Center the sphere to 0,0,0 global coordinates.
2) Get the up-vector of the airplane, which is a also the normal of the plane tangent to the sphere, by normalizing the position vector of the airplane.
3) Get the at-vector of the airplane by subtracting the previous airplane position from the current airplane position and normalizing it.
4) Get the right-vector of the airplane by taking the cross product of the up-vector and the at-vector.
5) Use these vectors to create a rotation matrix, a correction rotation matrix.
6) Locally rotate the airplane and create a local rotation matrix.
7) Get the final rotation of the airplane by multiplying the local rotation matrix times the correction rotation matrix.
8) Use the at-vector of the final rotation matrix as the directional vector of the airplane and translate the airplane along this vector scaled by the speed of the airplane.

## What is "Point movement on a surface of a sphere"?

"Point movement on a surface of a sphere" is the study of how a point or object moves on the surface of a sphere, taking into account factors such as curvature, distance, and direction.

## How is point movement on a surface of a sphere different from point movement on a flat surface?

The main difference is that on a sphere, the shortest distance between two points is not a straight line, but rather a geodesic curve. This means that the rules of Euclidean geometry do not apply and movement must be calculated using spherical geometry.

## What is the significance of studying point movement on a surface of a sphere?

Studying point movement on a surface of a sphere has many practical applications, such as in navigation, astronomy, and geodesy. It also has theoretical importance in understanding the geometry and physics of curved surfaces.

## What are some challenges in calculating point movement on a surface of a sphere?

One of the main challenges is accurately representing the curvature of the sphere and accounting for it in calculations. Also, since a sphere is a three-dimensional object, it can be difficult to visualize and apply traditional two-dimensional methods of calculation.

## What are some real-world examples of point movement on a surface of a sphere?

Some examples include the movement of satellites in orbit around the Earth, the navigation of ships or airplanes over the curved surface of the Earth, and the movement of celestial bodies in the night sky.

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