Finding the coordinates of a point on a sphere

In summary, the problem at hand is to find the xyz coordinates of a point C on the surface of a sphere. This can be done by first solving the analogous problem on a plane using Cartesian coordinates. However, on a sphere, we must take into account its non-Euclidean (elliptic) geometry. The equations used for the plane can be adapted for the sphere by considering spherical triangles. The known quantities in this problem are the radius of the sphere, the origin of the sphere, the xyz coordinates of points A and B, the arc distance from A to C and from B to C, and the angle between AB and BC. The equations for finding the xyz coordinates of point C will involve two unknowns, the azimuth and
  • #1
lulukoko
3
0
I have three points: A, B and C, which are all on the surface of the same sphere.
I need to find the xyz coordinates of C.
What I know:
- the radius of the sphere
- the origin of the sphere
- the xyz coordinates of A and B
- the arc distance from A to C and from B to C
- the angle between AB and BC
Any ideas?
Thanks!
 
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  • #2
A good start would be to first do the analogous exercise on a plane, which has Euclidean geometry, and then see what needs to be changed to make it work on a sphere, which has non-Euclidean (elliptic) geometry.

The analogous exercise for a plane is:

I have three points: A, B and C on a plane
I need to find the Cartesian coordinates (x and y) of C.
What I know:
- the origin of the plane
- the Cartesian coordinates of A and B
- the distance from A to C (call it a) and from B to C (call it b)
- the angle between AB and BC (call it alpha)


Hint: We have two unknowns - the x and y cords of C, which we call X and Y. We can write two equations in X and Y, a, b and alpha that equate the AC distance to a and the BC distance to b.
 
  • #3
I know how to do this on a plane, the trouble I am having is in, as you say, adapting the solution to a sphere.

On the plane I used simple trigonometry with the coordinates of an unknown point C(X,Y) being
X= a cos(alpha) + b sin(alpha)
Y= -a cos(alpha) + b sin (alpha)
with the coordinates of A being (a, b).

Doing this on the sphere is proving to be more difficult because it would require me to have the azimuth and the polar angles of my point C, whereas all I have is the angle between AB and BC on the surface of the sphere.

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In the image attached, you can see that I know the length of AB, BC and AC. I know the angle alpha. I don't know any other angles. I know the xyz coordinates of A and B. I DO also know the xyz coordinates of point D, which is on the z-axis.
 

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  • #4
lulukoko said:
On the plane I used simple trigonometry with the coordinates of an unknown point C(X,Y) being
X= a cos(alpha) + b sin(alpha)
Y= -a cos(alpha) + b sin (alpha)
with the coordinates of A being (a, b).
That doesn't look correct to me. How did you derive it?

In my setup there are seven known quantities and two unknowns. You have only used three of the known quantities, so the above cannot give a correct answer.

I also note that you have stated you are using a and b as coordinates, whereas I defined them as lengths.

I think if you first get the process for deriving equations for the unknowns and then solving them completely clear for the Euclidean case, it will be much easier to apply that to the elliptic case.
 
  • #5
The equations I used were the following:
Suppose you are rotating about the origin clockwise through an angle theta. Then the point (s,t) ends up at (u,v) where
u = s cos (theta) + t sin (theta)
v = -s sin (theta) + t cos(theta)

I derived it myself from basic trigonometry functions, but here is an example that used the same reasoning as I did: http://www2.cs.uregina.ca/~anima/408/Notes/ObjectModels/Rotation.htm
 
  • #6
That is for the case of rotation around the origin by angle theta. In the OP problem the angle alpha is not at the origin. It is the angle between two line segments AB and BC, neither of which is known to go through the origin or to point towards it.
 
  • #7
What you really need is the equation that describes the circle and sphere.
 
  • #8
I think spherical triangles would be useful here.
 

FAQ: Finding the coordinates of a point on a sphere

1. What is a point on a sphere?

A point on a sphere is a location that can be identified using two coordinates: latitude and longitude. These coordinates represent the angular distance from the equator and the prime meridian, respectively.

2. How do you find the coordinates of a point on a sphere?

To find the coordinates of a point on a sphere, you will need to know the radius of the sphere, the angle of inclination from the equator (latitude), and the angle of rotation from the prime meridian (longitude). You can use mathematical formulas or online calculators to determine the coordinates.

3. What is the purpose of finding the coordinates of a point on a sphere?

The purpose of finding the coordinates of a point on a sphere is to accurately locate and identify a specific location on the surface of the sphere. This is useful for navigation, mapping, and other scientific and mathematical applications.

4. Can you find the coordinates of a point on a sphere without knowing its radius?

No, the radius of the sphere is a necessary component in determining the coordinates of a point on the sphere. Without knowing the radius, it is impossible to accurately calculate the angular distances needed for the coordinates.

5. Are there different methods for finding the coordinates of a point on a sphere?

Yes, there are various methods for finding the coordinates of a point on a sphere. The most common method is using spherical trigonometry, but there are also other mathematical formulas and computer algorithms that can be used.

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