how map 2d grid points (x,y) onto sphere as 3d points (x,y,z)

Question

I have a set of 2d grid points (x,y) that I want to map/project onto a sphere as 3d points (x,y,z).

I realize there will be some warping towards the poles as abs(y) increases but my grid patch will only cover a portion of the sphere near the equator so severe warping will be avoided.

I'm having trouble finding the right equations for that.

Answer 1

Paraphrased from the wikipedia article on Mercator projection:

Given a "mapping sphere" of radius R,
the Mercator projection (x,y) of a given latitude and longitude is:
   x = R * longitude
   y = R * log( tan( (latitude + pi/2)/2 ) )

and the inverse mapping of a given map location (x,y) is:
  longitude = x / R
  latitude = 2 * atan(exp(y/R)) - pi/2

To get the 3D coordinates from the result of the inverse mapping:

Given longitude and latitude on a sphere of radius S,
the 3D coordinates P = (P.x, P.y, P.z) are:
  P.x = S * cos(latitude) * cos(longitude)
  P.y = S * cos(latitude) * sin(longitude)
  P.z = S * sin(latitude)

(Note that the "map radius" and the "3D radius" will almost certainly have different values, so I have used different variable names.)

Answer 2

I suppose that your (x,y) on the sphere are latitude, longitude.

If so, see http://tutorial.math.lamar.edu/Classes/CalcII/SphericalCoords.aspx.

There:

phi = 90 degree - latitude

theta = longitude

rho = radius of your sphere.