# Transformation from global spherical to local cylindrical

I have the coordinates of a vortex in a global spherical coordinate on the surface of a sphere defined in terms of latitude and longitude(earth centric). Now I need to transform to a local coordinate system that is cylindrical but one that is centered on the center of the vortex. So the radius in the local cylindrical coordinate system is the distance from the center of the vortex to a point on the vortex and the azimuth angle is a local angle.

What would be the scaling, rotation and translation transformations if any for this problem ?