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# Projecting a Circle on a Sphere to an Enclosing Cylinder

A cylindrical projection maps points on a sphere to a cylinder wrapped around the sphere at its equator.
Several cylindrical mappings are possible, depending on the function that maps on the sphere to on the enclosing cylinder. The most common ones are the equidistant and the equal area cylindrical projections.
This Demonstration shows those two projections of a spherical circle. You can move the circle on the sphere by rotating it about either the or axis.

### DETAILS

The transformation formulas used in this Demonstration are based on the Mathematica built-in function ToSphericalCoordinates and the conversion formulas in [1, 2].
For the equidistant projection, the following mapping is used to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the cylinder wrapped around it:  .
The equidistant projection to a plane uses the following mapping to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the 2D plane: .
For the equal area projection, the following mapping is used to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the cylinder wrapped around it: .
The equal-area projection to a plane uses the following mapping to convert the Cartesian coordinates of a point on the unit sphere to the corresponding point on the 2D plane: .
References
[1] E. W. Weisstein. "Cylindrical Equidistant Projection" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/CylindricalEquidistantProjection.html.
[2] E. W. Weisstein. "Cylindrical Equal-Area Projection" from MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/CylindricalEqual-AreaProjection.html.

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