How to find coordinates of a point on a 3D cylinder in Cartesian system if any one point on cylinder and dimensions of cylinder are known?

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Consider a cylinder of known dimensions inserted in 3D cartesian space. I know the cartesian coordinate of one point located on the surface of cylinder. Using this information I want to find out the coordinates of other points of the cylinder. Please help me with this.

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Your problem is underdetermined, since every rotation of the cylinder you seek around the given point is a valid solution, too