The cross-product of the normalized `PlanetToObject` and `CameraToObject` directions is perpendicular to both vectors and is, by definition, the normal of a plane that contains both. The cross product of the plane's normal (perpendicular to the plane) and `CameraToObject` (a vector in the plane) is a second vector in the same plane, perpendicular to the first; the net-result is a rotation of `CameraToObject` 90 degrees around the face normal.

Note: I may or may not have reversed the order of the cross-products, but you only need to worry about it at the very end; the yellow vector should point to the "right of the camera"; invert it if it doesn't.

This was one of the hardest screenshots I've tried to diagram over; I had given up on making it look right but will post it anyway, since I think you'll still get it.

[![enter image description here][1]][1]
  [1]: https://i.sstatic.net/qmquV.png