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A mathematician, in particular a Euclidean geometer.
Two dimensional geometry is called "plane geometry" meaning that it occurs on a single surface or plane. The objects used in plane geometry are called plane figures.
Presumably it is simply geometry. However, plane geometry is geometry which is limited to objects with up to two dimensions, that is, objects that lie in a plane - or on a flat surface.
Point ; Line ; Plane - Remember the Point-Line-Plane Postulate