yes, a plane is a falt surface, because infinite flat surface is a plane.
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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.
A plane is a flat surface with no thickness. Our world has three dimensions, but there are only two dimensions on a plane.
In geometry a flat surface is usually called a plane surface, or just a plane. You use it like this.1) If straight lines lie in the same plane they a said to be co-planar.2) If two co-planar straight lines never intersect, they are parallel to each other.
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.
In the mind. A plane is an abstract concept: it is perfectly even (flat) and infinitely long in every direction. A flat surface, such a s desktop or sheet of paper may model it, but these will only be approximations.