What is a surface called where a straight line joining two points lies wholly within that surface?

A surface where a straight line joining two points lies wholly within that surface is known as a plane. In geometry, a plane is defined as a flat, two-dimensional surface that extends infinitely in all directions. The characteristic feature of a plane is that if you take any two points on that surface, the straight line connecting those points will also lie entirely on the plane. This property distinguishes it from curves, rectangles, and polygons, where straight lines may not meet this criterion.

Curves, for instance, may bend and not allow a straight line between two points to remain entirely on the surface; for example, on a circular edge. Rectangles and polygons consist of straight edges but are not infinite surfaces; they have defined boundaries. These boundaries can impede a line from remaining entirely within them unless you are considering the interior, which makes the definition less applicable. Thus, the correct identification of a surface where any straight line between points belongs fully to that surface is a plane.