G. Review of Related Literature
Area is a quantity expressing the size of the contents of a region on a 2-dimensional surface. Points and lines have zero area, cf. space-filling curves. A region may have infinite area, for example the entire Euclidean plane. The 3-dimensional analog of area is volume. Although area seems to be one of the basic notions in geometry, it is not easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. For polygons in the Euclidean plane, one can proceed as follows:The area of a polygon in the Euclidean plane is a positive number such that: * The area of the unit square is equal to one.* Congruent polygons have equal areas. * If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas of these polygons. It remains to show that the notion of area thus defined does not depend on the way one subdivides a polygon into smaller parts.A typical way to introduce area is through the more advanced notion of Lebesgue measure. In the presence of the axiom of choice it is possible to prove the existence of shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach–Tarski paradox). The sets involved do not arise in practical matters.In three dimensions, the analog of area is called volume. The n dimensional analog, usually referred to as 'content', is defined by means of a measure or as a Lebesgue integral.
In geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. Today a polygon is more usually understood in terms of sides. Usually two edges meeting at a corner are required to form an angle that is not straight; otherwise, the line segments will be considered parts of a single edge. The basic geometrical notion has been adapted in various ways to suit particular purposes. For example in the computer graphics field, the term polygon has taken on a slightly altered meaning, more related to the way the shape is stored and manipulated within the computer.
In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue. Thus, a point is a 0-dimensional object. Because of their nature as one of the simplest geometric concepts, they are often used in one form or another as the fundamental constituents of geometry, physics, vector graphics, and many other fields. Points are most often considered within the framework of Euclidean geometry, where they are one of the fundamental objects. Euclid originally defined the point vaguely, as "that which has no part".
