Books I through IV and VI ofEuclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, thePythagorean theorem (Proposition 47), equality of angles andareas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.
Later, the plane was described in a so-calledCartesian coordinate system, acoordinate system that specifies eachpoint uniquely in aplane by a pair ofnumericalcoordinates, which are thesigned distances from the point to two fixedperpendicular directed lines, measured in the sameunit of length. Each reference line is called acoordinate axis or justaxis of the system, and the point where they meet is itsorigin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of theperpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
The idea of this system was developed in 1637 in writings by Descartes and independently byPierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery.[1] Both authors used a single (abscissa) axis in their treatments, with the lengths ofordinates measured along lines not-necessarily-perpendicular to that axis.[2] The concept of using a pair of fixed axes was introduced later, after Descartes'La Géométrie was translated into Latin in 1649 byFrans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.[3]
Later, the plane was thought of as afield, where any two points could be multiplied and, except for 0, divided. This was known as thecomplex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named afterJean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematicianCaspar Wessel (1745–1818).[4] Argand diagrams are frequently used to plot the positions of thepoles andzeroes of afunction in the complex plane.
"Plane coordinates" redirects here; not to be confused withCoordinate plane.
In mathematics,analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicularcoordinate axes are given which cross each other at theorigin. They are usually labeledx andy. Relative to these axes, the position of any point in two-dimensional space is given by an ordered pair of real numbers, each number giving the distance of that point from theorigin measured along the given axis, which is equal to the distance of that point from the other axis.
Another widely used coordinate system is thepolar coordinate system, which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray.
While a pair of real numbers suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for theirembedding in theambient space.
The regularmonogon (or henagon) {1} and regulardigon {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a2-sphere,2-torus, orright circular cylinder.
There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are calledstar polygons and share the samevertex arrangements of the convex regular polygons.
In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for allm such thatm <n/2 (strictly speaking {n/m} = {n/(n −m)}) andm andn arecoprime.
Thehypersphere in 2 dimensions is acircle, sometimes called a 1-sphere (S1) because it is a one-dimensionalmanifold. In a Euclidean plane, it has the length 2πr and thearea of itsinterior is
Another mathematical way of viewing two-dimensional space is found inlinear algebra, where the idea of independence is crucial. The plane has two dimensions because the length of arectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independentvectors.
The dot product of two vectorsA = [A1,A2] andB = [B1,B2] is defined as:[5]
A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vectorA is denoted by. In this viewpoint, the dot product of two Euclidean vectorsA andB is defined by[6]
Ingraph theory, aplanar graph is agraph that can beembedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.[9] Such a drawing is called aplane graph orplanar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to aplane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
^M.R. Spiegel; S. Lipschutz; D. Spellman (2009).Vector Analysis (Schaum's Outlines) (2nd ed.). McGraw Hill.ISBN978-0-07-161545-7.
^Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010,ISBN978-0-521-86153-3
^Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum's Outlines, McGraw Hill (USA), 2009,ISBN978-0-07-161545-7
^Trudeau, Richard J. (1993).Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64.ISBN978-0-486-67870-2. Retrieved8 August 2012.Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.
