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NonEuclidean Geometry In mathematics, non Euclidean geometry Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry Euclidean geometry lies at the intersection of metric geometry and affine geometry, non Euclidean geometry Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional nonEuclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called nonEuclidean geometry. The essential difference between the metric geometries is the nature of parallel lines [...More...]  "NonEuclidean Geometry" on: Wikipedia Yahoo 

Line Segment In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a halfopen line segment includes exactly one of the endpoints. Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or otherwise a diagonal [...More...]  "Line Segment" on: Wikipedia Yahoo 

Length In geometric measurements, length is the most extended dimension of an object.[1] In the International System of Quantities, length is any quantity with dimension distance. In other contexts, length is a measured dimension of an object. Length Length may be distinguished from height, which is vertical extent, and width or breadth, which are the distance from side to side, measuring across the object at right angles to the length. For example, it is possible to cut a length of wire shorter than the wire's width [...More...]  "Length" on: Wikipedia Yahoo 

Twodimensional Space Twodimensional space Twodimensional space or bidimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). In Mathematics, it is commonly represented by the symbol ℝ2. For a generalization of the concept, see dimension. Twodimensional space Twodimensional space can be seen as a projection of the physical universe onto a plane [...More...]  "Twodimensional Space" on: Wikipedia Yahoo 

Onedimensional Space In physics and mathematics, a sequence of n numbers can specify a location in ndimensional space. When n = 1, the set of all such locations is called a onedimensional space. An example of a onedimensional space is the number line, where the position of each point on it can be described by a single number.[1] In algebraic geometry there are several structures that are technically onedimensional spaces but referred to in other terms. A field k is a onedimensional vector space over itself. Similarly, the projective line over k is a onedimensional space. In particular, if k = ℂ, the complex numbers, then the complex projective line P1(ℂ) is onedimensional with respect to ℂ, even though it is also known as the Riemann sphere. More generally, a ring is a lengthone module over itself. Similarly, the projective line over a ring is a onedimensional space over the ring [...More...]  "Onedimensional Space" on: Wikipedia Yahoo 

Symmetry Symmetry Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement")[1] in everyday language refers to a sense of harmonious and beautiful proportion and balance.[2][3][a] In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling [...More...]  "Symmetry" on: Wikipedia Yahoo 

Similarity (geometry) Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other [...More...]  "Similarity (geometry)" on: Wikipedia Yahoo 

Congruence (geometry) In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.[1] More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object [...More...]  "Congruence (geometry)" on: Wikipedia Yahoo 

Vertex (geometry) In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet [...More...]  "Vertex (geometry)" on: Wikipedia Yahoo 

Sphere A sphere (from Greek σφαῖρα — sphaira, "globe, ball"[1]) is a perfectly round geometrical object in threedimensional space that is the surface of a completely round ball (viz., analogous to a circular object in two dimensions). Like a circle, which geometrically is an object in twodimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in threedimensional space.[2] This distance r is the radius of the ball, and the given point is the center of the mathematical ball [...More...]  "Sphere" on: Wikipedia Yahoo 

Orthogonal In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero selforthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis. By extension, orthogonality is also used to refer to the separation of specific features of a system [...More...]  "Orthogonal" on: Wikipedia Yahoo 

Diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal [...More...]  "Diagonal" on: Wikipedia Yahoo 

Curve In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that its curvature need not be zero.[a] Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right [...More...]  "Curve" on: Wikipedia Yahoo 

Angle 2D anglesRight Interior Exterior2D angle pairsAdjacent Vertical Complementary Supplementary Transversal3D anglesDihedralAn angle formed by two rays emanating from a vertex.In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[1] Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle Angle is also used to designate the measure of an angle or of a rotation [...More...]  "Angle" on: Wikipedia Yahoo 

Area Area Area is the quantity that expresses the extent of a twodimensional figure or shape, or planar lamina, in the plane [...More...]  "Area" on: Wikipedia Yahoo 

Polygon In elementary geometry, a polygon (/ˈpɒlɪɡɒn/) is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of the polygon is sometimes called its body. An ngon is a polygon with n sides; for example, a triangle is a 3gon. A polygon is a 2dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not selfintersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and other selfintersecting polygons [...More...]  "Polygon" on: Wikipedia Yahoo 