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Relation used in geometry
In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are infinite flat planes in the same
Parallel_(geometry)
Two geometries based on axioms closely related to those specifying Euclidean geometry
at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative
Non-Euclidean_geometry
Type of non-Euclidean geometry
mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean
Hyperbolic_geometry
Geometric axiom
In geometry, the parallel postulate is the fifth postulate in Euclid's Elements and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional
Parallel_postulate
Mathematical model of the physical space
exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today
Euclidean_geometry
Euclidean geometry without distance and angles
notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines
Affine_geometry
Geometry without the parallel postulate
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally
Absolute_geometry
System of moving vectors in differential geometry
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If
Parallel_transport
Straight figure with zero width and depth
between two parallel lines Distance from a point to a line Flat (geometry) Incidence (geometry) Line segment Generalised circle Locus Plane (geometry) Polyline
Line_(geometry)
Branch of mathematics
and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed
Geometry
Type of geometry
for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said
Projective_geometry
Non-Euclidean geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines
Elliptic_geometry
Study of geometries as axiomatic systems
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean
Foundations_of_geometry
Geometry of the surface of a sphere
hyperbolic geometry; each of these new geometries makes a different change to the parallel postulate. The principles of any of these geometries can be extended
Spherical_geometry
Geometry without using coordinates
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic
Synthetic_geometry
Line intersecting 2 coplanar lines at 2 points
In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether
Transversal_(geometry)
Topics referred to by the same term
Look up parallel in Wiktionary, the free dictionary. Parallel may refer to: Parallel (geometry), two lines in the Euclidean plane which never intersect
Parallel
Field of mathematics which studies incidence structures
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that
Incidence_geometry
Topics referred to by the same term
computation Parallel evolution, the independent emergence of a similar trait in different unrelated species Parallel (geometry), the property of parallel lines
Parallelism
Overview of and topical guide to geometry
Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive
Outline_of_geometry
Convex quadrilateral with at least one pair of parallel sides
century American geometry textbooks define a trapezium as having no parallel sides, a trapezoid as having exactly one pair of parallel sides, and a parallelogram
Trapezoid
Geometrical concept
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional
Cross_section_(geometry)
Type of metric geometry
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined
Taxicab_geometry
Branch of differential geometry and differential topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds
Symplectic_geometry
Branch of mathematics
space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's
Algebraic_geometry
Study of geometry using a coordinate system
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts
Analytic_geometry
Generalization of the concept of parallel lines
A parallel curve of a given (progenitor) curve is the envelope of a family of congruent (equal-radius) circles centered on the curve. It generalises the
Parallel_curve
Projection of a 3D object onto a plane via parallel rays
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane
Parallel_projection
Solid with 2 parallel n-gonal bases connected by n parallelograms
In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the
Prism_(geometry)
Geometric system with a finite number of points
intersect at a unique point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly
Finite_geometry
Branch of mathematics
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.
Differential_geometry
Index of articles associated with the same name
In geometry, Max Dehn introduced two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel
Dehn_plane
Mathematical treatise by Euclid
published a description of acute geometry (or hyperbolic geometry), a geometry which assumed a different form of the parallel postulate. It is in fact possible
Euclid's_Elements
Three-dimensional solid
one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also
Cylinder
Branch of geometry
Descriptive geometry is a type of technical drawing and the branch of geometry which allows the representation of three-dimensional objects in two dimensions
Descriptive_geometry
Modern formulation of Euclid's parallel postulate
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line
Playfair's_axiom
Property shared by codirectional lines
In geometry, direction, also known as spatial direction, vector direction or relative direction, is the common characteristic of all rays which coincide
Direction_(geometry)
Historical development of geometry
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry
History_of_geometry
Geometry where the axiom of Archimedes is negated
functions. In this geometry, there are significant differences from Euclidean geometry; in particular, there are infinitely many parallels to a straight line
Non-Archimedean_geometry
Type of program in computer graphics
Shaders act on data such as vertices and primitives, generate or morph geometries and fragments, and calculate the colors in a rendered image. Shaders can
Shader
Fundamental result in geometry
Euclidean geometry, the triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate
Sum_of_angles_of_a_triangle
Relationship between two lines that meet at a right angle
in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely
Perpendicular
Topics referred to by the same term
click, ǀ Lateral click, a character in African languages, ǁ Parallel (geometry), ∥ Parallel (operator), also ∥ Logical disjunction, || in several programming
Vertical_bar_(disambiguation)
Fundamental object of geometry
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical
Point_(geometry)
Geometrical term
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line R {\displaystyle R} through a point P {\displaystyle
Limiting_parallel
Planar movement within a Euclidean space without rotation
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction
Translation_(geometry)
Conic solid with a polygonal base
Prismatoids", Discrete & Computational Geometry, 18: 13–52, doi:10.1007/PL00009307. O'Leary, Michael (2010), Revolutions of Geometry, John Wiley & Sons, p. 10,
Pyramid_(geometry)
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate
Constructions in hyperbolic geometry
Constructions_in_hyperbolic_geometry
Mathematical operation modeling parallel resistors
The parallel operator ‖ {\displaystyle \|} (pronounced "parallel", following the parallel lines notation from geometry; also known as reduced sum, parallel
Parallel_(operator)
Artistic concept relating to perspective
viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing
Vanishing_point
Affine subspace of a Euclidean space
In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space. Particularly, in the case the parent space
Flat_(geometry)
Shape with three sides
three sides connected at three corners. It is one of the basic shapes in geometry and the simplest of polygons. The corners, also called vertices, are zero-dimensional
Triangle
Common elements of two or more sets
objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet
Intersection
Bottom of a geometric figure
In geometry, a base is a side of a polygon or a face of a polyhedron, particularly one oriented perpendicular to the direction in which height is measured
Base_(geometry)
Study of angle-preserving transformations
In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines
Inversive_geometry
Function in mathematics
curve or family of curves in a parallel and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data
Connection_(mathematics)
Type of mechanical system
Clavel - US Patent 4,976,582, 1990 R. Clavel, Delta: a fast robot with parallel geometry, Proc 18th Int Symp Ind Robots; Sydney, Australia (1988), pp. 91-100
Parallel_manipulator
Family of geometric objects with a common property
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane
Pencil_(geometry)
Geometric line segment whose endpoints lie on a circular arc
In geometry, a chord (from Latin chorda 'catgut, string') of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord
Chord_(geometry)
2D surface which extends indefinitely
geometry, and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel
Plane_(mathematics)
Mathematical space with two coordinates
Tristan (2021). Visual Differential Geometry and Forms. Princeton. ISBN 0-691-20370-9. Stillwell, John (1992). Geometry of Surfaces. Springer. doi:10
Two-dimensional_space
Lines with constant perpendicular distance between them
In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point.
Clifford_parallel
Relation between sides of a right triangle
theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of
Pythagorean_theorem
Mathematics of varieties with integer coordinates
geometry. The extensive development of algebraic geometry in the 20th century produced powerful tools to study these equations. Diophantine geometry is
Diophantine_geometry
Branch of algebraic geometry
arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around
Arithmetic_geometry
Branch of mathematics
Noncommutative geometry (NCG) is a branch of mathematics that studies geometric ideas through noncommutative algebras. In ordinary geometry, a space can
Noncommutative_geometry
Branch of geometry that studies combinatorial properties and constructive methods
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric
Discrete_geometry
Technique of illustration
object are produced (called primary views), with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative
Multiview orthographic projection
Multiview_orthographic_projection
Geometric model of the physical space
In geometry, a three-dimensional (3D) space is a mathematical space in which three values (termed coordinates) are required to determine the position of
Three-dimensional_space
Framework of distances and directions
type of geometry that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinite number of parallel lines pass
Space
Construct allowing differentiation of tangent vector fields of manifolds
In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector
Affine_connection
Branch of computer science
Computational geometry is a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry. Some purely geometrical
Computational_geometry
Common point(s) shared by two lines in Euclidean geometry
A } {\displaystyle \{A\}} . Non-Euclidean geometry describes spaces in which one line may not be parallel to any other lines, such as a sphere, and spaces
Line–line_intersection
Property of a mathematical space
back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William
Dimension
Geometric transformation that preserves lines but not angles nor the origin
In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines
Affine_transformation
Given a triangle and a point, constructs a second triangle with a special point
Pedoe: Geometry: A Comprehensive Course. Dover, 1970, pp. 35–36, 114–115 Daniel Pedoe: "On (what should be) a Well-Known Theorem in Geometry." The American
Maxwell's_theorem_(geometry)
Branch of geometry
geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry
Convex_geometry
Means of projecting three-dimensional objects in two dimensions
three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection
Orthographic_projection
Quadrilateral with two pairs of parallel sides
In Euclidean geometry, a parallelogram is a chiral simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing
Parallelogram
Euclidean space without distance and angles
to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry. As in Euclidean space, the fundamental
Affine_space
Set of points equidistant from a center
Small circles on the sphere that are parallel to the equator are circles of latitude (or parallels). In geometry unrelated to astronomical bodies, geocentric
Sphere
Hexahedron with parallelogram faces
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning).
Parallelepiped
Statement that is taken to be true
might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its
Axiom
Length of a line segment
ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's Elements, distances were not represented as numbers
Euclidean_distance
Property of objects which are scaled or mirrored versions of each other
Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry (where Wallis's postulate is false) similar triangles
Similarity_(geometry)
Axiomatically defined geometrical space
In geometry, an affine plane is a system of points and lines that satisfy the following axioms: Any two distinct points lie on a unique line. Given any
Affine plane (incidence geometry)
Affine_plane_(incidence_geometry)
Part of a line that is bounded by two distinct end points; line with two endpoints
can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these. The last
Line_segment
Straight line segment that passes through the centre of a circle
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It
Diameter
Ancient Greek mathematician (fl. 300 BC)
Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated
Euclid
Mathematical construct in engineering
, and use the parallel axis theorem to derive the second moment of area with respect to the x ′ {\displaystyle x'} axis. The parallel axis theorem states
Second_moment_of_area
In geometry, a slab is a region between two parallel lines in the Euclidean plane, or between two parallel planes in three-dimensional Euclidean space
Slab_(geometry)
In geometry, an axis-aligned object (axis-parallel, axis-oriented) is an object in n-dimensional space whose shape is aligned with the coordinate axes
Axis-aligned_object
Field of mathematics dealing with three-dimensional Euclidean spaces
Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional
Solid_geometry
Shape with four equal sides and angles
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles
Square
Design technique
descriptive geometry and is a two-dimensional representation of a three-dimensional object. It is a parallel projection (the lines of projection are parallel both
3D_projection
Quadrilateral with four right angles
In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular
Rectangle
Geometric transformation
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale
Scaling_(geometry)
Completion of the usual space with "points at infinity"
for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said
Projective_space
Circles of latitude
Following are circles of latitude between the 45th parallel north and the 50th parallel north: 46° Map all coordinates using OpenStreetMap Download coordinates
Circles of latitude between the 45th parallel north and the 50th parallel north
Circles_of_latitude_between_the_45th_parallel_north_and_the_50th_parallel_north
Study of complex manifolds and several complex variables
geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry
Complex_geometry
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