• Geometric shape covers standard terms for plane shapes

Types, methodologies, and terminologies of geometry

• Absolute geometry
  Absolute geometry
  Absolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832...
  
  
• Affine geometry
  Affine geometry
  In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
  
  
• Algebraic geometry
  Algebraic geometry
  Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
  
  
• Analytic geometry
  Analytic geometry
  Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
  
  
• Archimedes' use of infinitesimals
  Archimedes' use of infinitesimals
  The Method of Mechanical Theorems is a work by Archimedes which contains the first attested explicit use of infinitesimals. The work was originally thought to be lost, but was rediscovered in the celebrated Archimedes Palimpsest...
  
  
• Birational geometry
  Birational geometry
  In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
  
  
• Complex geometry
  Complex geometry
  In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
  
  
• Combinatorial geometry
• Computational geometry
  Computational geometry
  Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...
  
  
• Conformal geometry
  Conformal geometry
  In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
  
  
• Constructive solid geometry
  Constructive solid geometry
  Constructive solid geometry is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine objects...
  
  
• Contact geometry
  Contact geometry
  In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
  
  
• Convex geometry
  Convex geometry
  Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...
  
  
• Descriptive geometry
  Descriptive geometry
  Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art...
  
  
• Differential geometry
• Digital geometry
  Digital geometry
  Digital geometry deals with discrete sets considered to be digitized models or images of objects of the 2D or 3D Euclidean space.Simply put, digitizing is replacing an object by a discrete set of its points...
  
  
• Discrete geometry
  Discrete geometry
  Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
  
  
• Distance geometry
  Distance geometry
  Distance geometry is the characterization and study of sets of points based only on given values of the distances between member pairs. Therefore distance geometry has immediate relevance where distance values are determined or considered, such as in surveying, cartography and...
  
  
• Elliptic geometry
  Elliptic geometry
  Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
  
  
• Enumerative geometry
  Enumerative geometry
  In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.-History:...
  
  
• Epipolar geometry
  Epipolar geometry
  Epipolar geometry is the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points...
  
  
• Euclidean geometry
  Euclidean geometry
  Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
  
  
• Finite geometry
  Finite geometry
  A finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...
  
  
• Geometry of numbers
  Geometry of numbers
  In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
  
  
• Hyperbolic geometry
  Hyperbolic geometry
  In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
  
  
• Information geometry
  Information geometry
  Information geometry is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. It derives its name from the fact that the Fisher information is used as the Riemannian metric when considering the geometry of probability distribution families...
  
  
• Integral geometry
  Integral geometry
  In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...
  
  
• Inversive geometry
• Inversive ring geometry
  Inversive ring geometry
  In mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields....
  
  
• Klein geometry
  Klein geometry
  In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.For background and motivation...
  
  
• Lie sphere geometry
  Lie sphere geometry
  Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century...
  
  
• Non-Euclidean geometry
  Non-Euclidean geometry
  Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
  
  
• Numerical geometry
• Ordered geometry
  Ordered geometry
  Ordered geometry is a form of geometry featuring the concept of intermediacy but, like projective geometry, omitting the basic notion of measurement...
  
  
• Parabolic geometry
  Parabolic geometry (differential geometry)
  In differential geometry and the study of Lie groups, a parabolic geometry is a homogeneous space G/P which is the quotient of a semisimple Lie group G by a parabolic subgroup P...
  
  
• Plane geometry
• Projective geometry
  Projective geometry
  In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
  
  
• Quantum geometry
  Quantum geometry
  In theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales...
  
  
• Riemannian geometry
  Riemannian geometry
  Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
  
  
• Ruppeiner geometry
  Ruppeiner geometry
  Ruppeiner geometry is thermodynamic geometry using the language of Riemannian geometry to study thermodynamics. George Ruppeiner proposed it in 1979...
  
  
• Spherical geometry
  Spherical geometry
  Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
  
  
• Symplectic geometry
• Synthetic geometry
  Synthetic geometry
  Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...
  
  
• Systolic geometry
• Taxicab geometry
  Taxicab geometry
  Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their coordinates...
  
  
• Toric geometry
  Toric geometry
  In algebraic geometry, a toric variety or torus embedding is a normal variety containing an algebraic torus as a dense subset, such that the action of the torus on itself extends to the whole variety.-The toric variety of a fan:...
  
  
• Transformation geometry
  Transformation geometry
  In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. Felix Klein, who pioneered this point of view, was himself interested in mathematical education. It took many years, though, for his "modern" point of view to...
  
  
• Tropical geometry
  Tropical geometry
  Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry. Its leading ideas had appeared in different guises in previous works of George M...
  
  

Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, foundations

• Hilbert's axioms
  Hilbert's axioms
  Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
  
  
• Point
  Point (geometry)
  In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
  
  
• Locus
  Locus (mathematics)
  In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
  
  
• Line
  Line (mathematics)
  The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
  
  
• Parallel
  Parallel (geometry)
  Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
  
  
• Angle
  Angle
  In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
  
  
  • Concurrent lines
    Concurrent lines
    In geometry, two or more lines are said to be concurrent if they intersect at a single point.In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:...
    
    
  • Adjacent angles
    Adjacent angles
    In geometry, adjacent angles, often shortened as adj. ∠s, are angles that have a common ray coming out of the vertex going between two other rays. In other words, they are angles that are side by side, or adjacent.- Complementary adjacent angles :...
    
    
  • Central angle
    Central angle
    A central angle is an angle which vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is equal to the central angle itself...
    
    
  • Complementary angles
    Complementary angles
    In geometry, complementary angles are angles whose measures sum to 90°. If the two complementary angles are adjacent their non-shared sides form a right angle....
    
    
  • Inscribed angle
    Inscribed angle
    In geometry, an inscribed angle is formed when two secant lines of a circle intersect on the circle....
    
    
  • Internal angle
    Internal angle
    In geometry, an interior angle is an angle formed by two sides of a polygon that share an endpoint. For a simple, convex or concave polygon, this angle will be an angle on the 'inner side' of the polygon...
    
    
  • Supplementary angles
    Supplementary angles
    Supplementary angles are pairs of angles that add up to 180 degrees. Thus the supplement of an angle of x degrees is an angle of degrees....
    
    
  • Angle trisection
    Angle trisection
    Angle trisection is a classic problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one-third of a given arbitrary angle, using only two tools: an un-marked straightedge, and a compass....
    
    
• Congruence
  Congruence (geometry)
  In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
  
  
  • Reflection
    Reflection (mathematics)
    In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
    
    
  • Rotation
    Rotation (mathematics)
    In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
    
    
  • Coordinate rotations and reflections
    Coordinate rotations and reflections
    In geometry, 2D coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1...
    
    
  • Translation
    Translation (geometry)
    In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
    
    
  • Glide reflection
    Glide reflection
    In geometry, a glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result...
    
    
• Similarity
  Similarity (geometry)
  Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
  
  
  • Similarity transformation
  • Homothety
• Shear mapping

Euclidean plane geometry

• 2D computer graphics
  2D computer graphics
  2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them...
  
  
• 2D geometric model
  2D geometric model
  A 2D geometric model is a geometric model of an object as two-dimensional figure, usually on the Euclidean or Cartesian plane.Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and machine parts made of...
  
  
• Altitude
  Altitude (triangle)
  In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...
  
  
• Brahmagupta's formula
  Brahmagupta's formula
  In Euclidean geometry, Brahmagupta's formula finds the area of any quadrilateral given the lengths of the sides and some of the angles. In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle.- Basic form :...
  
  
• Compass and straightedge constructions
  • Squaring the circle
    Squaring the circle
    Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
    
    
• Complex geometry
  Complex geometry
  In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
  
  
• Conic section
  Conic section
  In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
  
  
  • Focus
    Focus (geometry)
    In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...
    
    
  • Circle
    Circle
    A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
    
    
    • List of circle topics
    • Thales' theorem
      Thales' theorem
      In geometry, Thales' theorem states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem...
      
      
    • Circumcircle
    • Concyclic
    • Incircle and excircles of a triangle
      Incircle and excircles of a triangle
      In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides...
      
      
    • Orthocentric system
      Orthocentric system
      In geometry, an orthocentric system is a set of four points in the plane one of which is the orthocenter of the triangle formed by the other three....
      
      
    • Monge's theorem
      Monge's theorem
      In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is inside one of the others, the three intersection points of the three pairs of external tangent lines are in fact collinear....
      
      
    • Power center
    • Nine-point circle
      Nine-point circle
      In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points defined from the triangle...
      
      
    • Circle points segments proof
    • Mrs. Miniver's problem
      Mrs. Miniver's problem
      Mrs. Miniver's problem is a geometry problem about circles. Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference of A and B .The problem derives from "A Country House Visit", one of Jan Struther's newspaper articles...
      
      
    • Isoperimetric theorem
    • Annulus
      Annulus (mathematics)
      In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
      
      
    • Ptolemaios' theorem
    • Steiner chain
      Steiner chain
      In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles , where n is finite and each circle in the chain is tangent to the previous and next circles in the chain...
      
      
  • Eccentricity
    Eccentricity (mathematics)
    In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...
    
    
  • Ellipse
    Ellipse
    In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
    
    
    • Semi-major axis
      Semi-major axis
      The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
      
      
  • Hyperbola
    Hyperbola
    In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
    
    
  • Parabola
    Parabola
    In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
    
    
  • Matrix representation of conic sections
  • Dandelin spheres
    Dandelin spheres
    In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin...
    
    
• Curve of constant width
  Curve of constant width
  In geometry, a curve of constant width is a convex planar shape whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines.More generally, any compact...
  
  
• Cyclic quadrilateral
  Cyclic quadrilateral
  In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. Other names for these quadrilaterals are chordal quadrilateral and inscribed...
  
  
• Euler's line
  Euler's line
  In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral; it passes through several important points determined from the triangle...
  
  
• Equilateral triangle
• Frieze group
  Frieze group
  A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  
• Golden angle
  Golden angle
  In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section; that is, into two arcs such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full...
  
  
• Heron's formula
  • Heronian triangle
    Heronian triangle
    In geometry, a Heronian triangle is a triangle whose sidelengths and area are all rational numbers. It is named after Hero of Alexandria. Any such rational triangle can be scaled up to a corresponding triangle with integer sides and area, and often the term Heronian triangle is used to refer to...
    
    
• Holditch's theorem
  Holditch's theorem
  In plane geometry, Holditch's theorem states that if a chord of fixed length is allowed to rotate around a convex closed curve, then the locus of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose area is less than that of the original curve by...
  
  
• Interactive geometry software
  Interactive geometry software
  Interactive geometry software are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting a few points and using them to define new objects such as lines, circles or other points...
  
  
• Parallel postulate
  Parallel postulate
  In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
  
  
• Pedal triangle
  Pedal triangle
  In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.More specifically, consider a triangle ABC, and a point P that is not one of the vertices A, B, C. Drop perpendiculars from P to the three sides of the triangle...
  
  
• Polygon
  Polygon
  In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
  
  
  • Star polygon
  • Pick's theorem
    Pick's theorem
    Given a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the...
    
    
  • Shape dissection
  • Bolyai–Gerwien theorem
    Bolyai–Gerwien theorem
    In geometry, the Bolyai–Gerwien theorem, named after Farkas Bolyai and Paul Gerwien, states that any two simple polygons of equal area are equidecomposable; i.e...
    
    
  • Poncelet–Steiner theorem
  • Polygon triangulation
    Polygon triangulation
    In computational geometry, polygon triangulation is the decomposition of a polygonal area P into a set of triangles, i.e., finding the set of triangles with pairwise non-intersecting interiors whose union is P....
    
    
• Pons asinorum
  Pons asinorum
  Pons asinorum is the name given to Euclid's fifth proposition in Book 1 of his Elements of geometry, also known as the theorem on isosceles triangles. It states that the angles opposite the equal sides of an isosceles triangle are equal...
  
  
• Pythagorean triangle
• Sangaku
  Sangaku
  Sangaku or San Gaku are Japanese geometrical puzzles in Euclidean geometry on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes.-History:The Sangaku were painted in color on wooden tablets and hung in the...
  
  
• Straightedge
  Straightedge
  A straightedge is a tool with an edge free from curves, or straight, used for transcribing straight lines, or checking the straightness of lines...
  
  
• Symmedian
  Symmedian
  Symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle , and reflecting the line over the corresponding angle bisector...
  
  
• Tessellation
  Tessellation
  A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
  
  
  • Prototile
    Prototile
    In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others...
    
    
  • Aperiodic tiling
    Aperiodic tiling
    An aperiodic tiling is a tiling obtained from an aperiodic set of tiles. Properly speaking, aperiodicity is a property of particular sets of tiles; any given finite tiling is either periodic or non-periodic...
    
    
    • Wang tile
      Wang tile
      Wang tiles , first proposed by mathematician, logician, and philosopher Hao Wang in 1961, are a class of formal systems...
      
      
    • Penrose tiling
      Penrose tiling
      A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original...
      
      
• Trapezoid
  Trapezoid
  In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...
  
  , trapezium
  Trapezium
  The word trapezium has several meanings:* - a quadrilateral with one pair of parallel sides ....
  
  
  • Isosceles trapezoid
    Isosceles trapezoid
    In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid...
    
    
• Triangle
  Triangle
  A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
  
  
  • Pythagorean theorem
    Pythagorean theorem
    In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
    
    
  • Triangle inequality
    Triangle inequality
    In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....
    
    
  • Pedoe's inequality
    Pedoe's inequality
    In geometry, Pedoe's inequality, named after Daniel Pedoe, states that if a, b, and c are the lengths of the sides of a triangle with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then...
    
    
• Trigonometry
  Trigonometry
  Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
  
  
  • List of trigonometry topics
• Wallpaper group
  Wallpaper group
  A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  

3-dimensional Euclidean geometry (solid geometry

Solid geometry

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry...

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• 3D projection
  3D projection
  3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.-...
  
  
• 3D computer graphics
  3D computer graphics
  3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
  
  
  • Binary space partitioning
    Binary space partitioning
    In computer science, binary space partitioning is a method for recursively subdividing a space into convex sets by hyperplanes. This subdivision gives rise to a representation of the scene by means of a tree data structure known as a BSP tree.Originally, this approach was proposed in 3D computer...
    
    
  • Ray tracing
  • Graham scan
    Graham scan
    The Graham scan is a method of computing the convex hull of a finite set of points in the plane with time complexity O. It is named after Ronald Graham, who published the original algorithm in 1972...
    
    
• Borromean rings
  Borromean rings
  In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.- Mathematical properties :...
  
  
• Cavalieri's principle
  Cavalieri's principle
  In geometry, Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura Cavalieri, is as follows:* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane...
  
  
• Cross section
  Cross section (geometry)
  In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
  
  
• Crystal
  Crystal
  A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
  
  
• Cuisenaire rods
  Cuisenaire rods
  Cuisenaire rods give students a hands-on elementary school way to learn elementary math concepts, such as the four basic arithmetic operations and working with fractions....
  
  
• Desargues' theorem
  Desargues' theorem
  In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
  
  
• Right circular cone
• Hyperboloid
• Napkin ring problem
  Napkin ring problem
  In geometry, it is somewhat surprising that the volume of a band of specified height around a sphere—the part that remains after a hole in the shape of a circular cylinder is drilled through the sphere—does not depend on the sphere's radius....
  
  
• Pappus's centroid theorem
  Pappus's centroid theorem
  In mathematics, Pappus' centroid theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution....
  
  
• Paraboloid
  Paraboloid
  In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
  
  
• Polyhedron
  Polyhedron
  In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
  
  
  • Defect
    Defect (geometry)
    In geometry, the defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would...
    
    
  • Dihedral angle
    Dihedral angle
    In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
    
    
  • Prism
    Prism (geometry)
    In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
    
    
  • Prismatoid
    Prismatoid
    In geometry, a prismatoid is a polyhedron where all vertices lie in two parallel planes....
    
    
  • Honeycomb
    Honeycomb (geometry)
    In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
    
    
  • Pyramid
    Pyramid (geometry)
    In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
    
    
  • Parallelepiped
    Parallelepiped
    In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
    
    
  • Tetrahedron
    Tetrahedron
    In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
    
    
    • Heronian tetrahedron
      Heronian tetrahedron
      A Heronian tetrahedron is a tetrahedron whose side lengths, face areas and volume are all rational numbers. The faces must therefore all be Heronian triangles....
      
      
  • Platonic solid
    Platonic solid
    In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
    
    
  • Archimedean solid
    Archimedean solid
    In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
    
    
  • Kepler-Poinsot polyhedra
  • Johnson solid
    Johnson solid
    In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
    
    
  • Uniform polyhedron
    Uniform polyhedron
    A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
    
    
  • Polyhedral compound
    Polyhedral compound
    A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram....
    
    
  • Hilbert's third problem
    Hilbert's third problem
    The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...
    
    
  • Deltahedron
    Deltahedron
    A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
    
    
  • Surface normal
    Surface normal
    A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
    
    
• 3-sphere
  3-sphere
  In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
  
  , spheroid
  Spheroid
  A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....
  
  , ellipsoid
  • Parabolic microphone
    Parabolic microphone
    A parabolic microphone is a microphone that uses a parabolic reflector to collect and focus sound waves onto a receiver, in much the same way that a parabolic antenna does with radio waves...
    
    
  • Parabolic reflector
    Parabolic reflector
    A parabolic reflector is a reflective device used to collect or project energy such as light, sound, or radio waves. Its shape is that of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis...
    
    
• Soddy's hexlet
  Soddy's hexlet
  In geometry, Soddy's hexlet is a chain of six spheres , each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, these three spheres are shown as an outer circumscribing sphere C , and two spheres A and B above and below the plane of their...
  
  
• Sphericon
  Sphericon
  The sphericon is a three-dimensional solid with one side and two edges, discovered by Colin Roberts, of Hertfordshire, England.It may be constructed from a bicone with an apex of 90 degrees, by splitting the bicone along a plane through both apexes, rotating one of the two halves by 90 degrees,...
  
  
• Stereographic projection
  Stereographic projection
  The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
  
  
• Stereometry

n-dimensional Euclidean geometry

• Ball

• • Convex hull
    Convex hull
    In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
    
    
• Coxeter group
  Coxeter group
  In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
  
  
• Euclidean distance
  Euclidean distance
  In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...
  
  
• Homothetic center
  Homothetic center
  In geometry, a homothetic center is a point from which at least two geometrically similar figures can be seen as a dilation/contraction of one another. If the center is external, the two figures are directly similar to one another; their angles have the same rotational sense...
  
  
• Hyperplane
  Hyperplane
  A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
  
  
• Lattice
  Lattice (group)
  In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
  
  
  • Ehrhart polynomial
    Ehrhart polynomial
    In mathematics, an integral polytope has an associated Ehrhart polynomial which encodes the relationship between the volume of a polytope and the number of integer points the polytope contains...
    
    
  • Leech lattice
    Leech lattice
    In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
    
    
  • Minkowski's theorem
    Minkowski's theorem
    In mathematics, Minkowski's theorem is the statement that any convex set in Rn which is symmetric with respect to the origin and with volume greater than 2n d contains a non-zero lattice point...
    
    
• Packing and covering
  • Sphere packing
    Sphere packing
    In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space...
    
    
    • Kepler conjecture
      Kepler conjecture
      The Kepler conjecture, named after the 17th-century German astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic...
      
      
    • Kissing number problem
      Kissing number problem
      In geometry, a kissing number is defined as the number of non-overlapping unit spheres that touch another given unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another...
      
      
  • Honeycomb
    Honeycomb (geometry)
    In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
    
    
    • Andreini tessellation
    • Uniform tessellation
      Uniform tessellation
      In geometry, a uniform tessellation is a vertex-transitive tessellations made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex....
      
      
    • Voronoi tessellation
    • Delaunay triangulation
      Delaunay triangulation
      In mathematics and computational geometry, a Delaunay triangulation for a set P of points in a plane is a triangulation DT such that no point in P is inside the circumcircle of any triangle in DT. Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the...
      
      
    • Quasicrystal
      Quasicrystal
      A quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry...
      
      
• Parallelogram law
  Parallelogram law
  In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals...
  
  
• Polytope
  Polytope
  In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
  
  
  • Schläfli symbol
  • Regular polytope
    Regular polytope
    In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
    
    
  • Regular Polytopes
    Regular Polytopes (book)
    Regular Polytopes is a mathematical geometry book written by Canadian mathematician H.S.M. Coxeter. Originally written in 1947, the book was updated and republished in 1963 and 1973....
    
    
• Sphere
  Sphere
  A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
  
  
• Quadric
  Quadric
  In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
  
  
  • Hypersphere
    Hypersphere
    In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
    
    , sphere
    Sphere
    A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
    
    
  • Spheroid
    Spheroid
    A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....
    
    
  • Ellipsoid
  • Hyperboloid
  • Paraboloid
    Paraboloid
    In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
    
    
  • Cone
    Cone (geometry)
    A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
    
    
• Torus
  Torus
  In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
  
  
• Root system
  Root system
  In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
  
  
• Similarity
  Similarity (geometry)
  Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
  
  
• Zonotope

Other Geometries (Not Euclidean)

• Projective geometry
  Projective geometry
  In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
  
  
  • Arc (projective geometry)
    Arc (projective geometry)
    In mathematics, a -arc in a finite projective plane π is a set of k points of \pi such that each line intersects A in at most d points, and there is at least one line that does intersect A in d points...
    
    
  • Desargues' theorem
    Desargues' theorem
    In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
    
    
  • Girard Desargues
  • Desarguesian plane
    Desarguesian plane
    In projective geometry a Desarguesian plane, named after Gérard Desargues, is a plane in which Desargues' theorem holds. The ordinary real projective plane is a Desarguesian plane. More generally any projective plane over a division ring is Desarguesian, and conversely Hilbert showed that any...
    
    
  • Line at infinity
    Line at infinity
    In geometry and topology, the line at infinity is a line that is added to the real plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.-Geometric formulation:In...
    
    
  • Point at infinity
  • Plane at infinity
    Plane at infinity
    In projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. The result of the addition is the projective 3-space, P^3...
    
    
  • Hyperplane at infinity
  • Projective line
    Projective line
    In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
    
    
  • Projective plane
    Projective plane
    In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
    
    
    • Oval (projective plane)
      Oval (projective plane)
      In mathematics, an oval in a projective plane is a set of points, no three collinear, such that there is a unique tangent line at each point...
      
      
    • Roman surface
      Roman surface
      The Roman surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry...
      
      
  • Projective space
    Projective space
    In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
    
    
  • Complex projective line
  • Complex projective plane
  • Fundamental theorem of projective geometry
  • Projective transformation
    • Möbius transformation
  • Cross-ratio
    Cross-ratio
    In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
    
    
  • Duality
    Duality (projective geometry)
    A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
    
    
  • Homogeneous coordinates
    Homogeneous coordinates
    In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
    
    
  • Pappus's hexagon theorem
    Pappus's hexagon theorem
    In mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...
    
    
  • Incidence
    Incidence (geometry)
    In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
    
    
  • Pascal's theorem
    Pascal's theorem
    In projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration.- Related results :This theorem...
    
    
• Affine geometry
  Affine geometry
  In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
  
  
  • Affine space
    Affine space
    In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
    
    
  • Affine transformation
    Affine transformation
    In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
    
    
• Finite geometry
  Finite geometry
  A finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...
  
  
• Differential geometry
  • Contact geometry
    Contact geometry
    In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
    
    
  • Riemannian geometry
    Riemannian geometry
    Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
    
    
  • Symplectic geometry
• Non-Euclidean plane geometry
  Non-Euclidean geometry
  Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
  
  
• Angle excess
  Angle excess
  Angle excess, also known as spherical excess is the amount by which the sum of the angles of a polygon on a sphere exceeds the sum of the angles of a polygon with the same number of sides in a plane. For instance, a plane triangle has an angle sum of 180°; an octant is a spherical triangle with...
  
  
• Hyperbolic geometry
  Hyperbolic geometry
  In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
  
  
  • Pseudosphere
    Pseudosphere
    In geometry, the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid....
    
    
  • Tractricoid
• Elliptic geometry
  Elliptic geometry
  Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
  
  
  • Spherical geometry
    Spherical geometry
    Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
    
    
• Minkowski space
  Minkowski space
  In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
  
  
• Thurston's conjecture

Numerical geometry

• Curve interpolation
• Parametric curve
  • Bézier curve
    Bézier curve
    A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case....
    
    
  • Spline
    Spline (mathematics)
    In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...
    
    
    • Hermite spline
      Hermite spline
      In the mathematical subfield of numerical analysis, a Hermite spline is a spline curve where each polynomial of the spline is in Hermite form.-See also:*Cubic Hermite spline*Hermite polynomials*Hermite interpolation...
      
      
      • Beta spline
        • B-spline
          B-spline
          In the mathematical subfield of numerical analysis, a B-spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. B-splines were investigated as early as the nineteenth century by Nikolai Lobachevsky...
          
          
    • Higher-order spline
    • NURBS
• Parametric surface
  Parametric surface
  A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence...
  
  
  • Parametric patch

Geometric algorithms

• Convex hull
  Convex hull
  In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
  
   construction
• Euclidean shortest path
  Euclidean shortest path
  The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles....
  
  
• Point in polygon
  Point in polygon
  In computational geometry, the point-in-polygon problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon...
  
  
• Point location
  Point location
  The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics, geographic information systems , motion planning, and computer aided design ....
  
  
• Hidden line removal
  Hidden line removal
  Hidden line removal is an extension of wireframe model rendering where lines covered by surfaces are not drawn.This is not the same as hidden face removal since this involves depth and occlusion while the other involves normals....
  
  

Various

• Coordinate-free treatment
  Coordinate-free treatment
  A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its ideas without reference to any particular coordinate system....
  
  
• Chirality
  Chirality (mathematics)
  In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
  
  
  • Handedness
    Handedness
    Handedness is a human attribute defined by unequal distribution of fine motor skills between the left and right hands. An individual who is more dexterous with the right hand is called right-handed and one who is more skilled with the left is said to be left-handed...
    
    
  • Relative direction
    Relative direction
    The most common relative directions are left, right, forward, backward, up, and down. No absolute direction corresponds to any of the relative directions. This is a consequence of the translational invariance of the laws of physics: nature, loosely speaking, behaves the same no matter what...
    
    
  • Mirror image
    Mirror image
    A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...
    
    
• Erlangen program
  Erlangen program
  An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...
  
  
• Four-dimensional space
• Group action
  Group action
  In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
  
  , invariant
  Invariant (mathematics)
  In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
  
  
• Hadwiger's theorem
  Hadwiger's theorem
  In integral geometry , Hadwiger's theorem characterises the valuations on convex bodies in Rn. It was proved by Hugo Hadwiger.-Valuations:...
  
  
• Infinitesimal transformation
  Infinitesimal transformation
  In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A...
  
  
• Pi
  Pi
  ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
  
  
• Polar sine
  Polar sine
  In mathematics, the polar sine of a vertex angle of a polytope is defined as follows. Let v1, ..., vn, n ≥ 2, be non-zero vectors from the vertex in the directions of the edges...
  
  
• Symmetry
  Symmetry
  Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
  
  , shape
  Shape
  The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
  
  , pattern
  Pattern
  A pattern, from the French patron, is a type of theme of recurring events or objects, sometimes referred to as elements of a set of objects.These elements repeat in a predictable manner...
  
  
  • Crystal system
    Crystal system
    In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals...
    
    
  • Frieze group
    Frieze group
    A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
    
    
  • Isometry
    Isometry
    In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
    
    
  • Lattice
    Lattice (group)
    In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
    
    
  • Point group
    Point group
    In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
    
    
  • Point groups in two dimensions
    Point groups in two dimensions
    In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O, including O itself...
    
    
  • Point groups in three dimensions
    Point groups in three dimensions
    In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group...
    
    
  • Space group
    Space group
    In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
    
    
  • Symmetry group
    Symmetry group
    The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
    
    
  • Translational symmetry
    Translational symmetry
    In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
    
    
  • Wallpaper group
    Wallpaper group
    A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
    
    
• Mathematics and fiber arts
  Mathematics and fiber arts
  Mathematical ideas have been used as inspiration for a number of fiber arts including quilt making, knitting, cross-stitch, crochet, embroidery and weaving...
  
  
• Van Hiele model - Prevailing theory of how children learn to reason in geometry

Applications

• Astronomy
  Astronomy
  Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
  
  
• Computer graphics
  Computer graphics
  Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
  
  
• Image analysis
  Image analysis
  Image analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques...
  
  
• Robot control
  Robot control
  -See also:* Control theory* Mobile robot navigation* Robot kinematics* Simultaneous localization and mapping* Robot locomotion* Motion planning* Robot learning* Vision Based Robot Control...
  
  
• The Strähle construction
  Strähle construction
  Strähle's construction is a geometric method for determining the lengths for a series of vibrating strings with uniform diameters and tensions to sound pitches in a specific rational tempered musical tuning. It was first published in the 1743 Proceedings of the Royal Swedish Academy of Sciences by...
  
   is used in the design of some musical instruments.
• Burmester's theory
  Burmester's theory
  Burmester's theory is named after Ludwig Burmester . Burmester introduced geometric techniques for synthesis of linkages in the late 19th century. His approach was to compute the geometric constraints of the linkage directly from the inventor's desired movement for a floating link...
  
  for the design of mechanical linkages