Outline of geometry

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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.

Contents

Types, methodologies, and terminologies of geometry.

General geometry concepts

General concepts

Measurements

Trigonometric functions

Vectors

Vector spaces and complex dimensions

Symmetry, shape and pattern

Euclidean geometry foundations

Euclidean plane geometry

3-dimensional Euclidean geometry

n-dimensional Euclidean geometry

Other geometries (not Euclidean)

Numerical geometry

Geometric algorithms

History of geometry

History of geometry

Generalizations

Geometric algorithms

Connections to other fields

Lists

See also

Further reading

Related Research Articles

In geometry, a polygon is a plane figure made up of line segments connected to form a closed polygonal chain.

<span class="mw-page-title-main">Quadrilateral</span> Polygon with four sides and four corners

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .

<span class="mw-page-title-main">Triangle</span> Shape with three sides

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.

<span class="mw-page-title-main">Rectangle</span> Quadrilateral with four right angles

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as  ABCD.

<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

<span class="mw-page-title-main">Shape</span> Form of an object or its external boundary

A shape is a graphical representation of an object's form or its external boundary, outline, or external surface; it is distinct from other object properties, such as color, texture, or material type. In geometry, shape excludes information about the object's location, scale, orientation and reflection. A figure is a representation including both shape and size.

<span class="mw-page-title-main">Cyclic quadrilateral</span> Quadrilateral whose vertices can all fall on a single circle

In Euclidean geometry, a cyclic quadrilateral or inscribed 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. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

<span class="mw-page-title-main">Hyperbolic geometry</span> Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice versa.

<span class="mw-page-title-main">Affine geometry</span> Euclidean geometry without distance and angles

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring the metric notions of distance and angle.

<span class="mw-page-title-main">Tessellation</span> Tiling of a plane in mathematics

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.

<span class="mw-page-title-main">Discrete geometry</span> 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 objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

<span class="mw-page-title-main">Midpoint</span> Point on a line segment which is equidistant from both endpoints

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

<span class="mw-page-title-main">Concyclic points</span> Points on a common circle

In geometry, a set of points are said to be concyclic if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its circumscribing circle or circumcircle. All concyclic points are equidistant from the center of the circle.

<span class="mw-page-title-main">Isosceles trapezoid</span> Trapezoid symmetrical about an axis

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure, or as a trapezoid whose diagonals have equal length. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length, and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure.

<span class="mw-page-title-main">Concurrent lines</span> Lines which intersect at a single point

In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point.

<span class="mw-page-title-main">Varignon's theorem</span> The midpoints of the sides of an arbitrary quadrilateral form a parallelogram

In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was published posthumously in 1731.

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

<span class="mw-page-title-main">Lexell's theorem</span> Characterizes spherical triangles with fixed base and area

In spherical geometry, Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle, called Lexell's circle or Lexell's locus, passing through each of the two points antipodal to the two base vertices.