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# Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. [1]

84 relations: Affine group, Affine transformation, Alfred S. Posamentier, Altitude (triangle), Angle bisector theorem, Area, Bijection, Birkhoff's axioms, Catenary, Cengage, Ceva's theorem, Circle, Compass-and-straightedge construction, Complex plane, Congruence (geometry), Corresponding sides and corresponding angles, Distance, Eccentricity (mathematics), Ellipse, Equilateral triangle, Euclid's Elements, Euclidean distance, Euclidean group, Euclidean plane isometry, Euclidean space, Exponential function, Function (mathematics), Geometric mean theorem, Geometry, George David Birkhoff, Group (mathematics), Hamming distance, Harold Scott MacDonald Coxeter, Hausdorff dimension, Hilbert's axioms, Homoeoid, Homothetic transformation, Hyperbola, Hyperbolic geometry, Hypotenuse, Intercept theorem, Isosceles triangle, Jaccard index, John Wallis, John Wiley & Sons, Lipschitz continuity, Logarithm, Logarithmic scale, Logarithmic spiral, Measure (mathematics), ... Expand index (34 more) »

## Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

## Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

## Alfred S. Posamentier

Alfred S. Posamentier (born October 18, 1942) is an American educator and a lead commentator on American math and science education, regularly contributing to The New York Times and other news publications.

## Altitude (triangle)

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex).

## Angle bisector theorem

In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle.

## Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Birkhoff's axioms

In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms.

## Catenary

In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.

## Cengage

Cengage is an educational content, technology, and services company for the higher education, K-12, professional, and library markets worldwide.

## Ceva's theorem

Ceva's theorem is a theorem about triangles in Euclidean plane geometry.

## Circle

A circle is a simple closed shape.

## Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

## Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

## Congruence (geometry)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

## Corresponding sides and corresponding angles

In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons.

## Distance

Distance is a numerical measurement of how far apart objects are.

## Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section.

## Ellipse

In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

## Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal.

## Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

## Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

## Euclidean group

In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space.

## Euclidean plane isometry

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Geometric mean theorem

The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the lengths of the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## George David Birkhoff

George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Hamming distance

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different.

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

## Hausdorff dimension

Hausdorff dimension is a measure of roughness in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a space, taking into account the distance between its points.

## Hilbert's axioms

Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry.

## Homoeoid

A homoeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D).

## Homothetic transformation

In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if) or reverse (if) the direction of all vectors.

## Hyperbola

In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set.

## Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

## Hypotenuse

In geometry, a hypotenuse (rarely: hypothenuse) is the longest side of a right-angled triangle, the side opposite of the right angle.

## Intercept theorem

The intercept theorem, also known as Thales' theorem (not to be confused with another theorem with that name) or basic proportionality theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels.

## Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

## Jaccard index

The Jaccard index, also known as Intersection over Union and the Jaccard similarity coefficient (originally coined coefficient de communauté by Paul Jaccard), is a statistic used for comparing the similarity and diversity of sample sets.

## John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

## Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

## Logarithmic scale

A logarithmic scale is a nonlinear scale used when there is a large range of quantities.

## Logarithmic spiral

A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature.

## Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

## Menelaus's theorem

Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry.

## Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

## Nearest neighbor search

Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point.

## Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

## Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

## Numerical taxonomy

Numerical taxonomy is a classification system in biological systematics which deals with the grouping by numerical methods of taxonomic units based on their character states.

## Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.

## Packing dimension

In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space.

## Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.

## 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.

## Polygon

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

## Proportionality (mathematics)

In mathematics, two variables are proportional if there is always a constant ratio between them.

## Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Rectangle

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.

## Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

## Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

## Rhombus

In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length.

## Right triangle

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).

## Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

## Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

## Scaling (geometry)

In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

## Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).

## Semantic similarity

Semantic similarity is a metric defined over a set of documents or terms, where the idea of distance between them is based on the likeness of their meaning or semantic content as opposed to similarity which can be estimated regarding their syntactical representation (e.g. their string format).

## Shape

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition.

## Solution of triangles

Solution of triangles (solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known.

## Square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.

## Synthetic geometry

Synthetic geometry (sometimes referred to as axiomatic or even pure geometry) is the study of geometry without the use of coordinates or formulas.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Transitive relation

In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.

## Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

## Translational symmetry

In geometry, a translation "slides" a thing by a: Ta(p).

## Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.

## Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

## References

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