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Geometry
 Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly "stretching", possibly with additional rotation, i.e., both have the same shape, or additionally the mirror image is taken, i.e., one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. Two triangles are similar if and only if they have the same three angles, the so-called "AAA" condition. However, since the sum of the interior angles in a triangle is fixed, as long as two angles are the same, all three are, called "AA". Image File history File links Download high-resolution version (936x648, 136 KB) Some similar geometric shapes -- and some that are not. ...
An example of congruence. ...
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ...
Shape (OE. sceap Eng. ...
This article is about the Twilight Zone episode. ...
Circle illustration This article is about the shape and mathematical concept of circle. ...
For other uses, see Square. ...
A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παÏαβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...
For other uses, see Ellipse (disambiguation). ...
In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ...
For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...
Similar triangles If triangle ABC is similar to triangle DEF, then this relation can be denoted as  In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°.
Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following: There are several meanings for the word deduction: Natural deduction Deductive reasoning Deductions in terms of taxation, such as Itemized deductions Standard deduction See also: Logic Venn diagram Inductive reasoning Both statistics and the scientific method rely on both induction and deduction. ...
    This idea can be extended to similar polygons with any number of sides. That is, given any two similar polygons, the corresponding sides are proportional. Look up polygon in Wiktionary, the free dictionary. ...
Angle/side similarities A concept commonly taught in high school mathematics is that of proving the "angle" and "side" theorems, which can be used to define two triangles as similar (or indeed, congruent).
In each of these three-letter acronyms, A stands for equal angles, and S for equal sides. For example, ASA refers to an angle, side and angle that are all equal and adjacent, in that order. In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...
Look up adjacent in Wiktionary, the free dictionary. ...
- AAA - Angle-Angle-Angle. If two triangles share three common angles, they are similar. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) Additionally, since the interior angles of a triangle have a sum of 180°, having two triangles with only two common angles (sometimes known as AA) implies similarity as well.
See also: Congruence (geometry) An example of congruence. ...
Similarity in Euclidean space One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean space is a function f from the space into itself that multiplies all distances by the same positive scalar r, so that for any two points x and y we have In mathematics, a dilation is a function f from a metric space into itself that satisfies the identity where d(x,y) is the distance from x to y and r is some positive real number. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...
 where "d(x,y)" is the Euclidean distance from x to y. Two sets are called similar if one is the image of the other under such a similarity. In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...
A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an isometry. In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
Viewing the complex plane as a 2-dimensional space over the reals, the 2D similarity transformations expressed in terms of the complex plane are f(z) = az + b and  , and all affine transformations are of the form  (a, b, and c complex). In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b...
Similarity in general metric spaces In a general metric space (X, d), an exact similitude is a function f from the metric space X into itself that multiplies all distances by the same positive scalar r, called f's contraction factor, so that for any two points x and y we have Image File history File links Sierpinski_triangle_(blue). ...
Image File history File links Sierpinski_triangle_(blue). ...
Sierpinski triangle The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński who described it in 1916. ...
In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...
 Weaker versions of similarity would for instance have f be a bi-Lipschitz function and the scalar r a limit In mathematics, a function f : D → R defined on a set D of real numbers with real values is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K ≥ 0 such that for all in D. The smallest such K is called the...
 This weaker version applies when the metric is an effective resistance on a topologically self-similar set.
A self-similar subset of a metric space (X, d) is a set K for which there exists a finite set of similitudes  with contraction factors  such that K is the unique compact subset of X for which
 These self-similar sets have a self-similar measure μDwith dimension D given by the formula In mathematics, a measure is a function that assigns a number, e. ...
 which is often (but not always) equal to the set's Hausdorff dimension and packing dimension. If the overlaps between the fs(K) are "small", we have the following simple formula for the measure: In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space. ...
Topology In topology, a metric space can be constructed by defining a similarity instead of a distance. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of dissimilarity: the closer the points, the lesser the distance). A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
Distance is a numerical description of how far apart objects are at any given moment in time. ...
The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are
- Positive defined:
 - Majored by the similarity of one element on itself (auto-similarity):
 and  More properties can be invoked, such as reflectivity ( ) or finiteness ( ). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).
Self-similarity Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set {.., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ..}. When this set is plotted on a logarithmic scale it has translational symmetry. A self-similar object is exactly or approximately similar to a part of itself. ...
A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. ...
A translation slides an object by a vector a: Ta(p) = p + a. ...
See also In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ...
In information theory, the Hamming distance between two strings of equal length is the number of positions for which the corresponding symbols are different. ...
The Jaccard index, also known as Jaccard distance and Jaccard similarity coefficent, is a statistic used for comparing the similarity and diversity of sample sets. ...
In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ...
Semantic similarity, variously also called semantic closeness/proximity/nearness, is a concept whereby a set of documents or terms within term lists are assigned a metric based on the likeness of their meaning / semantic content. ...
Data clustering is a common technique for statistical data analysis, which is used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics. ...
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