|
|
| (686 intermediate revisions by more than 100 users not shown) |
| Line 1: |
Line 1: |
| {{For|other uses|Similarity transformation (disambiguation){{!}}Similarity transformation|Similarity (disambiguation)}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
|
| |
|
| [[Image:Similar-geometric-shapes.svg|thumb|300px|Figures shown in the same color are similar]]
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| 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. More precisely, one can be obtained from the other by uniformly [[Scaling (geometry)|scaling]] (enlarging or shrinking), possibly with additional [[Translation (geometry)|translation]], [[Rotation (mathematics)|rotation]] and [[Reflection (mathematics)|reflection]]. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is [[congruence (geometry)|congruent]] to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level.
| | * Only registered users will be able to execute this rendering mode. |
| | * Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere. |
|
| |
|
| For example, all [[circle]]s are similar to each other, all [[square (geometry)|square]]s are similar to each other, and all [[equilateral triangles]] are similar to each other. On the other hand, [[ellipse]]s are ''not'' all similar to each other, [[rectangles]] are not all similar to each other, and [[isosceles triangle]]s are not all similar to each other.
| | Registered users will be able to choose between the following three rendering modes: |
|
| |
|
| If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
|
| |
|
| This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
|
| |
|
| ==Similar triangles==
| | '''source''' |
| In [[geometry]] two triangles, <math>\triangle ABC</math> and <math>\triangle A'B'C'</math>, are '''similar''' if and only if corresponding angles are [[Congruence (geometry)|congruent]] and the lengths of [[corresponding sides]] are [[Proportionality (mathematics)|proportional]].<ref>{{harvnb|Sibley|1998|loc=p. 35}}</ref> It can be shown that two triangles having congruent angles (''equiangular triangles'') are similar, that is, the corresponding sides can be proved to be proportional. This is known as the ''AAA similarity theorem''.<ref>{{harvnb|Stahl|2003|loc=p. 127}}. This is also proved in [[Euclid's Elements]], Book VI, Proposition 4.</ref> Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.<ref>For instance, {{harvnb|Venema|2006|loc=p. 122}} and {{harvnb|Henderson|Taimiṇa|2005|loc=p. 123}}</ref>
| | :<math forcemathmode="source">E=mc^2</math> --> |
|
| |
|
| There are several statements which are necessary and sufficient for two triangles to be similar:
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
|
| |
|
| 1. The triangles have two congruent angles,<ref>[[Euclid's elements]] Book VI Proposition 4.</ref> which in Euclidean geometry implies that all their angles are congruent.<ref>This statement is not true in [[Non-euclidean geometry]] where the triangle angle sum is not 180 degrees.</ref> That is:
| | ==Demos== |
|
| |
|
| :If <math> \angle BAC</math> is equal in measure to <math>\angle B'A'C'</math>, and <math>\angle ABC</math> is equal in measure to <math>\angle A'B'C'</math>, then this implies that <math>\angle ACB</math> is equal in measure to <math>\angle A'C'B'</math>. | | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
|
| |
|
| 2. All the corresponding sides have lengths in the same ratio:<ref>[[Euclid's elements]] Book VI Proposition 5</ref>
| |
|
| |
|
| : <math> {AB \over A'B'} = {BC \over B'C'} = {AC \over A'C'}</math>. This is equivalent to saying that one triangle (or its mirror image) is an [[Homothetic transformation|enlargement]] of the other. | | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
|
| |
|
| 3. Two sides have lengths in the same ratio, and the angles included between these sides have the same measure.<ref>[[Euclid's elements]] Book VI Proposition 6</ref> For instance:
| | ==Test pages == |
| : <math> {AB \over A'B'} = {BC \over B'C'} </math> and <math>\angle ABC</math> is equal in measure to <math>\angle A'B'C'</math>.
| |
| This is known as the ''SAS Similarity Criterion''.<ref>{{harvnb|Venema|2006|loc=p. 143}}</ref>
| |
|
| |
|
| When two triangles <math>\triangle ABC</math> and <math>\triangle A'B'C'</math> are similar, one writes<ref name=PL>[[Alfred Posamentier|Posamentier, Alfred S.]] and Lehmann, Ingmar. ''The Secrets of Triangles'', Prometheus Books, 2012.</ref>{{rp|p. 22}}
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
|
| |
|
| :<math>\triangle ABC\sim\triangle A'B'C' \, </math>.
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| There are several elementary results concerning similar triangles in Euclidean geometry:<ref>{{harvnb|Jacobs|1974|loc=pp. 384 - 393}}</ref>
| | ==Bug reporting== |
| * Any two [[equilateral triangle]]s are similar. | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| * Two triangles, both similar to a third triangle, are similar to each other ([[Transitive relation|transitivity]] of similarity of triangles).
| |
| * Corresponding [[altitude (triangle)|altitudes]] of similar triangles have the same ratio as the corresponding sides.
| |
| * Two [[right triangle]]s are similar if the [[hypotenuse]] and one other side have lengths in the same ratio.<ref>{{citation|title=Lessons in Geometry, Vol. I: Plane Geometry|first=Jacques|last=Hadamard|authorlink=Jacques Hadamard|publisher=American Mathematical Society|year=2008|isbn=9780821843673|at=Theorem 120, p. 125|url=http://books.google.com/books?id=SaZwAAAAQBAJ&pg=PA125}}.</ref> | |
| | |
| Given a triangle <math>\triangle ABC</math> and a line segment <math>\overline{DE}</math> one can, with [[ruler and compass construction|straightedge and compass]], find a point ''F'' such that <math>\triangle ABC \sim \triangle DEF</math>. The statement that the point ''F'' satisfying this condition exists is ''Wallis's Postulate''<ref>Named for [[John Wallis]] (1616-1703)</ref> and is logically equivalent to [[Euclid's Fifth Axiom|Euclid's Parallel Postulate]].<ref>{{harvnb|Venema|2006|loc=p. 122}}</ref> In [[hyperbolic geometry]] (where Wallis's Postulate is false) similar triangles are congruent.
| |
| | |
| In the axiomatic treatment of Euclidean geometry given by [[George David Birkhoff|G.D. Birkhoff]] (see [[Birkhoff's axioms]]) the SAS Similarity Criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of [[Hilbert's axioms]].<ref>{{harvnb|Venema|2006|loc=p. 143}}</ref>
| |
| | |
| ==Other similar polygons== | |
| | |
| The concept of similarity extends to [[polygon]]s with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are [[Proportionality (mathematics)|proportional]] and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all [[rhombus|rhombi]] would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all [[rectangle]]s would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.
| |
| | |
| ==Similar curves==
| |
| | |
| Several types of curves have the property that all examples of that type are similar to each other. These include:
| |
| *[[Circle]]s
| |
| *[[Parabola]]s
| |
| *[[Hyperbola]]s of a specific [[eccentricity (mathematics)|eccentricity]]
| |
| *[[Ellipse]]s of a specific eccentricity
| |
| *[[Catenary|Catenaries]]
| |
| *Graphs of the [[logarithm]] function for different bases
| |
| *Graphs of the [[exponential function]] for different bases
| |
| *[[Logarithmic spiral]]s
| |
| | |
| ==Similarity in Euclidean space==
| |
| | |
| A '''similarity''' (also called a '''similarity transformation''' or '''similitude''') of a [[Euclidean space]] is a [[bijection]] ''f'' from the space onto itself that multiplies all distances by the same positive [[real number]] ''r'', so that for any two points ''x'' and ''y'' we have
| |
| | |
| :<math>d(f(x),f(y)) = r d(x,y), \, </math>
| |
| | |
| where "''d''(''x'',''y'')" is the [[Euclidean distance]] from ''x'' to ''y''.<ref>{{harvnb|Smart|1998|loc=p. 92}}</ref> The [[Scalar (mathematics)|scalar]] ''r'' has many names in the literature including; the ''ratio of similarity'', the ''stretching factor'' and the ''similarity coefficient''. When ''r'' = 1 a similarity is called an [[Euclidean plane isometry|isometry]] (rigid motion). Two sets are called '''similar''' if one is the image of the other under a similarity.
| |
| | |
| Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments.<ref>{{harvnb|Yale|1968|loc=p. 47 Theorem 2.1}}</ref> Similarities preserve angles but do not necessarily preserve orientation, ''direct similitudes'' preserve orientation and ''opposite similitudes'' change it.<ref>{{harvnb|Pedoe|1988|loc=pp. 179-181}}</ref>
| |
| | |
| The similarities of Euclidean space form a [[Group (mathematics)|group]] under the operation of composition called the ''similarities group S''.<ref>{{harvnb|Yale|1968|loc=p. 46}}</ref> The direct similitudes form a [[normal subgroup]] of ''S'' and the [[Euclidean group]] ''E(n)'' of isometries also forms a normal subgroup.<ref>{{harvnb|Pedoe|1988|loc=p. 182}}</ref> The similarities group ''S'' is itself a subgroup of the [[affine group]], so every similarity is an [[affine transformation]].
| |
| <!-- FOR LATER INCLUSION
| |
| 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 orthogonal transformation. -->
| |
| | |
| One can view the Euclidean plane as the [[complex plane]],<ref>This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line.</ref> that is, as a 2-dimensional space over the [[real number|reals]]. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by <math>f(z)=az+b</math> (direct similitudes) and <math>f(z)=a\overline z+b</math> (opposite similitudes) where ''a'' and ''b'' are complex numbers, ''a'' ≠ 0. When |''a''| = 1, these similarities are isometries.
| |
| | |
| ==Ratios of sides, of areas, and of volumes==
| |
| {{Main|Square-cube law}}
| |
| The ratio between the [[Area (geometry)|areas]] of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by three, its area is multiplied by nine — i.e. by three squared). The altitudes of similar triangles are in the same ratio as corresponding sides. If a triangle has a side of length ''b'' and an altitude drawn to that side of length ''h'' then a similar triangle with corresponding side of length ''kb'' will have an altitude drawn to that side of length ''kh''. The area of the first triangle is, ''A'' = ''bh''/2, while the area of the similar triangle will be ''A*'' = ''(kb)(kh)''/2 = ''k''<sup>2</sup>''A''. Similar figures which can be decomposed into similar triangles will have areas related in the same way. The relationship holds for figures that are not rectifiable as well.
| |
| | |
| The ratio between the [[volume]]s of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27 — i.e. by three cubed).
| |
| | |
| Galileo's square–cube law concerns similar solids. If the ratio of similitude (ratio of corresponding sides) between the solids is ''k'', then the ratio of surface areas of the solids will be ''k''<sup>2</sup>, while the ratio of volumes will be ''k''<sup>3</sup>.
| |
| | |
| ==Similarity in general metric spaces== | |
| | |
| [[Image:Sierpinski triangle (blue).jpg|thumb|300px|[[Sierpinski triangle]]. A space having self-similarity dimension ln 3 / ln 2 = log<sub>2</sub>3, which is approximately 1.58. (from [[Hausdorff dimension]].)]]
| |
| | |
| In a general [[metric space]] (''X'', ''d''), an exact '''similitude''' is a [[function (mathematics)|function]] ''f'' from the metric space X into itself that multiplies all distances by the same positive [[scalar (mathematics)|scalar]] ''r'', called f's contraction factor, so that for any two points ''x'' and ''y'' we have
| |
| | |
| :<math>d(f(x),f(y)) = r d(x,y).\, \,</math>
| |
| | |
| Weaker versions of similarity would for instance have ''f'' be a bi-[[Lipschitz continuity|Lipschitz]] function and the scalar ''r'' a limit
| |
| | |
| :<math>\lim \frac{d(f(x),f(y))}{d(x,y)} = r. </math>
| |
| | |
| 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 <math>\{ f_s \}_{s\in S}</math> with contraction factors <math>0\leq r_s < 1 </math> such that ''K'' is the unique compact subset of ''X'' for which
| |
| | |
| :<math>\bigcup_{s\in S} f_s(K)=K. \,</math>
| |
| | |
| These self-similar sets have a self-similar [[Measure (mathematics)|measure]] <math>\mu^D</math>with dimension ''D'' given by the formula
| |
| | |
| :<math>\sum_{s\in S} (r_s)^D=1 \, </math>
| |
| | |
| which is often (but not always) equal to the set's [[Hausdorff dimension]] and [[packing dimension]]. If the overlaps between the <math>f_s(K)</math> are "small", we have the following simple formula for the measure:
| |
| | |
| :<math>\mu^D(f_{s_1}\circ f_{s_2} \circ \cdots \circ f_{s_n}(K))=(r_{s_1}\cdot r_{s_2}\cdots r_{s_n})^D.\,</math>
| |
| | |
| ==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).
| |
| | |
| The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are
| |
| # Positive defined: <math>\forall (a,b), S(a,b)\geq 0</math>
| |
| # Majored by the similarity of one element on itself ('''auto-similarity'''): <math>S (a,b) \leq S (a,a)</math> and <math>\forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b</math>
| |
| | |
| More properties can be invoked, such as '''reflectivity''' (<math>\forall (a,b)\ S (a,b) = S (b,a)</math>) or '''finiteness''' (<math>\forall (a,b)\ S(a,b) < \infty</math>). 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, ..} of numbers of the form <math>\{2^i, 3\cdot 2^i\}</math> where <math>i</math> ranges over all integers. When this set is plotted on a [[logarithmic scale]] it has one-dimensional [[translational symmetry]]: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two.
| |
| | |
| ==See also==
| |
| * [[Congruence (geometry)]]
| |
| * [[Hamming distance]] (string or sequence similarity)
| |
| * [[Inversive geometry#Dilations|inversive geometry]]
| |
| * [[Jaccard index]]
| |
| * [[Proportionality (mathematics)|Proportionality]]
| |
| * [[Semantic similarity]]
| |
| * [[Nearest neighbor search|Similarity search]]
| |
| * [[Similarity space]] on [[Numerical taxonomy]]
| |
| * [[Homoeoid]] (shell of concentric, similar ellipsoids)
| |
| * [[Solution of triangles]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{citation|first1=David W.|last1=Henderson|first2=Daina|last2=Taimina|title=Experiencing Geometry/Euclidean and Non-Euclidean with History|edition=3rd|publisher=Pearson Prentice-Hall|year=2005|isbn=978-0-13-143748-7}}
| |
| * {{citation|first=Harold R.|last=Jacobs|title=Geometry|year=1974|publisher=W.H. Freeman and Co.|isbn=0-7167-0456-0}}
| |
| * {{citation|first=Dan|last=Pedoe|title=Geometry/A Comprehensive Course|year=1988|origyear=1970|publisher=Dover|isbn=0-486-65812-0}}
| |
| * {{citation|first=Thomas Q.|last=Sibley|title=The Geometric Viewpoint/A Survey of Geometries|year=1998|publisher=Addison-Wesley|isbn=978-0-201-87450-1}}
| |
| * {{citation|first=James R.|last=Smart|title=Modern Geometries|edition=5th|publisher=Brooks/Cole|year=1998|isbn=0-534-35188-3}}
| |
| * {{citation|first=Saul|last=Stahl|title=Geometry/From Euclid to Knots|year=2003|publisher=Prentice-Hall|isbn=978-0-13-032927-1}}
| |
| * {{citation|first=Gerard A.|last=Venema|title=Foundations of Geometry|year=2006|publisher=Pearson Prentice-Hall|isbn=978-0-13-143700-5}}
| |
| * {{citation|first=Paul B.|last=Yale|title=Geometry and Symmetry|year=1968|publisher=Holden-Day}}
| |
| | |
| ==Further reading==
| |
| * Judith N. Cederberg (1989, 2001) ''A Course in Modern Geometries'', Chapter 3.12 Similarity Transformations, pp. 183–9, Springer ISBN 0-387-98972-2 .
| |
| * [[H.S.M. Coxeter]] (1961,9) ''Introduction to Geometry'', §5 Similarity in the Euclidean Plane, pp. 67–76, §7 Isometry and Similarity in Euclidean Space, pp 96–104, [[John Wiley & Sons]].
| |
| * Günter Ewald (1971) ''Geometry: An Introduction'', pp 106, 181, [[Wadsworth Publishing]].
| |
| * George E. Martin (1982) ''Transformation Geometry: An Introduction to Symmetry'', Chapter 13 Similarities in the Plane, pp. 136–46, Springer ISBN 0-387-90636-3 .
| |
| | |
| ==External links==
| |
| *[http://www.mathopenref.com/similartriangles.html Animated demonstration of similar triangles]
| |
| | |
| [[Category:Euclidean geometry]]
| |
| [[Category:Triangles]]
| |
