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| {{About|the quadrilateral shape|the album by Linda Perhacs|Parallelograms (album)}} | | {{Distinguish2|the [[Gauss–Markov theorem]] of mathematical statistics}} |
| {{Infobox Polygon | | {{merge|Ornstein–Uhlenbeck process|discuss=Talk:Gauss–Markov process#Merger proposal|date=March 2012}} |
| | name = Parallelogram | |
| | image = Parallelogram.svg
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| | caption = This parallelogram is a [[rhomboid]] as it has no right angles and unequal sides.
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| | type = [[quadrilateral]]
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| | edges = 4
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| | symmetry = [[Point reflection|C<sub>2</sub>]], [2]<sup>+</sup>, (22)
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| | area = ''b'' × ''h'' (base × height);<br>''ab'' sin θ | |
| | properties = [[convex polygon|convex]]}} | |
| In [[Euclidean geometry]], a '''parallelogram''' is a [[simple polygon|simple]] (non self-intersecting) [[quadrilateral]] with two pairs of [[Parallel (geometry)|parallel]] sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations. The three-dimensional counterpart of a parallelogram is a [[parallelepiped]].
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| The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.
| | '''Gauss–Markov stochastic processes''' (named after [[Carl Friedrich Gauss]] and [[Andrey Markov]]) are [[stochastic process]]es that satisfy the requirements for both [[Gaussian process]]es and [[Markov process]]es.<ref name=Rasmussen2006>{{cite book|last=C. E. Rasmussen & C. K. I. Williams,|title=Gaussian Processes for Machine Learning|date=2006|publisher=MIT Press|isbn=026218253X|page=Appendix B|url=http://www.gaussianprocess.org/gpml/chapters/RWB.pdf}}</ref><ref name=Lamon2008>{{cite book|last=Lamon|first=Pierre|title=3D-Position Tracking and Control for All-Terrain Robots|date=2008|publisher=Springer|isbn=978-3-540-78286-5|pages=93-95}}</ref> The stationary Gauss–Markov process is a very special case because it is unique, except for some trivial exceptions. |
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| ==Special cases==
| | Every Gauss–Markov process ''X''(''t'') possesses the three following properties: |
| *[[Rhomboid]] – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not [[right angle]]s<ref> http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf</ref>
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| *[[Rectangle]] – A parallelogram with four angles of equal size
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| *[[Rhombus]] – A parallelogram with four sides of equal length.
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| *[[Square (geometry)|Square]] – A parallelogram with four sides of equal length and angles of equal size (right angles).
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| ==Characterizations==
| | # If ''h''(''t'') is a non-zero scalar function of ''t'', then ''Z''(''t'') = ''h''(''t'')''X''(''t'') is also a Gauss–Markov process |
| A [[simple polygon|simple]] (non self-intersecting) [[quadrilateral]] is a parallelogram [[if and only if]] any one of the following statements is true:<ref>Owen Byer, Felix Lazebnik and Deirdre Smeltzer, ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, pp. 51-52.</ref><ref>Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.</ref>
| | # If ''f''(''t'') is a non-decreasing scalar function of ''t'', then ''Z''(''t'') = ''X''(''f''(''t'')) is also a Gauss–Markov process |
| *Two pairs of opposite sides are equal in length.
| | # There exists a non-zero scalar function ''h''(''t'') and a non-decreasing scalar function ''f''(''t'') such that ''X''(''t'') = ''h''(''t'')''W''(''f''(''t'')), where ''W''(''t'') is the standard [[Wiener process]]. |
| *Two pairs of opposite angles are equal in measure.
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| *The [[diagonal]]s bisect each other.
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| *One pair of opposite sides are [[Parallel (geometry)|parallel]] and equal in length.
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| *[[Adjacent angles]] are [[supplementary angles|supplementary]].
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| *Each diagonal divides the quadrilateral into two [[congruence (geometry)|congruent]] [[triangle]]s.
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| *The sum of the [[Square number|square]]s of the sides equals the sum of the squares of the diagonals. (This is the [[parallelogram law]].)
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| *It has [[rotational symmetry]] of order 2.
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| ==Properties==
| | Property (3) means that every Gauss–Markov process can be synthesized from the standard Wiener process (SWP). |
| * Diagonals of a parallelogram bisect each other,
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| *Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
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| *The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
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| *The area of a parallelogram is also equal to the magnitude of the [[vector cross product]] of two [[adjacent side (polygon)|adjacent]] sides.
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| *Any line through the midpoint of a parallelogram bisects the area.<ref>Dunn, J.A., and J.E. Pretty, "Halving a triangle", ''Mathematical Gazette'' 56, May 1972, p. 105.</ref>
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| *Any non-degenerate [[affine transformation]] takes a parallelogram to another parallelogram.
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| *A parallelogram has [[rotational symmetry]] of order 2 (through 180°). If it also has two lines of [[reflectional symmetry]] then it must be a rhombus or an oblong.
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| *The perimeter of a parallelogram is 2(''a'' + ''b'') where ''a'' and ''b'' are the lengths of adjacent sides.
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| *The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point. (This is an extension of [[Viviani's theorem]]). The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram.<ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>
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| ==Area formula== | | ==Properties of the Stationary Gauss-Markov Processes== |
| *A parallelogram with base ''b'' and height ''h'' can be divided into a [[trapezoid]] and a [[right triangle]], and rearranged into a [[rectangle]], as shown in the figure to the left. This means that the [[area]] of a parallelogram is the same as that of a rectangle with the same base and height.
| | A stationary Gauss–Markov process with [[variance]] <math>\textbf{E}(X^{2}(t)) = \sigma^{2}</math> and [[time constant]] <math>\beta^{-1}</math> has the following properties. |
| [[File:ParallelogramArea.svg|thumb|left|180px|alt=A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle|A parallelogram can be rearranged into a rectangle with the same area.]] | |
| <math>A = bh</math> | |
| [[File:Parallelogram area.svg|thumb|250px|The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram]] | |
| *The area ''K'' of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles.
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| :The area of the rectangle is
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| ::<math>A_\text{rect} = (B+A) \times H\,</math>
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| :and the area of a single orange triangle is | | Exponential [[autocorrelation]]: |
| ::<math>A_\text{tri} = \frac{1}{2} A \times H. \,</math>
| | :<math>\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}.\,</math> |
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| :Therefore, the area of the parallelogram is
| | A power [[spectral density]] (PSD) function that has the same '''shape''' as the [[Cauchy distribution]]: |
| ::<math>K = A_\text{rect} - 2 \times A_\text{tri} = ( (B+A) \times H) - ( A \times H) = B \times H</math> | |
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| *Another area formula, for two sides ''B'' and ''C'' and angle θ, is
| | :<math>\textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.\,</math> |
| ::<math>K = B \cdot C \cdot \sin \theta.\,</math>
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| *The area of a parallelogram with sides ''B'' and ''C'' (''B'' ≠ ''C'') and angle <math>\gamma</math> at the intersection of the diagonals is given by<ref>Mitchell, Douglas W., "The area of a quadrilateral", ''Mathematical Gazette'', July 2009.</ref>
| | (Note that the Cauchy distribution and this spectrum differ by scale factors.) |
| ::<math>K = \frac{|\tan \gamma|}{2} \cdot \left| B^2 - C^2 \right|.</math>
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| *When the parallelogram is specified from the lengths ''B'' and ''C'' of two adjacent sides together with the length ''D''<sub>1</sub> of (any) one diagonal, then the area can be found from [[Heron's formula]]. Specifically it is
| | The above yields the following spectral factorization: |
| ::<math>K=2\sqrt{S(S-B)(S-C)(S-D_1)}</math>
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| :where <math>S=(B+C+D_1)/2</math> and the leading factor 2 comes from the fact that the number of congruent triangles that the chosen diagonal divides the parallelogram into is two. | |
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| ===The area on coordinate system===
| | :<math>\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}} |
| Let vectors <math>\mathbf{a},\mathbf{b}\in\R^2</math> and let <math>V = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \end{bmatrix} \in\R^{2 \times 2}</math> denote the matrix with elements of '''a''' and '''b'''. Then the area of the parallelogram generated by '''a''' and '''b''' is equal to <math>|\det(V)| = |a_1b_2 - a_2b_1|\,</math>.
| | = \frac{\sqrt{2\beta}\,\sigma}{(s + \beta)} |
| | \cdot\frac{\sqrt{2\beta}\,\sigma}{(-s + \beta)}. |
| | </math> |
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| Let vectors <math>\mathbf{a},\mathbf{b}\in\R^n</math> and let <math>V = \begin{bmatrix} a_1 & a_2 & \dots & a_n \\ b_1 & b_2 & \dots & b_n \end{bmatrix} \in\R^{2 \times n}</math> Then the area of the parallelogram generated by '''a''' and '''b''' is equal to <math>\sqrt{\det(V V^\mathrm{T})}</math>.
| | which is important in Wiener filtering and other areas. |
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| Let points <math>a,b,c\in\R^2</math>. Then the area of the parallelogram with vertices at ''a'', ''b'' and ''c'' is equivalent to the absolute value of the determinant of a matrix built using ''a'', ''b'' and ''c'' as rows with the last column padded using ones as follows:
| | There are also some trivial exceptions to all of the above. |
| :<math>K = \left| \det \begin{bmatrix}
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| a_1 & a_2 & 1 \\
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| b_1 & b_2 & 1 \\
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| c_1 & c_2 & 1
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| \end{bmatrix} \right|. </math>
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| ==Proof that diagonals bisect each other==
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| [[File:Parallelogram1.svg|right|Parallelogram ABCD]]
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| To prove that the diagonals of a parallelogram bisect each other, we will use [[congruence (geometry)|congruent]] [[Triangle#Basic facts|triangle]]s:
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| :<math>\angle ABE \cong \angle CDE</math> ''(alternate interior angles are equal in measure)''
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| :<math>\angle BAE \cong \angle DCE</math> ''(alternate interior angles are equal in measure)''.
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| (since these are angles that a transversal makes with [[Parallel (geometry)|parallel lines]] ''AB'' and ''DC'').
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| Also, side ''AB'' is equal in length to side ''DC'', since opposite sides of a parallelogram are equal in length.
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| Therefore triangles ''ABE'' and ''CDE'' are congruent (ASA postulate, ''two corresponding angles and the included side'').
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| Therefore,
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| :<math>AE = CE</math>
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| :<math>BE = DE.</math>
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| Since the diagonals ''AC'' and ''BD'' divide each other into segments of equal length, the diagonals bisect each other.
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| Separately, since the diagonals ''AC'' and ''BD'' bisect each other at point ''E'', point ''E'' is the midpoint of each diagonal.
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| ==See also==
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| *[[Fundamental parallelogram]]
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| *[[Parallelogram law]]
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| ==References== | | ==References== |
| {{reflist}} | | {{reflist}} |
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| ==External links==
| | {{DEFAULTSORT:Gauss-Markov Process}} |
| {{Commons category|Parallelograms}} | | [[Category:Markov processes]] |
| *[http://www.elsy.at/kurse/index.php?kurs=Parallelogram+and+Rhombus&status=public Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)]
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| *{{MathWorld |urlname=Parallelogram |title=Parallelogram}}
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| *[http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/index.php Interactive Parallelogram --sides, angles and slope]
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| *[http://www.cut-the-knot.org/Curriculum/Geometry/AreaOfParallelogram.shtml Area of Parallelogram] at [[cut-the-knot]]
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| *[http://www.cut-the-knot.org/Curriculum/Geometry/EquiTriOnPara.shtml Equilateral Triangles On Sides of a Parallelogram] at [[cut-the-knot]]
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| *[http://www.mathopenref.com/parallelogram.html Definition and properties of a parallelogram] with animated applet
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| *[http://www.mathopenref.com/parallelogramarea.html Interactive applet showing parallelogram area calculation] interactive applet
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| [[Category:Quadrilaterals]]
| | {{probability-stub}} |
| [[Category:Elementary shapes]]
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Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes.[1][2] The stationary Gauss–Markov process is a very special case because it is unique, except for some trivial exceptions.
Every Gauss–Markov process X(t) possesses the three following properties:
- If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process
- If f(t) is a non-decreasing scalar function of t, then Z(t) = X(f(t)) is also a Gauss–Markov process
- There exists a non-zero scalar function h(t) and a non-decreasing scalar function f(t) such that X(t) = h(t)W(f(t)), where W(t) is the standard Wiener process.
Property (3) means that every Gauss–Markov process can be synthesized from the standard Wiener process (SWP).
Properties of the Stationary Gauss-Markov Processes
A stationary Gauss–Markov process with variance and time constant has the following properties.
Exponential autocorrelation:
A power spectral density (PSD) function that has the same shape as the Cauchy distribution:
(Note that the Cauchy distribution and this spectrum differ by scale factors.)
The above yields the following spectral factorization:
which is important in Wiener filtering and other areas.
There are also some trivial exceptions to all of the above.
References
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- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534