Extended Euclidean algorithm: Difference between revisions

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{{wiktionary}}
In [[geometry]], as a [[plane (geometry)|plane]] has one less [[dimension (mathematics and physics)|dimension]] than [[space]], a '''hyperplane''' is a subspace of one dimension less than its [[ambient space]].
 
A hyperplane of an ''n''-dimensional space is a [[flat (geometry)|flat]] [[subset]] with dimension ''n''&nbsp;&minus;&nbsp;1. By its nature, it separates the space into two [[Half-space (geometry)|half space]]s.
 
== Technical description ==
In [[geometry]], a '''hyperplane''' of an [[n-dimensional space|''n''-dimensional space]] ''V'' is a "flat" subset of dimension ''n''&nbsp;&minus;&nbsp;1, or equivalently, of [[codimension]]&nbsp;1 in&nbsp;''V''; it may therefore be referred to as an (''n''&nbsp;&minus;&nbsp;1)-[[flat (geometry)|flat]] of&nbsp;''V''. The space ''V'' may be a [[Euclidean space]] or more generally an [[affine space]], or a [[vector space]] or a [[projective space]], and the notion of hyperplane varies correspondingly; in all cases however, any hyperplane can be given in [[coordinate]]s as the solution of a single (due to the "codimension&nbsp;1" constraint) [[algebraic equation]] of degree&nbsp;1 (due to the "flat" constraint). If ''V'' is a vector space, one distinguishes "vector hyperplanes" (which are subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin; they can be obtained by [[translation (geometry)|translation]] of a vector hyperplane). A hyperplane in a Euclidean space separates that space into two [[Half-space (geometry)|half space]]s, and defines a [[reflection (mathematics)|reflection]] that fixes the hyperplane and interchanges those two half spaces.
 
== Dihedral angles ==
 
The [[dihedral angle]] between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding [[normal vector]]s. The product of the transformations in the two hyperplanes is a [[rotation (mathematics)|rotation]] whose axis is the [[Euclidean subspace|subspace]] of codimension&nbsp;2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.
 
==Special types of hyperplanes==
Several specific types of hyperplanes are defined with properties that are well suited for particular purposes. Some of these specializations are described here.
 
=== Affine hyperplanes ===
 
An '''affine hyperplane''' is an [[affine space|affine subspace]] of [[codimension]] 1 in an [[affine space]].
In [[Cartesian coordinates]], such a hyperplane can be described with a single [[linear equation]] of the following form (where at least one of the <math>a_i</math>'s is non-zero):
 
:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n = b.\ </math>
 
In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the [[connected component (topology)|connected component]]s of the [[complement (set theory)|complement]] of the hyperplane, and are given by the [[inequalities]]
 
:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n < b\ </math>
 
and
 
:<math>a_1x_1 + a_2x_2  + \cdots + a_nx_n > b.\ </math>
 
As an example, a point is a hyper plane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space. A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts (the complement of such a line is connected).
 
Any hyperplane of a Euclidean space has exactly two unit normal vectors.
 
Affine hyperplanes are used to define decision boundaries in many [[machine learning]] algorithms such as linear-combination (oblique) [[Decision tree learning|decision trees]], and [[Perceptron]]s.
 
=== Vector hyperplanes ===
 
In a vector space, a vector hyperplane is a [[Linear subspace|subspace]] of codimension&nbsp;1, only possibly shifted from the origin by a vector. Such a hyperplane is the solution of a single [[linear equation]].
 
===Projective hyperplanes===
'''Projective hyperplanes''', are used in projective geometry. Projective geometry can be viewed as affine geometry with [[vanishing point]]s (points at infinity) added. An affine hyperplane together with the associated points at infinity forms a projective hyperplane.  One special case of a projective hyperplane is the '''infinite''' or '''ideal hyperplane''', which is defined with the set of all points at infinity.
 
In real projective space, a hyperplane does not divide the space into two parts; rather, it takes two hyperplanes to separate points and divide up the space.  The reason for this is that in real projective space, the space essentially "wraps around" so that both sides of a lone hyperplane are connected to each other.
 
==See also==
*[[hypersurface]]
*[[decision boundary]]
*[[ham sandwich theorem]]
*[[arrangement of hyperplanes]]
*[[separating hyperplane theorem]]
*[[supporting hyperplane theorem]]
 
==References==
{{Reflist}}
* [[Charles W. Curtis]] (1968) ''Linear Algebra'', page 62, [[Allyn & Bacon]], Boston.
* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 7, Krieger, Huntington ISBN 0-88275-368-1 .
* Victor V. Prasolov & VM Tikhomirov (1997,2001) ''Geometry'', page 22, volume 200 in ''Translations of Mathematical Monographs'', [[American Mathematical Society]], Providence ISBN 0-8218-2038-9 .
 
== External links ==
*{{MathWorld|urlname=Hyperplane|title=Hyperplane}}
*{{MathWorld|urlname=Flat|title=Flat}}
 
{{Dimension topics}}
 
[[Category:Euclidean geometry]]
[[Category:Affine geometry]]
[[Category:Linear algebra]]
[[Category:Projective geometry]]

Latest revision as of 01:31, 10 October 2014

There are numerous articles on weight loss which have already been composed. We have read a amount of them which recommend a diet for those wishing to get rid of fat. Others have spoken of exercise. We have some whom suggest you hiring a personal trainer. The question, nevertheless, nonetheless remains. What is healthy weight reduction? What exactly is healthy fat for females?

Basically, a bmi chart is a calculation that uses height and weight to understand how much body fat an individual has. It is a graphical representation of the body, where weight is always on the horizontal axis and height on the vertical axis. Once you provide this chart to a doctor, he is able to tell we the appropriate answer to better your wellness. You is glad to locate that now online tools have made it much simpler for you to calculate your BMI. Usually individuals with heavy fat have BMI between 25 and 29.9, whereas the underweight people have BMI somewhere about 18.5. When you are aware of the BMI, you are able to change your lifestyle into a healthy one thus as to guarantee wise wellness.

There are many methods to get rid of weight, as you are no doubt aware. Ask your doctor for an opinion regarding what will be right for we bmi chart men. In general, the Dietary Guidelines recommend which you lose fat the old-fashioned way-by doing this slowly plus by changing the eating and exercise habits. If the BMI is inside the obese range, you will wish to consult a specialist to determine whether a unique weight-loss program, medicine, or operation is right for you. You may nonetheless should learn how to change your eating plus exercise habits, but these alone will not be enough to help we meet the goals.

One way is to watch their weight plus keep it. However fat is not the only factor that chooses whether a girl is overweight or no. A BMI Calculator helps a female measure her body mass index. This informs her what her ideal weight about her height is. A BMI Calculator female has become an indispensable tool for ladies.

By filling in the height plus weight in an this calculator, an individual could understand his BMI which enables him to deduce whether he is anorexic or not. While a healthy BMI is in the range 19 - 25, a BMI index of 17.5 is considered to be an casual indicator of anorexia nervosa. However, this is not an exact diagnosis for anorexia as some individuals can have a low BMI but NOT be anorexic. Anorexia is a psychological condition that has to be diagnosed only after an in-depth psychological evaluation plus laboratory tests, like, blood tests, electrocardiogram, and bone density test.

Second, tall people are at better risk of early mortality than persons of average height. Because the BMI tends to flag tall persons bmi chart women as being overweight, it's more valuable inside setting life insurance premiums than 1 would expect at first blush. With BMI as the rationale, insurance companies may charge slightly high (plus more realistic) life insurance costs for tall people, without appearing to be discriminatory.

31. Racing for a Charity: Millions of dollars a year are raised by runners for charity. It could make the race more meaningful when you have someone in mind as you run the miles. Supporting a good cause will also be a method into a sold-out race.

If you are wondering what the healthy weight is for a woman, then the BMI is the path to take. You could also check the circumference of the waist. If you believe you may be obese, then you need to begin eating healthy and exercise regularly so to lose those pounds plus become the healthy, modern we.

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