Dependent and independent variables: Difference between revisions

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[[Image:Midpoint.svg|282px|thumb|The midpoint of the segment ({{mvar|x}}<sub>1</sub>, {{mvar|y}}<sub>1</sub>) to ({{mvar|x}}<sub>2</sub>, {{mvar|y}}<sub>2</sub>)]]
 
In [[geometry]], the '''midpoint''' is the middle [[point (geometry)|point]] of a [[line segment]]. It is [[Distance|equidistant]] from both endpoints, and it is the [[centroid]] both of the segment and of the endpoints. It [[Bisection|bisects]] the segment.
 
==Formulas==
 
The midpoint of a segment in ''n''-dimensional space whose endpoints are <math>A = (a_1, a_2, \dots , a_n)</math> and <math> B = (b_1, b_2, \dots , b_n)</math> is given by
 
:<math>\frac{A+B}{2}.</math>
 
That is, the ''i''<sup>th</sup> coordinate of the midpoint (''i''=1, 2, ..., ''n'') is
 
:<math>\frac{a_i+b_i}{2}.</math>
 
==Construction==
Given two points of interest, finding the midpoint is one of the [[compass and straightedge constructions]].
The midpoint of a line segment can be located by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint. It is more challenging to locate the midpoint using only a compass, but it is still possible.<ref>{{cite web|url=http://mathworld.wolfram.com/Midpoint.html|title=Wolfram mathworld|date=29 September 2010}}</ref>
 
==Generalizations==
The midpoint is actually an [[affine geometry|affine]] [[invariant (mathematics)|invariant]]. Hence, [[#Formulas|aforementioned]] formulas for Cartesian coordinates are feasible for any [[affine coordinate system]].
 
The midpoint is not defined in [[projective geometry]]. Any point inside a [[projective range]] may be projectively mapped to any other point inside (the same or some else) projective range. Fixing one of such points as a midpoint actually defines an affine structure on the [[projective line]] containing that range. The [[projective harmonic conjugate]] of the two endpoints together with the midpoint is the [[point at infinity]].<ref>[[H. S. M. Coxeter]] (1955) ''The Real Projective Plane'', page 119</ref>
 
The definition of the midpoint of a segment may be extended to [[geodesic]] [[arc (geometry)|arcs]] on a [[Riemannian manifold]]. Note that, unlike affine case, the midpoint between two ''points'' may be defined not uniquely.
 
==See also==
 
* [[Midpoint polygon]]
* [[Median (geometry)]]
* [[Bisection#Line segment bisector|Segment bisector]]
* [[Numerical integration#Quadrature rules based on interpolating functions|Numerical integration]]
 
==References==
{{Reflist}}
 
==External links==
*[http://www.mathopenref.com/midpoint.html Animation] – showing the characteristics of the midpoint of a line segment
 
[[Category:Elementary geometry]]
[[Category:Affine geometry]]
[[Category:Analytic geometry]]

Latest revision as of 14:23, 18 September 2014

The use of e-cigs is just a practice that's getting ever more popular, and several people took up the use of ecigs being a cleanser alternative to cigarette smoking. The reason being e-cigs don't result in ash, and they do not create a person’s residence, garments, and automobile smelling poor. Among the biggest areas of starting to employ ecigs entails investing in a beginner kit, and there are certainly a amount of issues someone should know about buying a beginner kit. Including investing in a slightly better beginner kit, and there are certainly a few factors a person must have a part of their starter kit.

Purchasing a marginally greater kit

Among the most frequent faults that people create when first just starting to smoke ecigs is that they tend to choose the cheapest ecig that's available to them. This could create a individual certainly not trying out the practice, and it also means that a person WOn't ever purchase a nice ecig when they do take up the practice. This is the reason it is a good idea to buy a better one, that appears the way in which someone want to buy to, which is even a good concept to purchase one which can be gradually replaced with time.

What should really be included in the kit

An excellent starter kit may have things such as a charger that is user friendly. The kit also needs to incorporate instructions on how to complete the e-cigarette with liquid, a rag to clean it with, plus it must also possess a some extra ejuice or tubes. More [http://ciglites.metroblog.com/ check out the post right here].

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