midpoint
American Heritage Dictionary:
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mid·point
(mĭd'point') 
n.
n.
- Mathematics. The point of a line segment or curvilinear arc that divides it into two parts of the same length.
- A position midway between two extremes.
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midpoint
Roget's Thesaurus:
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midpoint
noun
- A point or an area equidistant from all sides of something: center, median, middle, midst. See edge/center.
Word Tutor:
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midpoint
IN BRIEF: In the middle or at the center.
We were ready to come home at the midpoint of the trip.
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Oxford Dictionary of Biochemistry:
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midpoint
- the point on a line, or a graph, that is equidistant from either end of the line or from two inflections in the graph.
- a point in time that is half way between the beginning and the end of an event, or a series of events.
| midnight zone, midkine, middle molecule | |
| midpoint potential, migratase, migrate |
Random House Word Menu:
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categories related to 'midpoint'
- Fixed Positions and Points - midpoint: point at or near the center or middle
Wikipedia on Answers.com:
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Midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints.
In statistics and use of histograms, a midpoint is known as class mark.
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Contents
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Formulas
The formula for determining the midpoint of two points x1 and x2 on a line is:
The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1) and (x2, y2) in Cartesian coordinates, is:
The formula for determining the midpoint of a segment in the space, with endpoints (x1, y1, z1) and (x2, y2, z2) is:
More generally, for an n-dimensional space with axes 
, the midpoint of an interval is given by:
Construction
Given two points, 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.[1]
Generalizations
The midpoint is actually an affine invariant. Hence, 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 another 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.[2]
The definition of the midpoint of a segment may be extended to geodesic arcs on a Riemannian manifold. Note that, unlike affine case, the midpoint between two points may be defined not uniquely.
See also
References
- ^ "Wolfram mathworld". 29 September 2010. http://mathworld.wolfram.com/Midpoint.html.
- ^ H. S. M. Coxeter (1955) The Real Projective Plane, page 119
External links
- Animation – showing the characteristics of the midpoint of a line segment
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
Translations:
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Midpoint
Nederlands (Dutch)
middelpunt, punt dichtbij het midden
Ελληνική (Greek)
n. - μέσο σημείο, μέσα (γραμμής, γεγονότος κ.λπ.), (στατισ.) μέση τιμή τάξεως
Italiano (Italian)
punto centrale
Português (Portuguese)
n. - ponto central (m)
Русский (Russian)
средняя точка, кульминация
Español (Spanish)
n. - punto medio, mediana
Svenska (Swedish)
n. - mittpunkt
中文（简体）(Chinese (Simplified))
中点, 正中央
中文（繁體）(Chinese (Traditional))
n. - 中點, 正中央
日本語 (Japanese)
n. - 中心点, 中点, 中間点
العربيه (Arabic)
(الاسم) نقطه وسط
עברית (Hebrew)
n. - נקודת האמצע, אמצע
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American Heritage Dictionary
The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.
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Roget's Thesaurus
Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 byHoughton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.
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Oxford Dictionary of Biochemistry
Oxford University Press. Oxford Dictionary of Biochemistry and Molecular Biology © 1997, 2000, 2006 All rights reserved.
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Random House Word Menu
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Wikipedia on Answers.com
This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Midpoint.
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