中間点、中点

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(空間・時間の)まん中

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/07/03 15:20:43」(JST)

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For other uses, see Midpoint (disambiguation).

The midpoint of the segment (

x

₁,

y

₁) to (

x

₂,

y

₂)

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

Formulas[edit]

The midpoint of a segment in n-dimensional space whose endpoints are and is given by

That is, the i^th coordinate of the midpoint (i=1, 2, ..., n) is

Construction[edit]

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[edit]

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.

References[edit]

^ "Wolfram mathworld". 29 September 2010.
^ H. S. M. Coxeter (1955) The Real Projective Plane, page 119

External links[edit]

Animation – showing the characteristics of the midpoint of a line segment

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English Journal

Perceptual and response-related components of unilateral neglect may evolve independently of one another: evidence from five single-case studies.

Toraldo A1, Laiacona M, Pagani R, Mandelli A, Capitani E.Author information 1a DSU, Psychology Section , Pavia University , Pavia , Italy.AbstractThe Milner Landmark Task allows the disentanglement of perceptual and response-related components of unilateral neglect. If these two components reflect separate functional systems, then cases should be observed in which the two components evolve differently across time. To test this hypothesis we surveyed a continuous series of 21 right hemisphere stroke patients. Five patients from the sample were affected by unilateral neglect at the outset and could be submitted to repeated administrations of the Landmark task in the first weeks post stroke. Two versions of the task were used, Landmark-Manual and Landmark-Verbal, differing in the type of response required. Two patients showed independent changes in the perceptual and the response-related component of neglect, hence confirming the view of separate functional systems underlying them. Dissociations between the task versions were found, witnessing a role of the type of response. Unexpectedly, one patient showed an initial leftward deviation of the subjective midpoint of the stimulus line, which later reversed to a classical rightward deviation. We interpreted such a pattern in terms of co-existing "productive" and "negative" components of perceptual neglect.
Neurocase.Neurocase.2014 Jun;20(3):241-59. doi: 10.1080/13554794.2012.741257. Epub 2012 Nov 19.
The Milner Landmark Task allows the disentanglement of perceptual and response-related components of unilateral neglect. If these two components reflect separate functional systems, then cases should be observed in which the two components evolve differently across time. To test this hypothesis we s
PMID 23157652

Midpoint cell method for hybrid (MPI+OpenMP) parallelization of molecular dynamics simulations.

Jung J1, Mori T, Sugita Y.Author information 1Computational Biophysics Research Team, RIKEN Advanced Institute for Computational Science, 7-1-26 Minatojima-minamimachi, Chuo-ku, Kobe, Hyogo Kobe, 640-0047, Japan, Fax: +81 48 467 4532.AbstractWe have developed a new hybrid (MPI+OpenMP) parallelization scheme for molecular dynamics (MD) simulations by combining a cell-wise version of the midpoint method with pair-wise Verlet lists. In this scheme, which we call the midpoint cell method, simulation space is divided into subdomains, each of which is assigned to a MPI processor. Each subdomain is further divided into small cells. The interaction between two particles existing in different cells is computed in the subdomain containing the midpoint cell of the two cells where the particles reside. In each MPI processor, cell pairs are distributed over OpenMP threads for shared memory parallelization. The midpoint cell method keeps the advantages of the original midpoint method, while filtering out unnecessary calculations of midpoint checking for all the particle pairs by single midpoint cell determination prior to MD simulations. Distributing cell pairs over OpenMP threads allows for more efficient shared memory parallelization compared with distributing atom indices over threads. Furthermore, cell grouping of particle data makes better memory access, reducing the number of cache misses. The parallel performance of the midpoint cell method on the K computer showed scalability up to 512 and 32,768 cores for systems of 20,000 and 1 million atoms, respectively. One MD time step for long-range interactions could be calculated within 4.5 ms even for a 1 million atoms system with particle-mesh Ewald electrostatics. © 2014 Wiley Periodicals, Inc.
Journal of computational chemistry.J Comput Chem.2014 May 30;35(14):1064-72. doi: 10.1002/jcc.23591. Epub 2014 Mar 23.
We have developed a new hybrid (MPI+OpenMP) parallelization scheme for molecular dynamics (MD) simulations by combining a cell-wise version of the midpoint method with pair-wise Verlet lists. In this scheme, which we call the midpoint cell method, simulation space is divided into subdomains, each of
PMID 24659253

Altered native stability is the dominant basis for susceptibility of α1-antitrypsin mutants to polymerization.

Irving JA1, Haq I2, Dickens JA1, Faull SV1, Lomas DA2.Author information 1*Cambridge Institute for Medical Research, Department of Medicine, University of Cambridge, Wellcome Trust/MRC Building, Hills Road, Cambridge CB2 0XY, U.K.2†Wolfson Institute for Biomedical Research, The Cruciform Building, University College London, Gower Street, London WC1E 6BT, U.K.AbstractSerpins are protease inhibitors whose most stable state is achieved upon transition of a central 5-stranded β-sheet to a 6-stranded form. Mutations, low pH, denaturants and elevated temperatures promote this transition, which can result in a growing polymer chain of inactive molecules. Different types of polymer are possible, but, experimentally only heat has been shown to generate polymers in vitro consistent with ex vivo pathological specimens. Many mutations that alter the rate of heat-induced polymerization have been described, but interpretation is problematic because discrimination is lacking between the effect of global changes in native stability and specific effects on structural mechanism. We show that the temperature midpoint (Tm) of thermal denaturation reflects the transition of α1-antitrypsin to the polymerization intermediate, and determine the relationship with fixed-temperature polymerization half-times (t0.5) in the presence of stabilizing additives [TMAO (trimethylamine N-oxide), sucrose and sodium sulfate], point mutations and disulfide bonds. Combined with a retrospective analysis of 31 mutants characterized in the literature, the results of the present study show that global changes to native state stability are the predominant basis for the effects of mutations and osmolytes on heat-induced polymerization, summarized by the equation: ln(t0.5,mutant/t0.5,wild-type)=0.34×ΔTm. It is deviations from this relationship that hold key information about the polymerization process.
The Biochemical journal.Biochem J.2014 May 15;460(1):103-15. doi: 10.1042/BJ20131650.
Serpins are protease inhibitors whose most stable state is achieved upon transition of a central 5-stranded β-sheet to a 6-stranded form. Mutations, low pH, denaturants and elevated temperatures promote this transition, which can result in a growing polymer chain of inactive molecules. Different ty
PMID 24552432

Japanese Journal

Current-Driven Vortex--Antivortex Pair Rotation in a Magnetic Thin Film with Multiple Contacts

Tsukahara Hiroshi,Arai Hiroko,Imamura Hiroshi
J Phys Soc Jpn 82(8), 084701-084701-4, 2013-08-15
… In contrast, if the distance between the contacts is so short that the trajectory of each VA pair rotation can overlap with each other, two VA pair rotations are merged into one VA pair rotation whose trajectory encloses the two contacts as if a large elliptical contact exists at their midpoint. …
NAID 150000107766

Simple Method for Improving the Sensitivity of Functional Near-Infrared Spectroscopy: A Simulation and Phantom Study

Fukuda Keiko,Koishi Kazuaki,Murayama Takanobu
Jpn J Appl Phys 52(7), 077001-077001-7, 2013-07-25
… The sharing aperture approach uses cancellation signals detected at apertures for irradiating light, and the additional aperture approach uses the cancellation signals detected at the midpoint between the irradiation and detection apertures in the conventional measurement. …
NAID 150000107359

Geomorphological Displacement as a Combined Process of Tectonics and Mass-Movement in the 2011 East Japan Earthquake(<Featured Articles>Mega-Earthquake and Geomorphic Hazards)

YAGI Hiroshi
地形 33(4), 391-400, 2012-10-25
… That implies converse plunge to the midpoint along the Idosawa fault was causing sag and drag depression. …
NAID 110009554474