osculation

WordNet

(mathematics) a contact of two curves (or two surfaces) at which they have a common tangent

PrepTutorEJDIC

《誇張して》接吻(キス)すること(kiss)

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/05/01 21:57:42」(JST)

wiki en

[Wiki en表示]

A curve C containing a point P where the radius of curvature equals r, together with the tangent line and the osculating circle touching C at P

In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve. That is, if F is a family of smooth curves, C is a smooth curve (not in general belonging to F), and p is a point on C, then an osculating curve from F at p is a curve from F that passes through p and has as many of its derivatives at p equal to the derivatives of C as possible.^[1]^[2]

The term derives from the Latinate root "osculate", to kiss, because the two curves contact one another in a more intimate way than simple tangency.^[3]

Examples

Examples of osculating curves of different orders include:

The tangent line to a curve C at a point p, the osculating curve from the family of straight lines. The tangent line shares its first derivative (slope) with C and therefore has first-order contact with C.^[1]^[2]^[4]
The osculating circle to C at p, the osculating curve from the family of circles. The osculating circle shares both its first and second derivatives (equivalently, its slope and curvature) with C.^[1]^[2]^[4]
The osculating parabola to C at p, the osculating curve from the family of parabolas, has third order contact with C.^[2]^[4]
The osculating conic to C at p, the osculating curve from the family of conic sections, has fourth order contact with C.^[2]^[4]

Generalizations

The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces. For instance an osculating plane to a space curve is a plane that has second-order contact with the curve. This is as high an order as is possible in the general case.^[5]

References

^ ^a ^b ^c Rutter, J. W. (2000), Geometry of Curves, CRC Press, pp. 174–175, ISBN 9781584881667 .
^ ^a ^b ^c ^d ^e Williamson, Benjamin (1912), An elementary treatise on the differential calculus: containing the theory of plane curves, with numerous examples, Longmans, Green, p. 309 .
^ Max, Black (1954–1955), "Metaphor", Proceedings of the Aristotelian Society, N.S. 55: 273–294 . Reprinted in Johnson, Mark, ed. (1981), Philosophical Perspectives on Metaphor, University of Minnesota Press, pp. 63–82, ISBN 9780816657971 . P. 69: "Osculating curves don't kiss for long, and quickly revert to a more prosaic mathematical contact."
^ ^a ^b ^c ^d Taylor, James Morford (1898), Elements of the Differential and Integral Calculus: With Examples and Applications, Ginn & Company, pp. 109–110 .
^ Kreyszig, Erwin (1991), Differential Geometry, Toronto University Mathematical Expositions 11, Courier Dover Publications, pp. 32–33, ISBN 9780486667218 .

English Journal

Kissing or "osculation" in frontotemporal dementia.

Mendez MF, Shapira JS.
The Journal of neuropsychiatry and clinical neurosciences.J Neuropsychiatry Clin Neurosci.2014 Summer;26(3):258-61. doi: 10.1176/appi.neuropsych.13060139.
The authors investigated the neuropsychiatry of kissing in frontotemporal dementia. Among 15 patients, two had compulsive social kissing, bitemporal involvement, and Klüver-Bucy symptoms, and four pursued kissing with sexually disinhibited behavior. Future research should clarify the neuropsychiatr
PMID 24922478

[Nitrogen accumulation and allocation in Calamagrostis angusifolia in different plant communities of Sanjiang Plain].

Sun ZG1, Liu JS, Yu JB.
Ying yong sheng tai xue bao = The journal of applied ecology.Ying Yong Sheng Tai Xue Bao.2009 Feb;20(2):277-84.
PMID 19459364

Gauge freedom in orbital mechanics.

Efroimsky M1.
Annals of the New York Academy of Sciences.Ann N Y Acad Sci.2005 Dec;1065:346-74.
PMID 16510420

Japanese Journal

1.2テスラー kinematic open MRI による肩インピンジメントの定量的評価

田崎篤,山川晃,星川吉光,野崎太希,二村昭元,秋田恵一,望月智之
肩関節 36(2), 685-690, 2012
… Despite the lack of pain, there were cases with osculation. …
NAID 130002103763

Duality for a class of minimal surfaces in R^n+1

東北數學雜誌. Second series 51(4), 585-601, 1999-12
… This is related to classical osculation duality. …
NAID 110000026897

Nd熱拡散Ti:LiNbO_3導波路レーザの作製と発振特性

Khan Md.Saad,津川英信,藤村昌寿,春名正光,西原浩
電子情報通信学会技術研究報告. OPE, 光エレクトロニクス 96(410), 25-30, 1996-12-12
LiNbO_3(LN)光集積回路の新たな展開を目指して,同一基板上に受動素子だけではなく能動素子をも組込むことができるような,希土類ドープLN導波路レーザの研究が行われている.我々は,X板 LiNbO_3に希土類元素Ndを熱拡散してドープした基板上で,光損傷の影響をほとんど受けないZ軸伝搬Ti拡散導波路を用いた導波路レーザを初めて実現した.波長809nmの光励起により発振波長1093nmのレーザ光 …
NAID 110003303016

Related Pictures

Amazing Osculation by jackfreak1994 on DeviantArt August | 2011 | Every Day for a Year OSCULATION PLANE PDF | Dream Pdf The Information on Osculation: A Brief Kisstory | You Monsters Are People. osculation - définition - C'est quoi ? Osculation 3 secret kiss - Wattpad #harimon | Explore harimon on DeviantArt Georgi Matevosjan | PeopleOfAr

[rutter-1] Rutter, J. W. (2000), Geometry of Curves, CRC Press, pp. 174–175, ISBN 9781584881667 .

[williamson-2] Williamson, Benjamin (1912), An elementary treatise on the differential calculus: containing the theory of plane curves, with numerous examples, Longmans, Green, p. 309 .

[3] Max, Black (1954–1955), "Metaphor", Proceedings of the Aristotelian Society, N.S. 55: 273–294 . Reprinted in Johnson, Mark, ed. (1981), Philosophical Perspectives on Metaphor, University of Minnesota Press, pp. 63–82, ISBN 9780816657971 . P. 69: "Osculating curves don't kiss for long, and quickly revert to a more prosaic mathematical contact."

[taylor-4] Taylor, James Morford (1898), Elements of the Differential and Integral Calculus: With Examples and Applications, Ginn & Company, pp. 109–110 .

[5] Kreyszig, Erwin (1991), Differential Geometry, Toronto University Mathematical Expositions 11, Courier Dover Publications, pp. 32–33, ISBN 9780486667218 .

匿名

検索

案内

案内