Osculating curve

Osculating curve
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 mathematics and geometry, an osculating curve is an extension of the concept of tangent. A tangent line to a curve is the straight line that shares the location and direction of the curve, while an osculating circle to the same curve shares the location, direction, and curvature.

Two curves are said to be osculating at a particular point if they share the same osculating circle, just as they are said to be tangent if they share the same tangent line. 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.

If two smooth curves are tangent at a point and also cross there, they are not only tangent but also osculating. The converse – osculating curves cross at the point of osculation – is not necessarily true, but holds in almost all cases.

See also

References

Weisstein, Eric W., "Osculating Curves" from MathWorld.


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Osculating circle — Kissing circles redirects here. For Descartes theorem on mutually tangent (kissing) circles, see Descartes theorem. An osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given… …   Wikipedia

  • Osculating circle of a curve — Circle Cir cle (s[ e]r k l), n. [OE. cercle, F. cercle, fr. L. circulus (Whence also AS. circul), dim. of circus circle, akin to Gr. kri kos, ki rkos, circle, ring. Cf. {Circus}, {Circum }.] [1913 Webster] 1. A plane figure, bounded by a single… …   The Collaborative International Dictionary of English

  • Osculating orbit — In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space (at a given moment of time) is the gravitational Kepler orbit (i.e. ellipse or other conic) that it would have about its central body (corresponding to… …   Wikipedia

  • Osculating plane — A space curve, Frenet–Serret frame, and the osculating plane (spanned by T and N). In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in… …   Wikipedia

  • Curve — For other uses, see Curve (disambiguation). A parabola, a simple example of a curve In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight.… …   Wikipedia

  • Curve fitting — best fit redirects here. For placing ( fitting ) variable sized objects in storage, see fragmentation (computer). Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,… …   Wikipedia

  • osculating circle — noun the circle that touches a curve (on the concave side) and whose radius is the radius of curvature • Syn: ↑circle of curvature • Hypernyms: ↑circle * * * noun : a circle which is tangent to a curve at a given point, which lies in the limiting …   Useful english dictionary

  • osculating circle — /ɒskjəleɪtɪŋ ˈsɜkəl/ (say oskyuhlayting serkuhl) noun the circle, the arc of which best approximates a curve at a given point on that curve …  

  • osculating plane — Math. the plane containing the circle of curvature of a point on a given curve. [1860 65] * * * …   Universalium

  • osculating circle — noun The circle that has the same tangent, and the same curvature at the point on the curve …   Wiktionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”