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 (in succession, from the first derivative) 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]
- ^ Rutter, J. W. (2000), Geometry of Curves, CRC Press, pp. 174–175, ISBN 9781584881667.
- ^ 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, New Series, 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."