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The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces. For instance an
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is a plane that has second-order contact with the curve. This is as high an order as is possible in the general case.
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An elementary treatise on the differential calculus: containing the theory of plane curves, with numerous examples
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In one dimension, analytic curves are said to osculate at a point if they share the first three terms of their
452:, Toronto University Mathematical Expositions, vol. 11, Courier Dover Publications, pp. 32–33,
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411:: "Osculating curves don't kiss for long, and quickly revert to a more prosaic mathematical contact."
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Elements of the
Differential and Integral Calculus: With Examples and Applications
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Plane curve with the greatest order of contact with another curve
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of points where the ellipses are especially close to each other.
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Osculating ellipses – The spiral is not drawn: we see it as the
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Examples of osculating curves of different orders include:
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from a given family that has the highest possible order of
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The term derives from the
Latinate root "osculate", to
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about that point. This concept can be generalized to
396:, University of Minnesota Press, pp. 63–82,
30:"Osculation" redirects here. For other uses, see
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380:Max, Black (1954–1955), "Metaphor",
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84:with another curve. That is, if
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428:Taylor, James Morford (1898),
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362:Williamson, Benjamin (1912),
132:equal to the derivatives of
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229:The osculating parabola to
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32:Osculate (disambiguation)
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248:The osculating conic to
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