Osculating plane

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at a point of a curve

The plane having contact of order with at (see Osculation). The osculating plane can also be defined as the limit of a variable plane passing through three points of as these points approach . Usually, a curve intersects the osculating plane at the point of contact (see Fig.).

Figure: o070560a

If is given by equations

then the equation of the osculating plane has the form

where are moving coordinates and , , are calculated at the point of contact. If all three coefficients of in the equation of the osculating plane vanish, then the osculating plane becomes indefinite (and can coincide with any plane through the tangent line).



[a1] R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 31–35
[a2] D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988) pp. 10ff
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Osculating plane. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article