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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.
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.