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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.〔. Reprinted in . (P. 69 ): "Osculating curves don't kiss for long, and quickly revert to a more prosaic mathematical contact."〕 ==Examples== Examples of osculating curves of different orders include: *The tangent line to a curve ''C'' at a point ''p'', the osculating curve from the family of straight lines. The tangent line shares its first derivative (slope) with ''C'' and therefore has first-order contact with ''C''.〔〔〔.〕 *The osculating circle to ''C'' at ''p'', the osculating curve from the family of circles. The osculating circle shares both its first and second derivatives (equivalently, its slope and curvature) with ''C''.〔〔〔 *The osculating parabola to ''C'' at ''p'', the osculating curve from the family of parabolas, has third order contact with ''C''.〔〔 *The osculating conic to ''C'' at ''p'', the osculating curve from the family of conic sections, has fourth order contact with ''C''.〔〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「osculating curve」の詳細全文を読む スポンサード リンク
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