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In the mathematical field of differential geometry, the affine geometry of curves is the study of curves in an affine space, and specifically the properties of such curves which are invariant under the special affine group In the classical Euclidean geometry of curves, the fundamental tool is the Frenet–Serret frame. In affine geometry, the Frenet–Serret frame is no longer well-defined, but it is possible to define another canonical moving frame along a curve which plays a similar decisive role. The theory was developed in the early 20th century, largely from the efforts of Wilhelm Blaschke and Jean Favard. == The affine frame == Let x(''t'') be a curve in R''n''. Assume, as one does in the Euclidean case, that the first ''n'' derivatives of x(''t'') are linearly independent so that, in particular, x(''t'') does not lie in any lower-dimensional affine subspace of Rn. Then the curve parameter ''t'' can be normalized by setting determinant : Such a curve is said to be parametrized by its ''affine arclength''. For such a parameterization, : determines a mapping into the special affine group, known as a special affine frame for the curve. That is, at each point of the, the quantities define a special affine frame for the affine space R''n'', consisting of a point x of the space and a special linear basis attached to the point at x. The pullback of the Maurer–Cartan form along this map gives a complete set of affine structural invariants of the curve. In the plane, this gives a single scalar invariant, the affine curvature of the curve. 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「Affine geometry of curves」の詳細全文を読む スポンサード リンク
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