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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Affine focal set ： ウィキペディア英語版 Affine focal set In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold ''M'' embedded in a smooth manifold ''N'' is the caustic generated by the affine normal lines. It can be realised as the bifurcation set of a certain family of functions. The bifurcation set is the set of parameter values of the family which yield functions with degenerate singularities. This is not the same as the bifurcation diagram in dynamical systems. Let us assume that ''M'' is an ''n''-dimensional smooth hypersurface in real (''n''+1)-space. We assume that ''M'' has no points where the second fundamental form is degenerate. We recall from the article affine differential geometry that there is a unique transverse vector field over ''M''. This is the affine normal vector field, or the Blaschke normal field. A special (i.e. det = 1) affine transformation of real (''n'' + 1)-space will carry the affine normal vector field of ''M'' onto the affine normal vector field of the image of ''M'' under the transformation. == Geometric interpretation == Let us consider a local parametrisation of ''M''. Let $U\; \backslash subset\; \backslash mathbb^$ be an open neighbourhood of 0 with coordinates $\backslash bold\; =\; (u\_1,\backslash ldots,u\_n)$, and let $\backslash bold\; :\; U\; \backslash to\; \backslash mathbb^$ be a smooth parametrisation of ''M'' in a neighbourhood of one of its points. The affine normal vector field will be denoted by $\backslash bold$. At each point of ''M'' it is transverse to the tangent space of ''M'', i.e. :$\backslash bold\; :\; U\; \backslash to\; T\_\backslash mathbb^.\; \backslash ,$ For a fixed $\backslash bold\_0\; \backslash in\; U$ the affine normal line to ''M'' at $\backslash bold(\backslash bold\_0)$ may be parametrised by ''t'' where :$t\; \backslash mapsto\; \backslash bold(\backslash bold\_0)\; +\; t\; \backslash bold(\backslash bold\_0).$ The affine focal set is given geometrically as the infinitesimal intersections of the ''n''-parameter family of affine normal lines. To calculate this we choose an affine normal line, say at point ''p''; then we look at the affine normal lines at points infinitesimally close to ''p'' an see if any intersect the one at ''p''. If we choose a point infinitesimally close to $\backslash bold\; \backslash in\; U$, then it may be expressed as $\backslash bold\; +\; d\backslash bold$ where $d\backslash bold$ represents the infinitesimal difference. Thus $\backslash bold(\backslash bold)$ and $\backslash bold(\backslash bold\; +\; d\backslash bold)$ will be our ''p'' and its neighbour. For ''t'' and $d\backslash bold$ we try to solve :$\backslash bold(\backslash bold)\; +\; t\; \backslash bold(\backslash bold)\; =\; \backslash bold(\backslash bold\; +\; d\backslash bold)\; +\; t\; \backslash bold(\backslash bold\; +\; d\backslash bold).$ This can be done by using power series expansions, and is not too difficult; it is lengthy and has thus been omitted. We recall from the article affine differential geometry that the affine shape operator ''S'' is a type (1,1)-tensor field on ''M'', and is given by $Sv\; =\; D\_v\backslash bold$, where ''D'' is the covariant derivative on real (''n'' + 1)-space (for those well read: it is the usual flat and torsion free connexion). We find that the solutions to $\backslash bold(\backslash bold)\; +\; t\; \backslash bold(\backslash bold)\; =\; \backslash bold(\backslash bold\; +\; d\backslash bold)\; +\; t\; \backslash bold(\backslash bold\; +\; d\backslash bold)$ are when 1/''t'' is an eigenvalue of ''S'' and that $d\backslash bold$ is a corresponding eigenvector. The eigenvalues of ''S'' are not always distinct: there may be repeated roots, there may be complex roots, and ''S'' may not always be diagonalisable. For $0\; \backslash le\; k\; \backslash le\; ()$, where $()$ denotes the greatest integer function, there will generically be (''n'' − 2''k'')-pieces of the affine focal set above each point ''p''. The −2''k'' corresponds to pairs of eigenvalues becoming complex (like the solution to $x^2\; +\; a\; =\; 0$ as ''a'' changes from negative to positive). The affine focal set need not be made up of smooth hypersurfaces. In fact, for a generic hypersurface ''M'', the affine focal set will have singularities. The singularities could be found by calculation, but that may be difficult, and we still have no idea of what the singularity looks like up to diffeomorphism. If we use some singularity theory then we get much more information. 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 focal set」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.