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In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated ''principal directions'' which give the directions in which the surface curves the most and the least. The theorem is named for Leonhard Euler who proved the theorem in . More precisely, let ''M'' be a surface in three-dimensional Euclidean space, and ''p'' a point on ''M''. A normal plane through ''p'' is a plane passing through the point ''p'' containing the normal vector to ''M''. Through each (unit) tangent vector to ''M'' at ''p'', there passes a normal plane ''P''''X'' which cuts out a curve in ''M''. That curve has a certain curvature κ''X'' when regarded as a curve inside ''P''''X''. Provided not all κ''X'' are equal, there is some unit vector ''X''1 for which ''k''1 = κ''X''1 is as large as possible, and another unit vector ''X''2 for which ''k''2 = κ''X''2 is as small as possible. Euler's theorem asserts that ''X''1 and ''X''2 are perpendicular and that, moreover, if ''X'' is any vector making an angle θ with ''X''1, then The quantities ''k''1 and ''k''2 are called the principal curvatures, and ''X''1 and ''X''2 are the corresponding principal directions. Equation () is sometimes called Euler's equation . ==See also== * Differential geometry of surfaces * Dupin indicatrix 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「Euler's theorem (differential geometry)」の詳細全文を読む スポンサード リンク
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