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In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the plane of curvature. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas. == Definition == Let ''C'' be a space curve parametrized by arc length and with the unit tangent vector t. If the curvature of ''C'' at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors : where the prime denotes the derivative of the vector with respect to the parameter . The torsion measures the speed of rotation of the binormal vector at the given point. It is found from the equation : which means : ''Remark'': The derivative of the binormal vector is perpendicular to both the binormal and the tangent, hence it has to be proportional to the principal normal vector. The negative sign is simply a matter of convention: it is a by-product of the historical development of the subject. The radius of torsion, often denoted by σ, is defined as : Geometric relevance: The torsion measures the turnaround of the binormal vector. The larger the torsion is, the faster rotates the binormal vector around the axis given by the tangent vector (). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function. 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「Torsion of a curve」の詳細全文を読む スポンサード リンク
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