For a curve with radius vector , the unit tangent vector is defined by
(1)
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(2)
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(3)
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where is a parameterization variable, is the arc length, and an overdot denotes a derivative with respect to , . For a function given parametrically by , the tangent vector relative to the point is therefore given by
(4)
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(5)
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To actually place the vector tangent to the curve, it must be displaced by . It is also true that
(6)
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(7)
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(8)
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where is the normal vector, is the curvature, is the torsion, and is the scalar triple product.