For a curve with radius vector , the unit tangent vector
is defined by
(1)
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(2)
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(3)
|
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)
|
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)
|
where
is the normal vector,
is the curvature,
is the torsion, and
is the scalar triple
product.