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Tangent Vector


For a curve with radius vector r(t), the unit tangent vector T^^(t) is defined by

T^^(t)=(r^.)/(|r^.|)
(1)
=(r^.)/(s^.)
(2)
=(dr)/(ds),
(3)

where t is a parameterization variable, s is the arc length, and an overdot denotes a derivative with respect to t, x^.=dx/dt. For a function given parametrically by (f(t),g(t)), the tangent vector relative to the point (f(t),g(t)) is therefore given by

x(t)=(f^.)/(sqrt(f^.^2+g^.^2))
(4)
y(t)=(g^.)/(sqrt(f^.^2+g^.^2)).
(5)

To actually place the vector tangent to the curve, it must be displaced by (f(t),g(t)). It is also true that

(dT^^)/(ds)=kappaN^^
(6)
(dT^^)/(dt)=kappa(ds)/(dt)N^^
(7)
[T^.,T^..,T^...]=kappa^5d/(ds)(tau/kappa),
(8)

where N is the normal vector, kappa is the curvature, tau is the torsion, and [A,B,C] is the scalar triple product.


See also

Binormal Vector, Curvature, Manifold Tangent Vector, Normal Vector, Tangent, Tangent Bundle, Tangent Plane, Tangent Space, Torsion Explore this topic in the MathWorld classroom

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References

Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 108-111, 1997.

Referenced on Wolfram|Alpha

Tangent Vector

Cite this as:

Weisstein, Eric W. "Tangent Vector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TangentVector.html

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