Tangent indicatrix
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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, the tangent indicatrix of a closed
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
is a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
on the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
intimately related to the curvature of the original curve. Let \gamma(t) be a closed curve with nowhere-vanishing tangent vector \dot. Then the tangent indicatrix T(t) of \gamma is the closed curve on the unit sphere given by T = \frac. The
total curvature In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: :\int_a^b k(s)\,ds. The total curvature of a closed curve i ...
of \gamma (the integral of curvature with respect to arc length along the curve) is equal to the
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
of T.


References

* Solomon, B. "Tantrices of Spherical Curves." ''American Mathematical Monthly'' 103, 30–39, 1996. Differential geometry Spherical geometry {{differential-geometry-stub