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''tantrix''
 
''tantrix''
  
The tangent indicatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100101.png" /> of a regular [[Curve|curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100102.png" /> is the curve of oriented unit vectors tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100103.png" />.
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The tangent indicatrix $  T $
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of a regular [[Curve|curve]] $  \gamma : {[ a,b ] } \rightarrow {\mathbf R  ^ {n} } $
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is the curve of oriented unit vectors tangent to $  \gamma $.
  
More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100104.png" /> is a differentiable curve whose velocity vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100105.png" /> never vanishes, then
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More precisely, if $  \gamma : {[ a,b ] } \rightarrow {\mathbf R  ^ {n} } $
 +
is a differentiable curve whose velocity vector $  {\dot \gamma  } = { {d \gamma } / {dt } } $
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never vanishes, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100106.png" /></td> </tr></table>
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$$
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T ( t ) = {
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\frac{ {\dot \gamma  } ( t ) }{\left | { {\dot \gamma  } ( t ) } \right | }
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} .
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$$
  
The tangent indicatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100107.png" /> of any regular curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100108.png" /> thus traces out a curve on the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t1100109.png" /> which, as a point set, is independent of the parametrization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001010.png" />. A direct computation shows that the  "speeds" of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001012.png" /> relate via the curvature function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001014.png" /> (cf. also [[Curvature|Curvature]]):
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The tangent indicatrix $T$
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of any regular curve in $\mathbf{R}^{n}$
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thus traces out a curve on the unit sphere $S^{n-1} \in \mathbf{R}^{n}$
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which, as a point set, is independent of the parametrization of $\gamma$.  
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A direct computation shows that the  "speeds" of $T$
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and $\gamma$ relate via the curvature function $\kappa$ of $\gamma$ (cf. also [[Curvature]]):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001015.png" /></td> </tr></table>
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$$
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\left | { {
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\frac{dT }{dt }
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} } \right | = \kappa ( t ) \left | { {
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\frac{d \gamma }{dt }
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} } \right | .
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$$
  
It follows immediately that the length of the tangent indicatrix on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001016.png" /> gives the total curvature (the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001017.png" /> with respect to arc-length; cf. also [[Complete curvature|Complete curvature]]) of the original curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110010/t11001018.png" />. Because of this, the tangent indicatrix has proven useful, among other things, in studying total curvature of closed space curves (see [[#References|[a1]]], p. 29 ff).
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It follows immediately that the length of the tangent indicatrix on $  S ^ {n - 1 } $
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gives the total curvature (the integral of $  \kappa $
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with respect to arc-length; cf. also [[Complete curvature]]) of the original curve $  \gamma $.  
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Because of this, the tangent indicatrix has proven useful, among other things, in studying total curvature of closed space curves (see [[#References|[a1]]], p. 29 ff).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.S. Chern,   "Studies in global analysis and geometry" , ''Studies in Mathematics'' , '''4''' , Math. Assoc. America  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B. Solomon,   "Tantrices of spherical curves"  ''Amer. Math. Monthly'' , '''103''' :  1  (1996)  pp. 30–39</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Arnol'd,   "The geometry of spherical curves and the algebra of quaternions"  ''Russian Math. Surveys'' , '''50''' :  1  (1995)  pp. 1–68  (In Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> S.S. Chern, "Studies in global analysis and geometry" , ''Studies in Mathematics'' , '''4''' , Math. Assoc. America  (1967)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> B. Solomon, "Tantrices of spherical curves"  ''Amer. Math. Monthly'' , '''103''' :  1  (1996)  pp. 30–39</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Arnol'd, "The geometry of spherical curves and the algebra of quaternions"  ''Russian Math. Surveys'' , '''50''' :  1  (1995)  pp. 1–68  (In Russian)</TD></TR>
 +
</table>

Latest revision as of 19:23, 26 March 2024


tantrix

The tangent indicatrix $ T $ of a regular curve $ \gamma : {[ a,b ] } \rightarrow {\mathbf R ^ {n} } $ is the curve of oriented unit vectors tangent to $ \gamma $.

More precisely, if $ \gamma : {[ a,b ] } \rightarrow {\mathbf R ^ {n} } $ is a differentiable curve whose velocity vector $ {\dot \gamma } = { {d \gamma } / {dt } } $ never vanishes, then

$$ T ( t ) = { \frac{ {\dot \gamma } ( t ) }{\left | { {\dot \gamma } ( t ) } \right | } } . $$

The tangent indicatrix $T$ of any regular curve in $\mathbf{R}^{n}$ thus traces out a curve on the unit sphere $S^{n-1} \in \mathbf{R}^{n}$ which, as a point set, is independent of the parametrization of $\gamma$. A direct computation shows that the "speeds" of $T$ and $\gamma$ relate via the curvature function $\kappa$ of $\gamma$ (cf. also Curvature):

$$ \left | { { \frac{dT }{dt } } } \right | = \kappa ( t ) \left | { { \frac{d \gamma }{dt } } } \right | . $$

It follows immediately that the length of the tangent indicatrix on $ S ^ {n - 1 } $ gives the total curvature (the integral of $ \kappa $ with respect to arc-length; cf. also Complete curvature) of the original curve $ \gamma $. Because of this, the tangent indicatrix has proven useful, among other things, in studying total curvature of closed space curves (see [a1], p. 29 ff).

References

[a1] S.S. Chern, "Studies in global analysis and geometry" , Studies in Mathematics , 4 , Math. Assoc. America (1967)
[a2] B. Solomon, "Tantrices of spherical curves" Amer. Math. Monthly , 103 : 1 (1996) pp. 30–39
[a3] V.I. Arnol'd, "The geometry of spherical curves and the algebra of quaternions" Russian Math. Surveys , 50 : 1 (1995) pp. 1–68 (In Russian)
How to Cite This Entry:
Tangent indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_indicatrix&oldid=17246
This article was adapted from an original article by B. Solomon (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article