Namespaces
Variants
Actions

Difference between revisions of "Tangent indicatrix"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
(details)
 
Line 27: Line 27:
 
$$
 
$$
  
The tangent indicatrix  $ T $
+
The tangent indicatrix  $T$
of any regular curve in $ \mathbf R ^ {n} $
+
of any regular curve in $\mathbf{R}^{n}$
thus traces out a curve on the unit sphere $ S ^ {n - 1 } \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 $.  
+
which, as a point set, is independent of the parametrization of $\gamma$.  
A direct computation shows that the  "speeds" of $ T $
+
A direct computation shows that the  "speeds" of $T$
and $ \gamma $
+
and $\gamma$ relate via the curvature function $\kappa$ of $\gamma$ (cf. also [[Curvature]]):
relate via the curvature function $ \kappa $
 
of $ \gamma $(
 
cf. also [[Curvature|Curvature]]):
 
  
 
$$  
 
$$  
Line 47: Line 44:
 
It follows immediately that the length of the tangent indicatrix on  $  S ^ {n - 1 } $
 
It follows immediately that the length of the tangent indicatrix on  $  S ^ {n - 1 } $
 
gives the total curvature (the integral of  $  \kappa $
 
gives the total curvature (the integral of  $  \kappa $
with respect to arc-length; cf. also [[Complete curvature|Complete curvature]]) of the original curve  $  \gamma $.  
+
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 [[#References|[a1]]], p. 29 ff).
 
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>
+
<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=55679
This article was adapted from an original article by B. Solomon (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article