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The image of a curve in the three-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867101.png" /> under a mapping from the points of the curve onto the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867102.png" /> by any of the following unit vectors: the tangent, the principal normal or the binormal of this curve. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867103.png" /> be the radius vector of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867104.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867105.png" /> be the natural parameter and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867106.png" /> be the radius vector of the spherical mapping of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867107.png" /> into the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s0867108.png" /> with its centre at the origin by means of one of the unit vectors listed. The equation of the spherical indicatrix of the tangent is defined by the equation
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<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/s/s086/s086710/s0867109.png" /></td> </tr></table>
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The image of a curve in the three-dimensional Euclidean space  $  \mathbf R  ^ {3} $
 +
under a mapping from the points of the curve onto the unit sphere  $  S  ^ {2} $
 +
by any of the following unit vectors: the tangent, the principal normal or the binormal of this curve. Let  $  \mathbf r = \mathbf r ( s) $
 +
be the radius vector of the curve  $  l $,
 +
let  $  s $
 +
be the natural parameter and let  $  \mathbf R = \mathbf R ( s) $
 +
be the radius vector of the spherical mapping of the curve  $  l $
 +
into the unit sphere  $  S  ^ {2} $
 +
with its centre at the origin by means of one of the unit vectors listed. The equation of the spherical indicatrix of the tangent is defined by the equation
 +
 
 +
$$
 +
\mathbf R ( s)  = 
 +
\frac{d \mathbf r }{ds}
 +
,
 +
$$
  
 
that of the spherical indicatrix of the principal normal by the equation
 
that of the spherical indicatrix of the principal normal by the equation
  
<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/s/s086/s086710/s08671010.png" /></td> </tr></table>
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$$
 +
\mathbf R ( s)  =
 +
\frac{d  ^ {2} \mathbf r / ds  ^ {2} }{| d  ^ {2} \mathbf r
 +
/ {ds  ^ {2} } | }
 +
 
 +
$$
  
 
and that of the spherical indicatrix of the binormal by the equation
 
and that of the spherical indicatrix of the binormal by the equation
  
<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/s/s086/s086710/s08671011.png" /></td> </tr></table>
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$$
 +
\mathbf R ( s)  =
 +
\frac{( d \mathbf r / ds ) \times ( d  ^ {2} \mathbf r / ds  ^ {2} ) }{| {d  ^ {2} \mathbf r } / {ds  ^ {2} } | }
 +
.
 +
$$
  
The tangent to the spherical indicatrix of the tangent is parallel to the principal normal of the curve at the corresponding values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086710/s08671012.png" />. The curvature and the torsion of the spherical indicatrix can be expressed in terms of the curvature and torsion of the curve itself. For every spherical indicatrix there is an infinite set of curves for which it is an indicatrix, i.e. a curve cannot be uniquely restored from its spherical indicatrix.
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The tangent to the spherical indicatrix of the tangent is parallel to the principal normal of the curve at the corresponding values s $.  
 +
The curvature and the torsion of the spherical indicatrix can be expressed in terms of the curvature and torsion of the curve itself. For every spherical indicatrix there is an infinite set of curves for which it is an indicatrix, i.e. a curve cannot be uniquely restored from its spherical indicatrix.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Ya. Vygodskii,  "Differential geometry" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Ya. Vygodskii,  "Differential geometry" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Struik,  "Differential geometry" , Addison-Wesley  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.E. Weatherburn,  "Differential geometry" , '''1''' , Cambridge Univ. Press  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Salmon,  "Analytische Geometrie des Raumes" , '''1–2''' , Teubner  (1898)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Struik,  "Differential geometry" , Addison-Wesley  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.E. Weatherburn,  "Differential geometry" , '''1''' , Cambridge Univ. Press  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Salmon,  "Analytische Geometrie des Raumes" , '''1–2''' , Teubner  (1898)</TD></TR></table>

Latest revision as of 08:22, 6 June 2020


The image of a curve in the three-dimensional Euclidean space $ \mathbf R ^ {3} $ under a mapping from the points of the curve onto the unit sphere $ S ^ {2} $ by any of the following unit vectors: the tangent, the principal normal or the binormal of this curve. Let $ \mathbf r = \mathbf r ( s) $ be the radius vector of the curve $ l $, let $ s $ be the natural parameter and let $ \mathbf R = \mathbf R ( s) $ be the radius vector of the spherical mapping of the curve $ l $ into the unit sphere $ S ^ {2} $ with its centre at the origin by means of one of the unit vectors listed. The equation of the spherical indicatrix of the tangent is defined by the equation

$$ \mathbf R ( s) = \frac{d \mathbf r }{ds} , $$

that of the spherical indicatrix of the principal normal by the equation

$$ \mathbf R ( s) = \frac{d ^ {2} \mathbf r / ds ^ {2} }{| d ^ {2} \mathbf r / {ds ^ {2} } | } $$

and that of the spherical indicatrix of the binormal by the equation

$$ \mathbf R ( s) = \frac{( d \mathbf r / ds ) \times ( d ^ {2} \mathbf r / ds ^ {2} ) }{| {d ^ {2} \mathbf r } / {ds ^ {2} } | } . $$

The tangent to the spherical indicatrix of the tangent is parallel to the principal normal of the curve at the corresponding values $ s $. The curvature and the torsion of the spherical indicatrix can be expressed in terms of the curvature and torsion of the curve itself. For every spherical indicatrix there is an infinite set of curves for which it is an indicatrix, i.e. a curve cannot be uniquely restored from its spherical indicatrix.

References

[1] M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian)

Comments

References

[a1] D.J. Struik, "Differential geometry" , Addison-Wesley (1957)
[a2] C.E. Weatherburn, "Differential geometry" , 1 , Cambridge Univ. Press (1961)
[a3] G. Salmon, "Analytische Geometrie des Raumes" , 1–2 , Teubner (1898)
How to Cite This Entry:
Spherical indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_indicatrix&oldid=17468
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article