View source for Spherical indicatrix

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

<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>

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>

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>

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Ya. Vygodskii,   "Differential geometry" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR></table>



====Comments====


====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>