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The straight line passing through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164401.png" /> of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164402.png" /> perpendicular to the osculating plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164403.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164404.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164405.png" /> is a parametrization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164406.png" />, then the vector equation of the binormal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164407.png" /> corresponding to the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164408.png" /> of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b0164409.png" /> has the form
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The straight line passing through a point $M_0$ of a curve $L$ perpendicular to the osculating plane to $L$ at $M_0$. If $\mathbf r=\mathbf r(t)$ is a parametrization of $L$, then the vector equation of the binormal at $M_0$ corresponding to the value $t_0$ of the parameter $t$ has the form
  
<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/b/b016/b016440/b01644010.png" /></td> </tr></table>
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$$\mathbf S(\lambda)=\mathbf r(t_0)+\lambda[\mathbf r'(t_0),\mathbf r''(t_0)].$$
  
  
  
 
====Comments====
 
====Comments====
This definition holds for space curves for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b01644011.png" /> does not depend linearly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b01644012.png" />, i.e. the curvature should not vanish.
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This definition holds for space curves for which $\mathbf r''(t_0)$ does not depend linearly on $\mathbf r'(t_0)$, i.e. the curvature should not vanish.
  
For curves in a higher-dimensional Euclidean space, the binormal is generated by the second normal vector in the Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]), which is perpendicular to the plane spanned by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b01644013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b01644014.png" /> and depends linearly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016440/b01644015.png" /> (cf. [[#References|[a1]]]).
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For curves in a higher-dimensional Euclidean space, the binormal is generated by the second normal vector in the Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]), which is perpendicular to the plane spanned by $\mathbf r'(t_0)$ and $\mathbf r''(t_0)$ and depends linearly on $\mathbf r'(t_0),\mathbf r''(t_0),\mathbf r'''(t_0)$ (cf. [[#References|[a1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''2''' , Publish or Perish  (1970)  pp. 1–5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''2''' , Publish or Perish  (1970)  pp. 1–5</TD></TR></table>

Revision as of 17:51, 30 July 2014

The straight line passing through a point $M_0$ of a curve $L$ perpendicular to the osculating plane to $L$ at $M_0$. If $\mathbf r=\mathbf r(t)$ is a parametrization of $L$, then the vector equation of the binormal at $M_0$ corresponding to the value $t_0$ of the parameter $t$ has the form

$$\mathbf S(\lambda)=\mathbf r(t_0)+\lambda[\mathbf r'(t_0),\mathbf r''(t_0)].$$


Comments

This definition holds for space curves for which $\mathbf r''(t_0)$ does not depend linearly on $\mathbf r'(t_0)$, i.e. the curvature should not vanish.

For curves in a higher-dimensional Euclidean space, the binormal is generated by the second normal vector in the Frénet frame (cf. Frénet trihedron), which is perpendicular to the plane spanned by $\mathbf r'(t_0)$ and $\mathbf r''(t_0)$ and depends linearly on $\mathbf r'(t_0),\mathbf r''(t_0),\mathbf r'''(t_0)$ (cf. [a1]).

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 2 , Publish or Perish (1970) pp. 1–5
How to Cite This Entry:
Binormal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binormal&oldid=17792
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article