Difference between revisions of "Binormal"
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| − | The straight line passing through a point | + | {{TEX|done}} |
| + | 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==== | ====Comments==== | ||
| − | This definition holds for space curves for which | + | 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 | + | 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> | ||
Latest revision as of 21:21, 7 June 2020
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
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