Namespaces
Variants
Actions

Difference between revisions of "Frénet trihedron"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (moved Frenet trihedron to Frénet trihedron over redirect: accented title)
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
''natural trihedron''
 
''natural trihedron''
  
The trihedral angle formed by the rays emanating from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f0417001.png" /> of a regular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f0417002.png" /> in the respective directions of the tangent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f0417003.png" />, the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f0417004.png" /> and the [[Binormal|binormal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f0417005.png" /> to the curve. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f0417006.png" /> coordinate axes, respectively, lie along the sides of the Frénet trihedron, then the equation of the curve in this coordinate system has the form
+
The trihedral angle formed by the rays emanating from a point $P$ of a regular curve $\gamma$ in the respective directions of the tangent $\tau$, the normal $\nu$ and the [[Binormal|binormal]] $\beta$ to the curve. If the $x,y,z$ coordinate axes, respectively, lie along the sides of the Frénet trihedron, then the equation of the curve in this coordinate system 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/f/f041/f041700/f0417007.png" /></td> </tr></table>
+
$$x=\Delta s-\frac{k_1^2\Delta s^3}{6}+o(\Delta s^3),$$
  
<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/f/f041/f041700/f0417008.png" /></td> </tr></table>
+
$$y=\frac{k_1\Delta s^2}{2}+\frac{k_1'\Delta s^3}{6}+o(\Delta s^3),$$
  
<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/f/f041/f041700/f0417009.png" /></td> </tr></table>
+
$$z=-\frac{k_1k_2}{6}\Delta s^3+o(\Delta s^3),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f04170010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f04170011.png" /> are the curvature and torsion of the curve, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f04170012.png" /> is the [[Natural parameter|natural parameter]]. The qualitative form of the projections of the curve onto the planes of the Frénet trihedron for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f04170013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041700/f04170014.png" /> can be seen in the figures.
+
where $k_1$ and $k_2$ are the curvature and torsion of the curve, and $s$ is the [[Natural parameter|natural parameter]]. The qualitative form of the projections of the curve onto the planes of the Frénet trihedron for $k_1\neq0$ and $k_2\neq0$ can be seen in the figures.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041700a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041700a.gif" />

Latest revision as of 16:53, 30 July 2014

natural trihedron

The trihedral angle formed by the rays emanating from a point $P$ of a regular curve $\gamma$ in the respective directions of the tangent $\tau$, the normal $\nu$ and the binormal $\beta$ to the curve. If the $x,y,z$ coordinate axes, respectively, lie along the sides of the Frénet trihedron, then the equation of the curve in this coordinate system has the form

$$x=\Delta s-\frac{k_1^2\Delta s^3}{6}+o(\Delta s^3),$$

$$y=\frac{k_1\Delta s^2}{2}+\frac{k_1'\Delta s^3}{6}+o(\Delta s^3),$$

$$z=-\frac{k_1k_2}{6}\Delta s^3+o(\Delta s^3),$$

where $k_1$ and $k_2$ are the curvature and torsion of the curve, and $s$ is the natural parameter. The qualitative form of the projections of the curve onto the planes of the Frénet trihedron for $k_1\neq0$ and $k_2\neq0$ can be seen in the figures.

Figure: f041700a

Figure: f041700b

Figure: f041700c

This trihedron was studied by F. Frénet (1847).


Comments

References

[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
How to Cite This Entry:
Frénet trihedron. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9net_trihedron&oldid=32587
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article