View source for Frénet 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

<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>

<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>

<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>

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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041700a.gif" />

Figure: f041700a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041700b.gif" />

Figure: f041700b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f041700c.gif" />

Figure: f041700c

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Hsiung,   "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR></table>