View source for Natural equation

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''of a curve''

A system of equations

$$k_1=\phi(s),\quad k_2=\psi(s),$$

defining the curvature $k_1$ and torsion $k_2$ of the curve as functions of the arc length parameter $s$ on the curve. For any regular functions $\phi(s)>0$ and $\psi(s)$ there exists a curve, unique up to translation in space, with [[Curvature|curvature]] $\phi(s)$ and [[Torsion|torsion]] $\psi(s)$. A necessary and sufficient condition for a curve to be in a plane is that its torsion vanishes identically. A necessary and sufficient condition for a curve to be a straight line (or a segment of a straight line) is that its curvature vanishes identically.



====Comments====
In the article above, $\phi$ must be positive in order to generate uniqueness of the curve; for existence $\phi(s)\geq0$ suffices (cf. [[#References|[a1]]], Sects. 8.5.8 and 8.6.15).

Instead of  "natural equation"  one also finds the phrase  "intrinsic equation of a curve". The representation of (certain special) plane curves by means of a relation $k_1=\phi(s)$ goes back to L. Euler.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976) {{ZBL|0326.53001}}</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  B. O'Neill,   "Elementary differential geometry" , Acad. Press  (1966)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''2''' , Publish or Perish  pp. 1–5</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.J. Struik,   "Differential geometry" , Addison-Wesley  (1950)  pp. Sect. 1–8</TD></TR>
</table>