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''of a curve''
 
''of a curve''
  
 
A system of equations
 
A system of equations
  
<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/n/n066/n066070/n0660701.png" /></td> </tr></table>
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$$k_1=\phi(s),\quad k_2=\psi(s),$$
  
defining the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660702.png" /> and torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660703.png" /> of the curve as functions of the arc length parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660704.png" /> on the curve. For any regular functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660706.png" /> there exists a curve, unique up to translation in space, with [[Curvature|curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660707.png" /> and [[Torsion|torsion]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660708.png" />. 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.
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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====
 
====Comments====
In the article above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n0660709.png" /> must be positive in order to generate uniqueness of the curve; for existence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n06607010.png" /> suffices (cf. [[#References|[a1]]], Sects. 8.5.8 and 8.6.15).
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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 curveintrinsic equation of a curve" . The representation of (certain special) plane curves by means of a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066070/n06607011.png" /> goes back to L. Euler.
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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====
 
====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)</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>
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<table>
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<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>
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<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>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  B. O'Neill,  "Elementary differential geometry" , Acad. Press  (1966)</TD></TR>
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<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>
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<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>
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</table>

Latest revision as of 19:04, 26 January 2024

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 $\phi(s)$ and 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. [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

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) Zbl 0326.53001
[a3] B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
[a4] M. Spivak, "A comprehensive introduction to differential geometry" , 2 , Publish or Perish pp. 1–5
[a5] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a6] D.J. Struik, "Differential geometry" , Addison-Wesley (1950) pp. Sect. 1–8
How to Cite This Entry:
Natural equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_equation&oldid=15274
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article