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Difference between revisions of "Curve of constant slope"

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A curve the tangent to which forms a constant angle with a fixed direction. A helical line is an example of this. The ratio between the torsion and the curvature of a curve of constant slope is constant. The spherical indicatrix of the tangents of a curve of constant slope is a circle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027390/c0273901.png" /> is the natural parametrization of a curve of constant slope, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027390/c0273902.png" /> (see [[#References|[2]]]). The evolutes of a plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027390/c0273903.png" /> are curves of constant slope, the tangents of which are inclined towards the plane of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027390/c0273904.png" /> at a constant angle (see [[#References|[1]]]). For every curve of constant slope there is a comoving cone connected with its accompanying trihedron, the vertex of which lies on the curve, while the generators describe developable surfaces.
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A curve the tangent to which forms a constant angle with a fixed direction. A [[helical line]] is an example of this. The ratio between the torsion and the curvature of a curve of constant slope is constant. The [[spherical indicatrix]] of the tangents of a curve of constant slope is a circle. If $\mathbf{r} = \mathbf{r}(s)$ is the [[Natural parameter|natural parametrization]] of a curve of constant slope, then $(\mathbf{r}'', \mathbf{r}''', \mathbf{r}'''') = 0$ (see [[#References|[2]]]). The evolutes of a plane curve $\gamma$ are curves of constant slope, the tangents of which are inclined towards the plane of the curve $\gamma$ at a constant angle (see [[#References|[1]]]). For every curve of constant slope there is a comoving cone connected with its accompanying trihedron, the vertex of which lies on the curve, while the generators describe [[developable surface]]s.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , '''1''' , Springer  (1921)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.R. Forsyth,  "Lectures on the differential geometry of curves and surfaces" , Cambridge  (1912)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.E. Appell,  ''Arch. Math. Phys.'' , '''64''' :  1  (1879)  pp. 19–23</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , '''1''' , Springer  (1921)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.R. Forsyth,  "Lectures on the differential geometry of curves and surfaces" , Cambridge  (1912)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  P.E. Appell,  ''Arch. Math. Phys.'' , '''64''' :  1  (1879)  pp. 19–23</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
For the various concepts defined for curves in three-dimensional space mentioned above, such as trihedron, cf. [[Differential geometry|Differential geometry]].
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For the various concepts defined for curves in three-dimensional space mentioned above, such as trihedron, cf. [[Differential geometry]].
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Latest revision as of 07:12, 1 December 2016

A curve the tangent to which forms a constant angle with a fixed direction. A helical line is an example of this. The ratio between the torsion and the curvature of a curve of constant slope is constant. The spherical indicatrix of the tangents of a curve of constant slope is a circle. If $\mathbf{r} = \mathbf{r}(s)$ is the natural parametrization of a curve of constant slope, then $(\mathbf{r}'', \mathbf{r}''', \mathbf{r}'''') = 0$ (see [2]). The evolutes of a plane curve $\gamma$ are curves of constant slope, the tangents of which are inclined towards the plane of the curve $\gamma$ at a constant angle (see [1]). For every curve of constant slope there is a comoving cone connected with its accompanying trihedron, the vertex of which lies on the curve, while the generators describe developable surfaces.

References

[1] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , 1 , Springer (1921)
[2] A.R. Forsyth, "Lectures on the differential geometry of curves and surfaces" , Cambridge (1912)
[3] P.E. Appell, Arch. Math. Phys. , 64 : 1 (1879) pp. 19–23


Comments

For the various concepts defined for curves in three-dimensional space mentioned above, such as trihedron, cf. Differential geometry.

How to Cite This Entry:
Curve of constant slope. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curve_of_constant_slope&oldid=39868
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article