Difference between revisions of "Rotation surface"
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''surface of rotation, rotational surface'' | ''surface of rotation, rotational surface'' | ||
− | A surface generated by the rotation of a plane curve | + | A surface generated by the rotation of a plane curve $ L $ |
+ | around an axis in its plane. If $ L $ | ||
+ | is defined by the equations $ \rho = \rho ( u) $, | ||
+ | $ z = z( u) $, | ||
+ | the position vector of the surface of rotation is $ \mathbf r = \{ \rho ( u) \cos v, \rho ( u) \sin v, z( u) \} $, | ||
+ | where $ u $ | ||
+ | is the parameter of the curve $ L $, | ||
+ | $ \rho $ | ||
+ | is the distance between a point on the surface and the axis $ z $ | ||
+ | of rotation and $ v $ | ||
+ | is the angle of rotation. The line element of the surface of rotation is | ||
− | + | $$ | |
+ | ds ^ {2} = \ | ||
+ | ( \rho ^ {\prime 2 } + z ^ {\prime 2 } ) \ | ||
+ | du ^ {2} + \rho ^ {2} dv ^ {2} . | ||
+ | $$ | ||
− | The [[Gaussian curvature|Gaussian curvature]] is | + | The [[Gaussian curvature|Gaussian curvature]] is $ K = - z ^ \prime M/ \rho N ^ {4} $, |
+ | the [[Mean curvature|mean curvature]] is $ H = ( z ^ \prime N ^ {2} - \rho M)/ 2 \rho N ^ {3} $, | ||
+ | where $ M = z ^ \prime \rho ^ {\prime\prime} - z ^ {\prime\prime} \rho ^ \prime $, | ||
+ | $ N = \sqrt {\rho ^ {\prime 2 } + z ^ {\prime 2 } } $. | ||
+ | The lines $ u = \textrm{ const } $ | ||
+ | are called parallels of the surface of rotation and are circles located in a plane normal to the axis of rotation, with their centres on this axis. The lines $ v = \textrm{ const } $ | ||
+ | are called meridians; they are all congruent to the rotating curve and lie in planes passing through the axis of rotation. The meridians and the parallels of a surface of rotation are its curvature lines and form an [[Isothermal net|isothermal net]]. | ||
A surface of rotation allows for a [[Deformation|deformation]] into another surface of rotation, under which its net of curvature lines is preserved and therefore is a principal base of the deformation. The umbilical points (cf. [[Umbilical point|Umbilical point]]) of a surface of rotation are characterized by the property that the centre of curvature of the meridian lies on the axis of rotation. The product of the radius of a parallel by the cosine of the angle of intersection of the surface of rotation with the parallel is constant along a geodesic (Clairaut's theorem). | A surface of rotation allows for a [[Deformation|deformation]] into another surface of rotation, under which its net of curvature lines is preserved and therefore is a principal base of the deformation. The umbilical points (cf. [[Umbilical point|Umbilical point]]) of a surface of rotation are characterized by the property that the centre of curvature of the meridian lies on the axis of rotation. The product of the radius of a parallel by the cosine of the angle of intersection of the surface of rotation with the parallel is constant along a geodesic (Clairaut's theorem). | ||
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The metric of a surface of rotation can be presented in the form | The metric of a surface of rotation can be presented in the form | ||
− | + | $$ \tag{1 } | |
+ | ds ^ {2} = \Lambda ^ {2} ( r) | ||
+ | ( dx ^ {2} + dy ^ {2} ) ,\ \ | ||
+ | r ^ {2} = x ^ {2} + y ^ {2} . | ||
+ | $$ | ||
− | For the existence of metrics of the form (1) and for isometric immersions of these in | + | For the existence of metrics of the form (1) and for isometric immersions of these in $ \mathbf R ^ {n} $ |
+ | as surfaces of rotation see [[#References|[1]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.Kh. Sabitov, , ''Abstracts Coll. Diff. Geom. (August 1989, Eger, Hungary)'' pp. 47–48</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.Kh. Sabitov, , ''Abstracts Coll. Diff. Geom. (August 1989, Eger, Hungary)'' pp. 47–48</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | |||
====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) pp. 145</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish pp. 1–5</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> K. Leichtweiss, "Einführung in die Differentialgeometrie" , Springer (1973)</TD></TR></table> | <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) pp. 145</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish pp. 1–5</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> K. Leichtweiss, "Einführung in die Differentialgeometrie" , Springer (1973)</TD></TR></table> |
Revision as of 08:12, 6 June 2020
surface of rotation, rotational surface
A surface generated by the rotation of a plane curve $ L $ around an axis in its plane. If $ L $ is defined by the equations $ \rho = \rho ( u) $, $ z = z( u) $, the position vector of the surface of rotation is $ \mathbf r = \{ \rho ( u) \cos v, \rho ( u) \sin v, z( u) \} $, where $ u $ is the parameter of the curve $ L $, $ \rho $ is the distance between a point on the surface and the axis $ z $ of rotation and $ v $ is the angle of rotation. The line element of the surface of rotation is
$$ ds ^ {2} = \ ( \rho ^ {\prime 2 } + z ^ {\prime 2 } ) \ du ^ {2} + \rho ^ {2} dv ^ {2} . $$
The Gaussian curvature is $ K = - z ^ \prime M/ \rho N ^ {4} $, the mean curvature is $ H = ( z ^ \prime N ^ {2} - \rho M)/ 2 \rho N ^ {3} $, where $ M = z ^ \prime \rho ^ {\prime\prime} - z ^ {\prime\prime} \rho ^ \prime $, $ N = \sqrt {\rho ^ {\prime 2 } + z ^ {\prime 2 } } $. The lines $ u = \textrm{ const } $ are called parallels of the surface of rotation and are circles located in a plane normal to the axis of rotation, with their centres on this axis. The lines $ v = \textrm{ const } $ are called meridians; they are all congruent to the rotating curve and lie in planes passing through the axis of rotation. The meridians and the parallels of a surface of rotation are its curvature lines and form an isothermal net.
A surface of rotation allows for a deformation into another surface of rotation, under which its net of curvature lines is preserved and therefore is a principal base of the deformation. The umbilical points (cf. Umbilical point) of a surface of rotation are characterized by the property that the centre of curvature of the meridian lies on the axis of rotation. The product of the radius of a parallel by the cosine of the angle of intersection of the surface of rotation with the parallel is constant along a geodesic (Clairaut's theorem).
The only minimal surface of rotation is the catenoid. A ruled surface of rotation is a one-sheet hyperboloid or one of its degeneracies: a cylinder, a cone or a plane. A surface of rotation with more than one axis of rotation is a sphere or a plane.
The metric of a surface of rotation can be presented in the form
$$ \tag{1 } ds ^ {2} = \Lambda ^ {2} ( r) ( dx ^ {2} + dy ^ {2} ) ,\ \ r ^ {2} = x ^ {2} + y ^ {2} . $$
For the existence of metrics of the form (1) and for isometric immersions of these in $ \mathbf R ^ {n} $ as surfaces of rotation see [1].
References
[1] | I.Kh. Sabitov, , Abstracts Coll. Diff. Geom. (August 1989, Eger, Hungary) pp. 47–48 |
Comments
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) pp. 145 |
[a3] | M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5 |
[a4] | K. Leichtweiss, "Einführung in die Differentialgeometrie" , Springer (1973) |
Rotation surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation_surface&oldid=48591