Difference between revisions of "Principal curvature"
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− | < | + | The [[Normal curvature|normal curvature]] of a surface in a principal direction, i.e. in a direction in which it assumes an extremal value. The principal curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746602.png" /> are the roots of the quadratic equation |
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− | + | <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/p/p074/p074660/p0746603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table> | |
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− | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746606.png" /> are the coefficients of the [[First fundamental form|first fundamental form]], while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p0746609.png" /> are the coefficients of the [[Second fundamental form|second fundamental form]] of the surface, computed at the given point. | |
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− | are the | ||
− | + | The half-sum of the principal curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466011.png" /> of the surface gives the [[Mean curvature|mean curvature]], while their product is equal to the [[Gaussian curvature|Gaussian curvature]] of the surface. Equation (*) may be written as | |
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− | + | <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/p/p074/p074660/p07466012.png" /></td> </tr></table> | |
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− | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466013.png" /> is the mean, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466014.png" /> is the Gaussian curvature of the surface at the given point. | |
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− | + | The principal curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466016.png" /> are connected with the normal curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466017.png" />, taken in an arbitrary direction, by means of Euler's formula: | |
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− | + | <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/p/p074/p074660/p07466018.png" /></td> </tr></table> | |
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− | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466019.png" /> is the angle formed by the selected direction with the principal direction for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466020.png" />. | |
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====Comments==== | ====Comments==== | ||
− | In the case of an | + | In the case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466021.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466022.png" /> of Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466023.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466024.png" /> principal curvatures and principal directions are defined as follows. |
− | dimensional submanifold | ||
− | of Euclidean | ||
− | space | ||
− | principal curvatures and principal directions are defined as follows. | ||
− | Let | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466025.png" /> be a unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466026.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466027.png" />. The Weingarten mapping (shape operator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466030.png" /> in direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466031.png" /> is given by the tangential part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466033.png" /> is the [[Covariant differential|covariant differential]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466035.png" /> is a local extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466036.png" /> to a unit normal vector field. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466037.png" /> does not depend on the chosen extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466038.png" />. The principal curvatures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466039.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466040.png" /> in direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466041.png" /> are given by the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466042.png" />, the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466043.png" /> define the higher mean curvatures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466044.png" />, which include as extremal cases the mean curvature as the trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074660/p07466045.png" /> and the Lipschitz–Killing curvature as its determinant. |
− | be a unit normal to | ||
− | at | ||
− | The Weingarten mapping (shape operator) | ||
− | of | ||
− | at | ||
− | in direction | ||
− | is given by the tangential part of | ||
− | where | ||
− | is the [[Covariant differential|covariant differential]] in | ||
− | and | ||
− | is a local extension of | ||
− | to a unit normal vector field. | ||
− | does not depend on the chosen extension of | ||
− | The principal curvatures of | ||
− | at | ||
− | in direction | ||
− | are given by the eigen values of | ||
− | the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of | ||
− | define the higher mean curvatures of | ||
− | which include as extremal cases the mean curvature as the trace of | ||
− | and the Lipschitz–Killing curvature as its determinant. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)</TD></TR><TR><TD valign="top">[a5]</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">[a6]</TD> <TD valign="top"> H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish pp. 1–5</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)</TD></TR><TR><TD valign="top">[a5]</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">[a6]</TD> <TD valign="top"> H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish pp. 1–5</TD></TR></table> |
Revision as of 14:52, 7 June 2020
The normal curvature of a surface in a principal direction, i.e. in a direction in which it assumes an extremal value. The principal curvatures and are the roots of the quadratic equation
(*) |
where , and are the coefficients of the first fundamental form, while , and are the coefficients of the second fundamental form of the surface, computed at the given point.
The half-sum of the principal curvatures and of the surface gives the mean curvature, while their product is equal to the Gaussian curvature of the surface. Equation (*) may be written as
where is the mean, and is the Gaussian curvature of the surface at the given point.
The principal curvatures and are connected with the normal curvature , taken in an arbitrary direction, by means of Euler's formula:
where is the angle formed by the selected direction with the principal direction for .
Comments
In the case of an -dimensional submanifold of Euclidean -space principal curvatures and principal directions are defined as follows.
Let be a unit normal to at . The Weingarten mapping (shape operator) of at in direction is given by the tangential part of , where is the covariant differential in and is a local extension of to a unit normal vector field. does not depend on the chosen extension of . The principal curvatures of at in direction are given by the eigen values of , the principal directions by its eigen directions. The (normalized) elementary symmetric functions of the eigen values of define the higher mean curvatures of , which include as extremal cases the mean curvature as the trace of and the Lipschitz–Killing curvature as its determinant.
References
[a1] | N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965) |
[a2] | B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973) |
[a3] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) |
[a4] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) |
[a5] | M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145 |
[a6] | H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60 |
[a7] | D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German) |
[a8] | B. O'Neill, "Elementary differential geometry" , Acad. Press (1966) |
[a9] | M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5 |
Principal curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_curvature&oldid=49376