Lines on Surfaces and Curvature of Surfaces

Find the principal radii of curvature of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at the point $(0,0,c)$.

Find the principal radii of curvature of the paraboloid $\dfrac {x^{2}}{p}+\dfrac {y^{2}}{q}=2z$ at the point $(0,0,0)$.

Find the principal radii of curvature of the helicoid $x=u\cos v$, $y=u\sin v$, $z=mv$.

Find the principal radii of curvature of the paraboloid $xy=az$.

Find the principal radii of curvature of the surface $e^{z}\cos x=\cos y$.

Find the principal radii of curvature of the surface $\sin z=\sh x\cdot \sh y$.

Find the principal radii of curvature of the catenoid $x^{2}+y^{2}=a^{2}\ch^{2}\dfrac {z}{a}$.

Prove that if $\rho_{1}$ and $\rho_{2}$ are the principal radii of curvature at some point of an ellipsoid, then on the curve of intersection of the ellipsoid with a concentric sphere, $\sqrt[4]{\rho_{1}\rho_{2}}=m(\rho_{1}+\rho_{2})$, where $m$ is a constant.

An umbilical point, or point of rounding, is a point of a surface at which the principal radii of curvature are equal: $\rho_{1}=\rho_{2}$. Find the umbilical points of the ellipsoid

The same (find the umbilical points, see problem 2391) for the ellipsoid

Find the total (Gaussian) curvature of the paraboloid

Prove that the mean curvature of the surface $z=\ln \cos x-\ln \cos y$ equals zero.

Prove that the total (Gaussian) curvature of a paraboloid is constant at the points of its intersection with the corresponding elliptic cylinder.

Find the total and mean curvatures of the surface of revolution $z=f(\rho )$, where $\rho =\sqrt{x^{2}+y^{2}}$.

Prove that the surface of revolution of the tractrix

about the axis $Ox$ has constant negative curvature, equal to $-\dfrac {1}{a^{2}}$.

Through the vertex of the paraboloid of revolution $y^{2}+z^{2}=2px$, curves are drawn on its surface. Find the locus of their centers of curvature.

Through a point on a surface, $n$ normal sections are drawn, the angle between consecutive sections being $\dfrac {2\pi }{n}$. Prove that the arithmetic mean of the curvatures of the sections does not depend on either the number $n$ or the direction of the first section ($n\geqslant 2$).

A curve on a surface that makes, at each of its points, the greatest angle with the plane $xOy$ possible at that point on the surface is called a line of steepest descent. Prove that the tangent to it, the normal to the surface, and the line through the given point parallel to $Oz$, lie in one plane.

Prove that the projections onto the plane $xOy$ of the contour lines of a surface meet the projections of the lines of steepest descent at a right angle.

Prove that the curves whose projections onto the plane $xOy$ have the equation $x^{a^{2}}=Cy^{b^{2}}$, where $C$ is a constant, are the lines of steepest descent for the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$.

Two surfaces

whose equations differ in the value of $\lambda$, are called confocal. Prove that the curves of intersection of an ellipsoid with the hyperboloids of one and of two sheets confocal to it are lines of curvature for the ellipsoid.

Prove that for the helicoid of problem 2385, along each line of curvature one of the quantities $\ln \left(u+\sqrt{u^{2}+m^{2}}\right)\pm v$ remains constant.

A curve on a surface whose principal normal everywhere coincides with the normal to the surface is called a geodesic. Show that on the surface of a sphere, an arc of a great circle is a geodesic.

Show that on the surface

the points of a geodesic satisfy the condition

where $c=\mathrm{const}$.

Find the analogous condition for the geodesics of the surface of revolution