Consider the curveand rotate it about the lineto form the following surface.

The objective is to show the points obtained by rotation of the origin (0, 0) of the curve are planar points of the surface.

A point on a surface is said to be planar if the lines of curvature are constant.

Parameterize the surface with a circle of radiusas,

It is required to determine the asymptotic curves and the lines of curvature of

,,and show that its mean curvature is zero.

Letbe the surface with global parameterizationgiven by,

.

For the computation of the coefficientse,f,g, it is enough to findN(and thus),.

Use these to compute the following:

Introduce the values just obtained in, where.

Similarly,

Thus, the Gaussian curvature is given as,

Thus the asymptotic curves are:

The lines of curvature are.

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