Question

2. Find the principal curvatures and principal directions of the surface obtained by rotating the curve x coshz in the z-plane around the z-axis

Answer 1

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AS FOR GIVEN DATA..

Find the principal curvatures and principal directions of the surface obtained by rotating the curve x=coshz in the xz-plane around the z-axis

EXPLANATION ::

The surface obtained by rotating the curve
Cosh:
in the xz-plane around the z-axis is called the catenoid and it is parametrized as follows:
r(u, v)(cosh u cos v, coshu sin v, u), u E R, vE(-T, T

And thus we get the following:

(u v)(sinhu , 1), r, (u, v)=(cosh u sinv, -cosh u cos v, 0 cos v, sinhu sin

(-cosh u cos v, -cosh u sinv, sinhucosh u)

This gives,
= coshu V(sinhu)? +1 =(cosh u) r, X r

As
cos v, -sin v, sinhu) N r Xr cosh u

Now, fix a point
pr(uo, vo.
and then the coordinate curves at p are given as follows:

a:(-e, e) S
and
8:(-e, e)S
such that
а(0) — г(ио + t, to). B) — г(ио. To tt)

(and thus
a(0)r(ua, vo) = B(0) = p
)

Then, $\alpha'(0)=r_u(u_0,v_0),\: \beta'(0)=r_v(u_0,v_0)$

d cos vo,-sinvo, sinh(uo + t)) cosh (uot) dt dt t-0

cosh2(usnhugcos vo, sinhuosin vo, 1)

And thus
a' (0) cosh2 (uo, dNp(a'(0))

Therefore, the coordinate curve $\alpha$ is principal direction with principal curvature
cosh uo

Now,
d -(-сos (го + t), — sin (vo + t), sinh(uо)) cosh (uo dt dt t-0

(coshuosin vo -coshucos vo, 0) 0) cos v0 sin vo, cosh (uo) cosh2(uo

And thus
'(0) dNp(3'(0))cosh2 (uo

Therefore, the coordinate curve $\beta$ is principal direction with principal curvature
cosh uo

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