Gauss and Mean Curvature

Question

Find the Gauss and mean curvature of the Helicoid given by x = u cos φ, y = u sin φ and z = cφ.

The helicoid is given by r = r(u, φ) = (u cos φ, u sin φ, cφ,).

I was able to calculate

r_u = (cos φ,sin φ, 0)

r_φ = (−u sin φ, u cos φ, c)

r_uu = (0, 0, 0)

r_φφ = (−u cos φ, −u sin φ, 0)

r_uφ = (− sin φ, cos φ, 0)

We know that the mean and Gauss curvature is given by

Mean curvature = (2FM − LG − NE)/[2*(F² − EG)]

Gauss curvature = (M² − LN)/(F² − EG)

I calculated E = 1, F=0, G=u² + c²

Answer 1

To find the mean and Gauss curvature of the helicoid, we need to find the remaining coefficients of the first and second fundamental form.

The first fundamental form coefficients are:

E = 1

F = 0

G = u^2 + c^2

The second fundamental form coefficients can be found using the following formula:

L = ruu · n

M = ruφ · n

N = rφφ · n

where n is the unit normal to the surface, which can be found as:

n = (ru × rφ) / |ru × rφ|

We have already calculated ru, rφ, ruu, rφφ, and ruφ in the previous step, so we can use these to find n and the second fundamental form coefficients:

n = (-c cos φ, -c sin φ, u) / sqrt(u^2 + c^2)

L = ruu · n = 0

M = ruφ · n = -u / sqrt(u^2 + c^2)

N = rφφ · n = 0

Now we can plug these coefficients into the formulas for mean and Gauss curvature:

Mean curvature = (2FM - LG - NE) / [2(E*G - F^2)]

= (-2u / sqrt(u^2 + c^2)) / (2(u^2 + c^2))

Gauss curvature = (M^2 - LN) / (EG - F^2)

= (-u^2 / (u^2 + c^2)^2) / (u^2 + c^2)

Simplifying these expressions, we get:

Mean curvature = -u / (u^2 + c^2)^(3/2)

Gauss curvature = -1 / (u^2 + c^2)

So the mean curvature is a function only of u and the Gauss curvature is a constant.