10/15/2003 Lecture: Second Fundamental Form of Surfaces of Revolution

1 + 1

2 + 31 + 1

We denote the surface patch by x[u,v] and define it as follows:

x[u_, v_]  :=  {f[u] Cos[v],  f[u] Sin[v],  g[u]}

x[u, v]

The following denotes the partial derivative of x[u,v] with respect to u

D[x[u, v], u]

The following denotes the partial derivative of x[u,v] with respect to v

D[x[u, v], v]

We now take the cross product of D[x[u,v],u] and D[x[u,v],v] (lines 7 and 8 above respectively)

Cross[%7, %8]

We now simplify the last line to obtain the cross product of D[x[u,v],u] with D[x[u,v],v]

Simplify[%]

We now calculate the dot product of the last line with itself

% . %

And then simplify the result

Simplify[%]

However, we recall that f'[u]^2 + g'[u]^2 = 1because we are assuming the unit speed condition. Therefore, the unit normal is

%10/f[u]

Let's call the unit normal nu

nu  =  %

nu

Let's calculate the derivative of x[u,v] with respect to u twice.

D[x[u, v], u, u]

Now, let's calculate the component of the 2nd Fundamental Form called L

%  .  nu

Simplify the result

Simplify[%]

Now, let's calculate M

D[x[u, v], u, v] . nu

Let's calculate N

D[x[u, v], v, v] . nu

And simplify the result

Simplify[%]

Now let's work out the case of the sphere

f[u_]  :=  Cos[u]

g[u_]  :=  Sin[u]

x[u, v]

Let's plot the sphere

ParametricPlot3D[x[u, v], {u, 0, 2  Pi}, {v, 0, Pi}]