Gaussian Curvature of a Sphere

Hi all,

I am trying to find the Gaussian Curvature of the sphere with equation:

X(theta, phi) = (Rcos(theta)sin(phi) , Rsin(theta)sin(phi) , Rcos(theta)

I have found the coefficients of the first fundamental form:

E = R^2*sin(phi)^2

F = 0

G = R^2

But where should I go from here?

My (very unhelpful!) notes just jump to k = 1/R^2

Thanks for the help!

Re: Gaussian Curvature of a Sphere

What is the **definition** of "Gaussian Curvature"?

Re: Gaussian Curvature of a Sphere

k = eg - f^2 / (EG - F^2) ?

Re: Gaussian Curvature of a Sphere

the easiest way to get from where you are to where you want to be is to compute the coefficients of the second fundamental form (see here: Second fundamental form - Wikipedia, the free encyclopedia)

to do this, you are first going to have to compute the (unit) normal vector field, given by:

and then compute the coefficients of the second fundamental form as follows:

you can then calculate the Gaussian curvature as:

.

Re: Gaussian Curvature of a Sphere

Thanks for the help so far Deveno.

I am trying to calculate the unit normal and am unsure where to go from here:

dX/dtheta x dX/dphi = i j k

-Rsin(theta)sin(phi) Rcos(theta)sin(phi) 0

Rcos(theta)cos(phi) Rsin(theta)cos(phi) -Rsin(phi)

Of which the determinant is:

=i(-R^2cos(theta)sin(phi)^2) - j(R^2sin(theta)sin(phi)^2) + k(-R^2sin(theta)^2sin(phi)cos(phi) - R^2cos(theta)^2sin(phi)cos(phi))

I am not sure how to simplify this to get a final answer for N.

Thanks again for your help.

Re: Gaussian Curvature of a Sphere

for the cross-product i always use the formula:

.

the way i remember this is:

1st coordinate, leave out "1" subscripts, 2-->3 is positive (the positive cycle goes like this: 1-->2-->3-->1-->2-->3 etc.) (so 3-->2 is negative) so the u_{2}v_{3} term is positive.

2nd coordinate, leave out "2", the positive term is u_{3}v_{1} (see above).

3rd coordinate, well, there's only one possible way to "balance it out" now.

i find this easier to remember than the "determinant formula":

.

(by the way, you wrote down the wrong "third coordinate" in your parametrization, it should be: not theta. you did get the correct E,F and G, though)

a lengthy calculation shows that:

which is what you got (you can simplfy your "**k**" coordinate by taking out the common factor of ).

this is a normal vector, but it's not a UNIT vector, so we need to "normalize" it (how's that for an odd pun?) by dividing by its norm, which is:

this is almost as much work as finding the cross-product, first we find that:

taking the square root, we get that the magnitude of our normal vector is so that:

(a word about the signs, here: our parameterization takes horizontal rays vectors going left-to-right to "lattitude" arcs going counter-clockwise (west-to-east), and vertical rays going down-to-up to "longitude" arcs going from the north pole to the south pole, so the "outward" normal (using a right-hand-rule for the cross-product) in this case points "towards" the center of the sphere (inwards). we could have chosen a different parameterization to make the normal vector **n** simply be:

which would have been more intuitively obvious, geometrically (using -θ instead of θ, for example)).

i leave it to you to compute the second derivatives, and thus L,M and N.