What is the Gaussian Curvature of a Cone at its Vertex?

[lavenderblue]

Hi,

I know that you can determine that the Gaussian curvature of a cone tends to infinity at the vertex, but seeing as the curvature anywhere else on the cone is zero, how is this possible?

 

[arkajad]

Hi,

I know that you can determine that the Gaussian curvature of a cone tends to infinity at the vertex

Where do you know it from?

 

[lavenderblue]

I was told this by a GR lecturer. But I'm not sure of the mathematics!

 

[arkajad]

Did you look at the definition in, say Wikipedia? What is the behavior of the two principal curvatures for the cone as you approach the apex? What do you think?

 

[lavinia]

Hi,

I know that you can determine that the Gaussian curvature of a cone tends to infinity at the vertex, but seeing as the curvature anywhere else on the cone is zero, how is this possible?

Since the Gauss curvature of a cone is zero away from its vertex it does not tend to infinity.
But .. one could imagine a parabaloid like surface that deforms smoothly into a cone with the Gauss curvature of the tip increasing without limit.

 

[lavenderblue]

I was told that geometry of the cone is flat with K=0 everywhere except z=0. Do I use the expression for Gaussian curvature for a parabaloid?

 

[lavinia]

I was told that geometry of the cone is flat with K=0 everywhere except z=0. Do I use the expression for Gaussian curvature for a parabaloid?

You don not need to do a computation in my opinion. As the surface flattens into a cone the tip is approximated by a sphere of decreasing radius.

 

[arkajad]

I was told that geometry of the cone is flat with K=0 everywhere except z=0.

At z=0 it is just undefined. Depending on how you want to define it there - you will get any number you want. I think your teacher had in mid one of the two principal curvatures.