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

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Discussion Overview

The discussion revolves around the Gaussian curvature of a cone, particularly at its vertex. Participants explore the mathematical implications of curvature in relation to the cone's geometry, questioning how the curvature can be infinite at the vertex while being zero elsewhere on the surface.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that the Gaussian curvature of a cone tends to infinity at the vertex, while being zero elsewhere.
  • Others question the source of this information, suggesting it may come from a lecture or a definition found in resources like Wikipedia.
  • One participant proposes that while the Gaussian curvature is zero away from the vertex, it does not necessarily imply that it tends to infinity, suggesting a smooth deformation from a paraboloid to a cone.
  • Another participant mentions that the geometry of the cone is flat with K=0 everywhere except at the vertex, questioning whether the expression for Gaussian curvature for a paraboloid should be used.
  • Some participants note that at the vertex (z=0), the curvature is undefined and can yield various values depending on the definition used.
  • There is a suggestion that the tip of the cone can be approximated by a sphere of decreasing radius as the surface flattens into a cone.

Areas of Agreement / Disagreement

Participants express differing views on the nature of Gaussian curvature at the vertex of a cone, with no consensus reached on the implications of curvature being infinite at that point or how to define it mathematically.

Contextual Notes

Limitations include the dependence on definitions of curvature at the vertex and the unresolved mathematical steps regarding the transition from a paraboloid to a cone.

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

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