What is the Surface of Revolution with Constant Curvature +1?

In summary, the conversation is about finding a surface of revolution with a Gauss curvature of +1 at all points, which is not a sphere. The surface is parametrized as \psi (t, \theta ) = ( x(t), y(t) cos \theta , y(t) sin \theta ) and the equation for K is K = \frac{x' (x'' y' - x' y'')}{y(x'^2 + y'^2)^2}. The attempt at a solution involves the curve \alpha (t) = (x(t),y(t)) not having unit speed and possibly being able to produce a surface of revolution with constant curvature +1 that is not a sphere. However, the question
  • #1
mooshasta
31
0

Homework Statement



I'm trying to find a surface of revolution with Gauss curvature K of +1 at all points, which doesn't lie in a sphere.


Homework Equations



The surface is parametrized as [itex]\psi (t, \theta ) = ( x(t), y(t) cos \theta , y(t) sin \theta ) [/itex]

I have the equation
[tex]
K = \frac{x' (x'' y' - x' y'')}{y(x'^2 + y'^2)^2}
[/tex]


The Attempt at a Solution



I am thinking it has to do with the curve [itex] \alpha (t) = (x(t),y(t)) [/itex] not having unit speed, but I am kind of stuck as to where to go from there.


Thanks!
 
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  • #2
Is there one? The only way out I can think of is to make it disconnected. A surface of constant guassian curvature is locally isometric to a sphere.
 
  • #3
I know that for a unit-speed [itex]\alpha (t)[/itex], the equation reduces to [itex]K = \frac{-y''}{y}[/itex], which does clearly represent a sphere.

The way the question is worded on my homework seems to point to the fact that that reduction only applies to unit-speed curves, which is why I think that perhaps an [itex]\alpha (t)[/itex] that doesn't have unit-speed perhaps can give a surface of revolution with constant curvature +1 that isn't a sphere... but maybe there's another "gimmick" that I'm overlooking...

Anyways, thanks for your help!
 

What is the surface of constant curvature?

The surface of constant curvature is a mathematical concept that describes a surface whose curvature remains the same at every point. This means that the surface is either completely flat (zero curvature) or has a constant amount of curvature throughout.

What are some examples of surfaces with constant curvature?

Some examples of surfaces with constant curvature include a sphere (positive curvature), a saddle (negative curvature), and a flat plane (zero curvature).

How is the curvature of a surface determined?

The curvature of a surface is determined by measuring the amount of bending or curving at each point on the surface. This can be done using mathematical equations or through physical measurements.

What are the applications of surfaces with constant curvature?

The concept of surfaces with constant curvature has many applications in mathematics and physics. It is used in the study of geometry, differential equations, and the theory of relativity. It also has practical applications in fields such as computer graphics, architectural design, and aerospace engineering.

Can the surface of constant curvature exist in the real world?

While some surfaces with constant curvature, such as a sphere, can exist in the real world, others, like a flat plane with zero curvature, are theoretical concepts. However, the study of surfaces with constant curvature is important for understanding the world around us and has many practical applications.

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