Closed plane curve

[Lee7]

Homework Statement

How can I draw a closed plane curve with positive curvature that is not convex

The Attempt at a Solution

I was thinking drawing it like a banana but more curved, will that do?

 

[haruspex]

Is it allowed to cross itself?

 

[Lee7]

since it is not convex I think it can cross itself.

 

[Ray Vickson]

Homework Statement

How can I draw a closed plane curve with positive curvature that is not convex

The Attempt at a Solution

I was thinking drawing it like a banana but more curved, will that do?

A banana curve does not have positive curvature everywhere.

 

[Lee7]

Ray - You are right! So how can I draw one that is positive curvature?

 

[Ray Vickson]

Ray - You are right! So how can I draw one that is positive curvature?

Well, that is your assignment!

 

[Lee7]

Can you give a hint please?

 

[Lee7]

If it were convex then it is just a sphere? But a non-convex with positive curvature is confusing me.

 

[LCKurtz]

Can you give a hint please?

If it were convex then it is just a sphere? But a non-convex with positive curvature is confusing me.

You mean "circle". You were given a hint -- let the curve cross itself. Try it.

 

[Lee7]

Yup, I am meant circle. Will an number 8 satisfy it where the top is a bit more curved?

 

[LCKurtz]

No. As soon as you start curving the other way your curvature goes through zero.

 

[Lee7]

Wow, this is harder than I thought.

 

[Lee7]

So it must cross itself? Am I overthinking this?

 

[LCKurtz]

It isn't hard; it's almost trivial. Take a pencil and start drawing. Don't change curvature direction, use the hint, and close the curve.