# Closed plane curve

1. Sep 18, 2013

### Lee7

1. The problem statement, all variables and given/known data

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

3. The attempt at a solution

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

Last edited: Sep 18, 2013
2. Sep 18, 2013

### haruspex

Is it allowed to cross itself?

3. Sep 18, 2013

### Lee7

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

4. Sep 18, 2013

### Ray Vickson

A banana curve does not have positive curvature everywhere.

5. Sep 18, 2013

### Lee7

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

6. Sep 18, 2013

### Ray Vickson

7. Sep 18, 2013

### Lee7

Can you give a hint please?

8. Sep 18, 2013

### Lee7

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

9. Sep 18, 2013

### LCKurtz

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

10. Sep 18, 2013

### Lee7

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

11. Sep 18, 2013

### LCKurtz

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

12. Sep 18, 2013

### Lee7

Wow, this is harder than I thought.

13. Sep 18, 2013

### Lee7

So it must cross itself? Am I overthinking this?

14. Sep 18, 2013

### 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.