Closed plane curve

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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?
 
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  • #2
haruspex
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Is it allowed to cross itself?
 
  • #3
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since it is not convex I think it can cross itself.
 
  • #4
Ray Vickson
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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.
 
  • #5
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Ray - You are right! So how can I draw one that is positive curvature?
 
  • #6
Ray Vickson
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Ray - You are right! So how can I draw one that is positive curvature?
Well, that is your assignment!
 
  • #7
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Can you give a hint please?
 
  • #8
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If it were convex then it is just a sphere? But a non-convex with positive curvature is confusing me.
 
  • #9
LCKurtz
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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.
 
  • #10
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Yup, I am meant circle. Will an number 8 satisfy it where the top is a bit more curved?
 
  • #11
LCKurtz
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No. As soon as you start curving the other way your curvature goes through zero.
 
  • #12
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Wow, this is harder than I thought.
 
  • #13
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So it must cross itself? Am I overthinking this?
 
  • #14
LCKurtz
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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.
 

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