Draw a Closed Plane Curve w/ Positive Curvature

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SUMMARY

The discussion focuses on the challenge of drawing a closed plane curve with positive curvature that is not convex. Participants clarify that a banana shape does not maintain positive curvature throughout and that a simple circle would be convex. The consensus is that the curve must cross itself to achieve the desired properties. A suggestion is made to start drawing without changing the curvature direction to successfully create the curve.

PREREQUISITES
  • Understanding of closed plane curves
  • Knowledge of curvature concepts in geometry
  • Familiarity with convex vs. non-convex shapes
  • Basic drawing skills for geometric shapes
NEXT STEPS
  • Research the properties of closed curves in differential geometry
  • Explore examples of non-convex curves with positive curvature
  • Study the implications of self-intersecting curves
  • Learn about curvature continuity and its applications in design
USEFUL FOR

Students in geometry, mathematicians exploring curvature concepts, and artists interested in geometric design principles.

Lee7
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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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Is it allowed to cross itself?
 
since it is not convex I think it can cross itself.
 
Lee7 said:

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.
 
Ray - You are right! So how can I draw one that is positive curvature?
 
Lee7 said:
Ray - You are right! So how can I draw one that is positive curvature?

Well, that is your assignment!
 
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.
 
Lee7 said:
Can you give a hint please?

Lee7 said:
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
Yup, I am meant circle. Will an number 8 satisfy it where the top is a bit more curved?
 
  • #11
No. As soon as you start curving the other way your curvature goes through zero.
 
  • #12
Wow, this is harder than I thought.
 
  • #13
So it must cross itself? Am I overthinking this?
 
  • #14
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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