Undergrad Negative curvature in low dimensions

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SUMMARY

The discussion centers on the concept of negative curvature in low-dimensional geometry, specifically referencing hyperbolas and parabolas. Participants debate the visual representation of negative curvature, with one user asserting that if a function f(x) has positive curvature, then -f(x) exhibits negative curvature. The conversation highlights the complexity of understanding geometric properties and the need for clear communication in mathematical discussions.

PREREQUISITES
  • Understanding of basic geometric concepts, particularly curvature
  • Familiarity with functions and their properties in mathematics
  • Knowledge of hyperbolas and parabolas in coordinate geometry
  • Basic comprehension of mathematical communication and terminology
NEXT STEPS
  • Research the properties of hyperbolas and their curvature characteristics
  • Study the mathematical definition of curvature in differential geometry
  • Explore the implications of negative curvature in various geometric contexts
  • Learn about hypergeometry and its applications in modern mathematics
USEFUL FOR

This discussion is beneficial for mathematicians, geometry enthusiasts, and students seeking to deepen their understanding of curvature and its implications in low-dimensional spaces.

Paige_Turner
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TL;DR
Can a plane curve have negative curvature?
It's probably the hyperbola, but I don't see how it's curvature is negative. It looks like 2 parabolas.
thanx, paige turner
 
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Hello and :welcome: !

You may be able to see what "it" looks like, but we are mere humans and not telepathic.
Pleas present a complete case.

And if all you want is an answer: yes, curve can have a negative curvature.

##\ ##
 
If f(x) has positive curvature, then -f(x) has negative curvature. It is possible for a curve to curve downward
 
BvU said:
Hello and :welcome: !

> You may be able to see what "it" looks like, but we are mere humans and not telepathic.
Well, I'm autistic--not human at all. Ask my friends. The subject line indicates that "it" is a plane curve with negative curvature.

> Please present a complete case.

A case for what? I don't have an agenda.I don't have the answer. I just want to understand WITW is going on around me, and what's "around me" is the universe.

> if all you want is an answer

Um... what else could I want? I asked a question.

I just got here and I'm already in trouble with the mods. If past is indeed prologue, I will be banned and not have the slightest idea why--even when I'm polite and only talk about hypergeometry.
 
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