Describing Curvature of a Non-Uniform Curve Using Second Derivative Average?

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Discussion Overview

The discussion revolves around the challenge of describing the curvature of a non-uniform curve using second derivatives, particularly when dealing with a discrete set of data points rather than a continuous function. Participants explore various methods and concepts related to curvature, variance, and interpolation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests using the average of the second derivative over a segment to quantify how "curvy" it is, positing that a value closer to zero indicates a straighter segment.
  • Another participant references the radius of curvature but notes that it applies to continuous distributions, which they do not have.
  • Some participants question the relevance of discussing curvature in the context of a discrete set of points, with one suggesting that variance might be a more appropriate measure for identifying the flattest part of a distribution.
  • There is a mention of using discrete approximations to curvature and the possibility of applying smoothing techniques to the data.
  • A later reply emphasizes the need to consider the properties of the interpolating function when working with discrete data points.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate methods for analyzing curvature in discrete datasets, with no consensus reached on a specific approach or solution.

Contextual Notes

The discussion highlights limitations related to the application of continuous curvature concepts to discrete data, as well as the need for interpolation methods that accommodate the characteristics of the data set.

ice109
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suppose a curve is not uniformly curved and i would like to describe how "curvy" a segment of this curve is? how would i do to this? i imagine i can take the second derivative and find the average of it over the entire segment and the closer the average is to zero the straight the segment is but immediately I'm dealing with a set of data points.
 
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Variance? I don't even know why you would be talking about curvature of a set of points.
 
Emmanuel114 said:
Variance? I don't even know why you would be talking about curvature of a set of points.
yea variance is probably it, i want to find the flattest part of a distrubtion of points
 
ice109 said:
this is for continuous distributions which i do not have.
You can always use discrete approximations to curvature, and/or apply some smoothing.
 
Since you are dealing with a discrete set of points, and wish to make some sort of interpolation, you have to decide what niceties you want the interpolating function to have.
 

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