How to Linearize an R vs θ graph (distance vs angle)

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    Angle Graph Linearize
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Discussion Overview

The discussion revolves around the challenge of linearizing a graph that represents distance versus angle, specifically in the context of a displacement versus angle graph. Participants explore various methods and considerations for achieving linearization, including the applicability of linear approximations and the behavior of the function over different domains.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants question the necessity and feasibility of linearizing the graph, asking what motivates the desire for linearization and whether it is possible.
  • One participant suggests that while the function is not globally linear, it can be approximated locally by calculating the slope at a specific point, referencing the use of derivatives.
  • Another participant introduces the concept of Taylor Series as a method to assess the accuracy of the linear approximation over a given domain.
  • There is a proposal to use linear regression to find the best linear fit over defined segments of the graph, suggesting multiple linear approximations over different intervals.
  • A participant notes that the graph tends to follow the equation Y = sin(2x) but mentions that the initial points may be outliers affecting linearization.

Areas of Agreement / Disagreement

Participants express varying opinions on the feasibility of linearization, with some suggesting methods while others remain skeptical about the overall applicability of linearization to the function in question. The discussion does not reach a consensus on the best approach.

Contextual Notes

Limitations include the potential inaccuracy of linear approximations over larger domains and the presence of outliers that may affect the linearity of the graph.

Who May Find This Useful

This discussion may be of interest to individuals exploring mathematical modeling, particularly in the context of approximating non-linear functions, as well as those studying the application of calculus in real-world scenarios.

Bianca526
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TL;DR
I wanted to ask if there was any way I could linearize this graph? Its a displacement vs angle graph and Im having difficulty trying to figure out how to linearize it.
Screen Shot 2020-12-11 at 3.45.44 PM.png
 
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In pieces maybe?
 
If I may ask, how would I do that?
 
Why do you want to linearize it? And more importantly, what makes you think that it can be linearized?
 
I'll assume that you mean you want to approximate it's value with a linear function?

In a global sense, this certainly isn't a linear function. So, over a large domain, you can't linearize it with any reasonable accuracy.

However, any well behaved function can be approximated by a line in a small enough domain. It's as simple as calculating the slope of the curve at the point you want to approximate around, then, combined with the value at that point you can solve for the linear approximation; y = m⋅x + b. Do you know about derivatives? That is the key, that is what m is.

Then the real problem comes when you want to know how accurate your approximation is over some domain around the point. For that you'll want to learn about Taylor Series.

Alternatively, you can define your domain as a certain number of points and use linear regression to calculate the best linear fit.

In either case Wikipedia is a good place to get more information.
 
Bianca526 said:
If I may ask, how would I do that?

One linear approximation from (e.g.) 0 to 10. Another from 10-20. Another from 20-30. And so on.
 
Bianca526 said:
Summary:: I wanted to ask if there was any way I could linearize this graph? Its a displacement vs angle graph and I am having difficulty trying to figure out how to linearize it.

View attachment 274151
It tends to follow Y = sin (2xangle) apart from the first few points which seem to be outliers from a straight line.
 

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