Can a Velocity Vector be Integrated to Parameterize y=x^2?

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Discussion Overview

The discussion revolves around the integration of a velocity vector to parameterize the curve defined by y=x². Participants explore the implications of normalizing the velocity vector and the requirements for arc-length parameterization.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents a normalized velocity vector for the curve y=x² and questions its integration to obtain a parameterization.
  • Another participant argues that integrating the normalized vector will not yield the original curve, as it represents a unit tangent vector rather than the actual derivative of the position vector.
  • A participant suggests that to achieve arc-length parameterization, a transformation involving the angle from the atan2 function is necessary.
  • Another participant proposes defining a function s(t) to maintain a unit vector in the context of arc-length parameterization, leading to a differential equation to solve.
  • One participant raises a concern about maintaining a constant speed of 1 and the potential for imaginary results when taking arccos and arcsin of certain inputs.
  • A request is made for assistance in setting up and solving the differential equation relevant to the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the integration of the normalized velocity vector and the approach to arc-length parameterization. The discussion remains unresolved with multiple competing perspectives on how to proceed.

Contextual Notes

There are limitations regarding the assumptions made about the velocity vector and the conditions under which the parameterization is valid. The discussion also highlights the dependence on the definitions of the functions involved and the potential for complex results in certain scenarios.

Who May Find This Useful

This discussion may be of interest to those studying differential equations, parameterization of curves, and the mathematical principles underlying motion along a path in a plane.

rabbed
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Hi

(1/sqrt(4t²+1), 2t/sqrt(4t²+1)) gives a unit tangent to the curve y=x^2 at point (t,t^2).
Viewing the vector as velocity, shouldn't I be able to integrate it and get a parameterization for y=x^2?
 
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If you try to integrate it with respect to ##t## you will not recover the curve since you have normalised the vector. What you have is
$$
\frac{\frac{d\vec x}{dt}}{\left\lvert\frac{d\vec x}{dt}\right\rvert},
$$
not ##d\vec x/dt##.
 
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Ah, okay.
So if I want to arc-length parameterize, I would need to convert "sqrt(4t^2+1)" from the expression sqrt(4t^2+1)*(cos(atan2(2t,1)), sin(atan2(2t,1))) into some change of the angle atan2(2t,1)..
 
If you want to parametrise with the arc length you should define a function s(t) such that ##d\vec x/ds= (d\vec x/dt)(dt/ds)## is a unit vector. This will give you a differential equation to solve.
 
Okay,
but to have a constant speed of 1, changing the velocity angle is what you would actually do at each point to stay on the curve, right?
And this would involve taking arccos and arcsin of some input r where it's possible that |r| > 1, so that the result can be imaginary?
 
Can you show me how you would set up and solve the differential equation in my case?
 

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