- #1
Jhenrique
- 685
- 4
First, I'd like that you read this littler article (http://mathworld.wolfram.com/NaturalEquation.html). The solution given by Euler that coonects the system cartesian (x, y) with the curvature κ of the "cesaro system" (s, κ), is that the derivative of the cartesian tangential angle φ* wrt arc length s results the curvature κ.
However, in the 2D plane is definied the polar tangential angle ψ* too. Thus if I want express a curve given s and κ in terms of r and θ (or vice-versa) I need to establish a connection between the curvature κ and the polar system. So, would be correct to say that the derivative of ψ wrt s is equal to κ?
*
https://en.wikipedia.org/wiki/Tangential_angle
https://en.wikipedia.org/wiki/Subtangent
https://en.wikipedia.org/wiki/List_...rtesian_coordinates_from_Ces.C3.A0ro_equation
EDIT:
However, in the 2D plane is definied the polar tangential angle ψ* too. Thus if I want express a curve given s and κ in terms of r and θ (or vice-versa) I need to establish a connection between the curvature κ and the polar system. So, would be correct to say that the derivative of ψ wrt s is equal to κ?
*
https://en.wikipedia.org/wiki/Tangential_angle
https://en.wikipedia.org/wiki/Subtangent
https://en.wikipedia.org/wiki/List_...rtesian_coordinates_from_Ces.C3.A0ro_equation
EDIT:
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