Intrinsic coordinates

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Important points in intrinsic coordinates

Intrinsic coordinates is a coordinate system which defines points upon a curve partly by the nature of the tangents to the curve at that point. A point is given as (s, ψ) where s is the length of the curve from a set point (often the origin, in the case of the diagram on the right, point A) and ψ is the angle which the tangent to the curve at that point makes with the x-axis; ψ = f(s) is the intrinsic equation of the curve.

This is arguably not a true coordinate system, since it describes points on a curve relative to the curve itself rather than the space in which it is embedded, and points not on the curve have no intrinsic coordinates.

Inspection reveals three properties regarding the rate of change of the variables, namely:

$\frac{dy}{dx} = \tan \psi$

$\frac{dx}{ds} = \cos \psi$

$\frac{dy}{ds} = \sin \psi$

[edit] Radius of curvature

The radius of curvature, ρ, at a point is a measure of the radius of the arc which can be created by the extrapolation of that point. If this value is positive then the curve turns anticlockwise as s increases; if negative, the curve turns clockwise. It is given by: $\rho = \frac{ds}{d\psi}.$

It can be proven that the following is true:

$\rho = \frac {\big( 1 + (\frac{dy}{dx})^2\big)^{3/2}}{\frac {d^2y}{dx^2}}.$

This allows the radius of curvature of a line to be found from only Cartesian coordinates.

Another useful formula can relate the above to parametric form:

$\frac{ds}{d\psi} = \frac {\big({\dot{x}^2 + \dot{y}^2}\big)^{3/2}}{\dot {x}\ddot{y} - \dot{y}\ddot{x}},$

where

$\dot{x} = \frac{dx}{dt}\ \mbox{and}\ \ddot{x} = \frac{d^2x}{dt^2}.$

[edit] Conversion

To convert a cartesian equation y = f(x) to an intrinsic equation, differentiate it to get dy/dx. Then find the arc length (see formula - requires the derivative), integrating from 0 to x. Then convert x to ψ using the dy/dx relationship above by expressing s in terms of dy/dx.

The general expression for finding the intrinsic equation is:

$s = \int_\,\sqrt{1 + \tan^2 \psi}\frac{dx}{d\psi}\,d\psi$