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; s = f(ψ) is the intrinsic equation of the curve.

This coordinate system has limited use, it may break down entirely when straight lines are considered, but inspection reveals three properties regarding the rate of change of its variables, namely:

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

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

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

Radius of curvature

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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 bends upwards, and if the value is negative, the curve bends downward. It is given by: $\backslash rho\; =\; \backslash frac\{ds\}\{d\backslash psi\}.$

It can be proved that the following is true:

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

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:

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

where

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

Conversion

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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.

Coordinate systems