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the curve doesn't bend "upwards" or "downwards" depending on the sign of rho, it turns clockwise or anticlockwise |
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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.
The general expression for finding the intrinsic equation is:
<math>s = \int_\,\sqrt{1 + \tan^2 \psi}\frac{dx}{d\psi}\,d\psi</math>
This integration is only practical where <math>frac{dx}{\psi}</math> is shown in terms of <math>\psi</math>
[[Category:Coordinate systems]]
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