Chemistry - Intrinsic coordinates

Introduction

Instrisic 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 origin.

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

$\frac{dy}{dx} = \tan \Psi$ $\frac{dx}{ds} = \cos \Psi$ $\frac{dy}{ds} = \sin \Psi$

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

It can be proved that the following is true:

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

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, note that $\dot{x} = \frac{dx}{dt}$ and $\ddot{x} = \frac{d^2x}{dt^2}$

$\frac{ds}{d\Psi} = \frac {\sqrt{\dot{x}^2 + \dot{y}^2}^3}{\dot {x}\ddot{y} - \dot{y}\ddot{x}}.$

Categories: Coordinate systems