polar tangential angle


The angle, under which a polar curve is by a line through the origin, is called the polar tangential angle belonging to the intersectionMathworldPlanetmath point on the curve.

Given a polar curve

r=r(φ) (1)

in polar coordinates r,φ,  we derive an expression for the tangentPlanetmathPlanetmathPlanetmath of the polar tangential angle ψ, using the classical differentialMathworldPlanetmath geometric method.

The point P of the curve given by (1) corresponds to the polar angleMathworldPlanetmathφ=POA  and the polar radius r=OP.  The “near” point P corresponds to the polar angle  φ+dφ=POA  and the polar radius  r+dr=OP.  In the diagram, PQ is the arc of the circle with O as centre and OP as radius.  Thus, in the triangle-like figure PPQ we have

PQPQ=(r+dr)dφdr=r+drdrdφ. (2)

This figure can be regarded as an infinitesimalMathworldPlanetmathPlanetmath right triangleMathworldPlanetmath with the catheti PQ and PQ.  Accordingly, their ratio (2) is the tangent of the acute angleMathworldPlanetmathPlanetmath P of the triangle.  Because the addend dr in the last numerator in negligible compared with the addend r, it can be omitted.  Hence we get the tangent

tanψ=rdrdφ

of the polar tangential angle, i.e.

tanψ=r(φ)r(φ). (3)
Title polar tangential angle
Canonical name PolarTangentialAngle
Date of creation 2013-03-22 19:02:32
Last modified on 2013-03-22 19:02:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Derivation
Classification msc 51-01
Classification msc 53A04
Related topic LogarithmicSpiral