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


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