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 intersection point on the curve.
Given a polar curve
r=r(φ) | (1) |
in polar coordinates r,φ, we derive an expression for the tangent of the polar tangential angle ψ, using the classical differential
geometric method.
The point P of the curve given by (1) corresponds to the polar angle φ=∠POA and the polar radius r=OP. The “near” point P′ corresponds to the polar angle φ+dφ=∠P′OA and the polar radius r+dr=OP′. In the diagram, P′Q is the arc of the circle with O as centre and OP′ as radius. Thus, in the triangle-like figure PP′Q we have
P′QPQ=(r+dr)dφdr=r+drdrdφ. | (2) |
This figure can be regarded as an infinitesimal right triangle
with the catheti P′Q and PQ. Accordingly, their ratio (2) is the tangent of the acute angle
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 |