umair butt

Posted on Sunday, 22 February, 2004 - 12:58 am:

Q. Show that the curve with intrinsic equation

s = ψ

is a circle, radius 1

Michael McLoughlin

Posted on Sunday, 22 February, 2004 - 11:46 am:

The radius of curvature, R, is given by:

Latex image click or follow link to see src

Therefore, the curve is a circle of radius 1. This is because the radius of curvature is the radius of the circle that best fits the curve at a given point (see http://www.ies.co.jp/math/java/calc/curve/curve.html ). We see, therefore, that the radius of the circle that best fits the curve at every point has radius 1. This shows it is a circle.

umair butt

Posted on Sunday, 22 February, 2004 - 02:50 pm:

yes but is there a way to mathematically prove it?

Nicola Coles

Posted on Sunday, 22 February, 2004 - 03:27 pm:

Hi Umair,

If $s = ψ$ ,

$\cos s = \cos ψ$

$dx / ds = \cos ψ$

Therefore, $dx / ds = \cos s$

$\sin s = x + a$

$\sin s = \sin ψ$

$dy / dx = \sin ψ$

Therefore, $dy / dx = \sin s$

$- \cos s = y + b$

Therefore, $(x + a)^{2} + (y + b)^{2} = 1$

which is the equation of a circle, radius 1.

Hope that helps,

Nicola