Thanks Gareth! lituus <http://gosper.org/lituus.gif> Now what's a convincing portrayal of the sector area constancy? Hey Eric, gifs in MathWorld? —rwg On Sun, Oct 20, 2019 at 4:10 PM rwg <rwg@ma.sdf.org> wrote:
-------- Original Message -------- Subject: Re: [math-fun] lituus? Date: 2019-10-17 19:06 From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Reply-To: math-fun <math-fun@mailman.xmission.com>
On 14/10/2019 12:42, Bill Gosper wrote:
MathWorld: "The lituus is the locus of the point [image: P] moving such that the area <http://mathworld.wolfram.com/Area.html> of a circular sector <http://mathworld.wolfram.com/CircularSector.html> remains constant." What circle? What sector? Pic? —rwg
It's r^2 theta = const.
So for a point P at radius r and angle theta, take a circular sector spanning angles 0..theta and radius r. As the spiral winds inward around the origin, the radius decreases just enough to keep that area the same. For "almost all" of the curve we have theta>2pi and of course we need to count the area with multiplicity.
So e.g. if we take the constant to be 1, then at angle 2pi (i.e., the first time the curve crosses the positive x-axis) we have r = sqrt(1/2pi) and that circle is covered just once; at angle 4pi where it crosses again we have r = sqrt(1/4pi) so the circle is sqrt(2) times smaller in radius but is covered twice.