Lituus
A lituus is a spiral in which the angle is inversely proportional to the square of the radius (as expressed in polar coordinates).
- [math]\displaystyle{ r^2\theta = k \, }[/math]
The JavaScript code to produce this picture
<jsxgraph width="500" height="500" box="box1">
var b1 = JXG.JSXGraph.initBoard('box1', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
var k = b1.createElement('slider', [[1,8],[5,8],[0,1,4]]);
var c = b1.createElement('curve', [function(phi){return Math.sqrt(k.Value()/phi); }, [0, 0],0, 8*Math.PI],
{curveType:'polar', strokewidth:4});
</jsxgraph>
Other curves
A rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, these curves can all be expressed by a polar equation of the form
- [math]\displaystyle{ \!\,r=\cos(k\theta). }[/math]
If k is an integer, the curve will be rose shaped with
- 2k petals if k is even, and
- k petals if k is odd.
When k is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π. When k is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length [math]\displaystyle{ 2\pi }[/math] for [math]\displaystyle{ k }[/math] even, and [math]\displaystyle{ \pi }[/math] for [math]\displaystyle{ k }[/math] odd.)
The quadrifolium is a type of rose curve with n=2. It has polar equation:
- [math]\displaystyle{ r = \cos(2\theta), \, }[/math]
with corresponding algebraic equation
- [math]\displaystyle{ (x^2+y^2)^3 = (x^2-y^2)^2. \, }[/math]
