# Lituus: Difference between revisions

A lituus is a spiral in which the angle is inversely proportional to the square of the radius (as expressed in polar coordinates).

$\displaystyle{ r^2\theta = k \, }$

### 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

$\displaystyle{ \!\,r=\cos(k\theta). }$

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 $\displaystyle{ 2\pi }$ for $\displaystyle{ k }$ even, and $\displaystyle{ \pi }$ for $\displaystyle{ k }$ odd.)

The quadrifolium is a type of rose curve with n=2. It has polar equation:

$\displaystyle{ r = \cos(2\theta), \, }$

with corresponding algebraic equation

$\displaystyle{ (x^2+y^2)^3 = (x^2-y^2)^2. \, }$