Rose

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] \!\,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]2\pi[/math] for [math]k[/math] even, and [math]\pi[/math] for [math]k[/math] odd.)

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

[math] r = \cos(2\theta), \,[/math]

with corresponding algebraic equation

[math] (x^2+y^2)^3 = (x^2-y^2)^2. \, [/math]

The JavaScript code to produce this picture

<jsxgraph width="500" height="500" box="box2">
 var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
 var f = b2.createElement('slider', [[1,8],[5,8],[0,4,8]]);
 var len = b2.createElement('slider', [[1,7],[5,7],[0,2,2]],{name:'len'}); 
 var k = b2.createElement('slider', [[1,6],[5,6],[0,2,10]],{snapWidth:0.2,name:'k'});
 var c = b2.createElement('curve', [function(phi){return f.Value()*Math.cos(k.Value()*phi); }, [0, 0],0, function(){return len.Value()*Math.PI;}],
             {curveType:'polar', strokewidth:2});      
</jsxgraph>

External links

• http://en.wikipedia.org/wiki/Rose_(mathematics)
• http://en.wikipedia.org/wiki/Quadrifolium