# Difference between revisions of "Rose"

### 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

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

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

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

$r = \cos(2\theta), \,$

with corresponding algebraic equation

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

### 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]]);
var k = b2.createElement('slider', [[1,6],[5,6],[0,2,10]]);
var c = b2.createElement('curve', [function(phi){return f.Value()*Math.cos(Math.floor(k.Value())*phi); }, [0, 0],0, function(){return len.Value()*Math.PI;}],
{curveType:'polar', strokewidth:2});
</jsxgraph>