Rose

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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 for k even, and π 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

var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,boundingbox: [-10, 10, 10, -10]});
var f = b2.create('slider', [[1,8],[6,8],[0,4,8]]);
var len = b2.create('slider', [[1,7],[6,7],[0,2,8]],{snapWidth:1,name:'len'});
var k = b2.create('slider', [[1,6],[6,6],[0,2,12]],{snapWidth:0.2,name:'k'});
var c = b2.create('curve', [function(phi){return f.Value()*Math.cos(k.Value()*phi); }, [0, 0],0, function(){return len.Value()*Math.PI;}],
{curveType:'polar', strokewidth:2});

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