Trochoid

The Trochoid curve (blue) and its dual curve (red). The equation of the trochoid is

[math]\displaystyle{ x = a\phi-b\sin(\phi) }[/math]

[math]\displaystyle{ y = a-b\cos(\phi) }[/math]

References

• http://en.wikipedia.org/wiki/Trochoid
• http://en.wikipedia.org/wiki/Dual_curve

The underlying JavaScript code

board = JXG.JSXGraph.initBoard('jsxgbox', {boundingbox:[-10,10,10,-10], axis:true});
board.suspendUpdate();
var D = JXG.Math.Numerics.D;
var a = board.create('slider', [[1,-1],[8,-1],[-5,1,5]], {style:6,name:'a'});
var b = board.create('slider', [[1,-2],[8,-2],[-5,1,5]], {style:6,name:'b'});
var x = function(phi) { return a.Value()*phi-b.Value()*Math.sin(phi); }
var y = function(phi) { return a.Value()-b.Value()*Math.cos(phi); }
var c1 = board.create('curve', [x,y,-Math.PI*4,Math.PI*4],{strokeWidth:3});
 
var dualCurve = function(x,y,board) {
    var X = function(phi) { return D(y)(phi)/(y(phi)*D(x)(phi)-x(phi)*D(y)(phi)); }
    var Y = function(phi) { return D(x)(phi)/(x(phi)*D(y)(phi)-y(phi)*D(x)(phi)); }
    return [X,Y];
}
var dual = dualCurve(x,y,board);
var c2 = board.create('curve', [dual[0],dual[1],-Math.PI*1,Math.PI*1],{strokeWidth:3, strokeColor:'red'});
board.unsuspendUpdate();