Semicubical parabola

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A semicubical parabola is a curve defined parametrically as

[math]\displaystyle{ x = t^2 }[/math]
[math]\displaystyle{ y = at^3 }[/math]

  • From Wikipedia:

The semicubical parabola was discovered in 1657 by William Neile who computed its arc length; it was the first algebraic curve (excluding the line) to be rectified. It is unique in that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods.

  • From MathDL:

1659: Hendrik van Heuraet sent van Schooten his rectification of the semi-cubical parabola. This was published---his only publication---in the second Latin edition of Descartes' Geometrie. This broke the spell of Aristotle's dictum that curved lines could not in principle be compared with straight lines.


The underlying JavaScript code

var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-1,2,3,-2], keepaspectratio:true, axis:true});
var a = brd.create('slider',[[-0.5,1.8],[1,1.8],[-5,0.20,5]], {name:'a'});

var p = brd.create('curve',
             [function(t){ return t*t;},
              function(t){ return a.Value()*t*t*t;},
              -2, 2
             {strokeWidth:1, strokeColor:'black'});