Curves

Hypocycloid

Main
Parametric Cartesian equation:
x=(ab)cos(t)+bcos((a/b1)t),y=(ab)sin(t)bsin((a/b1)t)x = (a - b) \cos(t) + b \cos((a/b - 1)t), y = (a - b) \sin(t) - b \sin((a/b - 1)t)

Description

There are four curves which are closely related. These are the epicycloid, the epitrochoid, the hypocycloid and the hypotrochoid and they are traced by a point PP on a circle of radius bb which rolls round a fixed circle of radius aa.

For the hypocycloid, an example of which is shown above, the circle of radius bb rolls on the inside of the circle of radius aa. The point PP is on the circumference of the circle of radius bb. For the examplea=5a = 5 and b=3b = 3.

These curves were studied by Dürer (1525), Desargues (1640), Huygens (1679), Leibniz, Newton (1686), de L'Hôpital (1690), Jacob Bernoulli (1690), la Hire (1694), Johann Bernoulli (1695), Daniel Bernoulli (1725), Euler (1745, 1781).

Special cases are a=3ba = 3b when a tricuspoid is obtained and a=4ba = 4b when an astroid is obtained.

If a=(n+1)ba = (n + 1)b where nn is an integer, then the length of the epicycloid is 8nb8nb and its area is πb2(n2n)πb^{2}(n^{2} - n).

The pedal curve, when the pedal point is the centre, is a rhodonea curve.

The evolute of a hypocycloid is a similar hypocycloid - look at the evolute of the hypocycloid above to see it is a similar hypocycloid but smaller in size.

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Xah Lee