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Curves

Hypotrochoid

Main
Parametric Cartesian equation:
x=(a−b)cos⁡(t)+ccos⁡((a/b−1)t),y=(a−b)sin⁡(t)−csin⁡((a/b−1)t)x = (a - b) \cos(t) + c \cos((a/b -1)t), y = (a - b) \sin(t) - c \sin((a/b -1)t)

View the interactive version of this curve.

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 hypotrochoid, 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 at distance cc from the centre of the circle of radius bb. For this examplea=5,b=7a = 5, b = 7and c = 2.22.2.

These curves were studied by la Hire, Desargues, Leibniz, Newton and many others.

Other Web site:

Xah Lee

Associated Curves

Definitions of the Associated curves

  • Evolute
  • Involute 1
  • Involute 2
  • Inverse curve wrt origin
  • Inverse wrt another circle
  • Pedal curve wrt origin
  • Pedal wrt another point
  • Negative pedal curve wrt origin
  • Negative pedal wrt another point
  • Caustic wrt horizontal rays
  • Caustic curve wrt another point