# Curves

### Hypotrochoid

- Parametric Cartesian equation:
- $x = (a - b) \cos(t) + c \cos((a/b -1)t), y = (a - b) \sin(t) - c \sin((a/b -1)t)$

### 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 $P$ on a circle of radius $b$ which rolls round a fixed circle of radius $a$.For the hypotrochoid, an example of which is shown above, the circle of radius $b$ rolls on the inside of the circle of radius $a$. The point $P$ is at distance $c$ from the centre of the circle of radius $b$. For this example$a = 5, b = 7$and c = $2.2$.

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

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Xah Lee