# Curves

### Epitrochoid

- 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 epitrochoid, an example of which is shown above, the circle of radius $b$ rolls on the outside of the circle of radius $a$. The point $P$ is at distance $c$ from the centre of the circle of radius $b$. For the example $a = 5, b = 3$ and $c = 5$ (so $P$ goes inside the circle of radius $a$).

An example of an epitrochoid appears in Dürer's work

*Instruction in measurement with compasses and straight edge*(1525). He called them spider lines because the lines he used to construct the curves looked like a spider.

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

**Other Web site:**

Xah Lee