Hypocycloid

Definition

A hypocycloid is the curve traced by a fixed point on a smaller circle that rolls without slipping inside a larger circle. Its parametric equations can be derived and analyzed using calculus.

5 Must Know Facts For Your Next Test

1. The parametric equations of a hypocycloid are $x(\theta) = (a - b) \cos(\theta) + b \cos((a - b)\theta / b)$ and $y(\theta) = (a - b) \sin(\theta) - b \sin((a - b)\theta / b)$ where $a$ is the radius of the larger circle and $b$ is the radius of the smaller circle.
2. When $b = a/2$, the hypocycloid becomes an astroid, which has four cusps.
3. Hypocycloids can be used to model gears and other mechanical systems with rolling components.
4. The length of one arc of a hypocycloid can be calculated using integral calculus, considering one complete rotation inside the larger circle.
5. Special cases of hypocycloids include ellipses and straight lines when specific ratios between $a$ and $b$ are chosen.

Review Questions

Related terms

Parametric Equations:

Equations that express coordinates as functions of one or more variables.

Astroid:

A special type of hypocycloid with four cusps formed when $b = a/2$.

Epicycloid: The curve traced by a point on a smaller circle that rolls without slipping outside a larger circle.