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.
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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.
When $b = a/2$, the hypocycloid becomes an astroid, which has four cusps.
Hypocycloids can be used to model gears and other mechanical systems with rolling components.
The length of one arc of a hypocycloid can be calculated using integral calculus, considering one complete rotation inside the larger circle.
Special cases of hypocycloids include ellipses and straight lines when specific ratios between $a$ and $b$ are chosen.
Review Questions
What are the parametric equations for a hypocycloid?
Explain what happens to the shape of a hypocycloid when the ratio between the radii of the circles changes.
How would you derive the length of an arc in a hypocycloid?
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.