Hypocycloid — Definition, Formula & Examples

A hypocycloid is the curve traced by a fixed point on a small circle that rolls without slipping along the inside of a larger circle. The shape of the curve depends on the ratio of the two circles' radii.

Given a fixed circle of radius $R$ and an interior rolling circle of radius $r$ , a hypocycloid is the locus of a point on the circumference of the rolling circle. When the ratio $R/r = k$ is a positive integer, the curve is a closed figure with exactly $k$ cusps.

Key Formula

x(t) = (R - r)\cos t + r\cos\!\left(\frac{R - r}{r}\,t\right), \qquad y(t) = (R - r)\sin t - r\sin\!\left(\frac{R - r}{r}\,t\right)

Where:

$R$ = Radius of the fixed outer circle
$r$ = Radius of the rolling inner circle
$t$ = Parameter (angle), typically ranging over $[0, 2\pi]$ for integer $R/r$

How It Works

As the smaller circle rolls inside the larger one, the tracing point sweeps out a curve with pointed tips called cusps. The number of cusps equals

k = R/r

when

k

is an integer. For

k = 3

, you get a deltoid (three cusps). For

k = 4

, you get an astroid (four cusps). If

k

is not rational, the curve never closes and eventually fills a ring-shaped region.

Worked Example

Problem: Find the parametric equations for the astroid, where R = 4 and r = 1.

Step 1: Substitute R = 4 and r = 1 into the general formulas.

x(t) = (4-1)\cos t + 1\cdot\cos\!\left(\frac{4-1}{1}\,t\right) = 3\cos t + \cos 3t

Step 2: Do the same for y(t).

y(t) = 3\sin t - \sin 3t

Step 3: These can be simplified using trigonometric identities to the well-known astroid form.

x^{2/3} + y^{2/3} = R^{2/3} = 4^{2/3} \approx 2.52

Answer: The astroid has parametric equations

x = 3\cos t + \cos 3t

y = 3\sin t - \sin 3t

, and the implicit Cartesian equation

x^{2/3} + y^{2/3} = 4^{2/3}

Why It Matters

Hypocycloids appear in gear design (the Spirograph toy generates them), in the profile of certain cutting tools, and in kinematics problems in engineering. The astroid specifically arises in envelope problems in differential geometry and optics.

Common Mistakes

Mistake: Confusing a hypocycloid with an epicycloid.

Correction: A hypocycloid is generated by rolling inside a fixed circle; an epicycloid is generated by rolling outside. The sign difference in the parametric equations reflects this distinction.