Hypotrochoid

The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are Template:Math).

In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius Template:Mvar rolling around the inside of a fixed circle of radius Template:Mvar, where the point is a distance Template:Mvar from the center of the interior circle.

The parametric equations for a hypotrochoid are:

$\begin{array}{cc}& x(\theta )=(R-r)\cos \theta +d\cos \left(\frac{R-r}{r}\theta \right)\\ & y(\theta )=(R-r)\sin \theta -d\sin \left(\frac{R-r}{r}\theta \right)\end{array}$

where Template:Mvar is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because Template:Mvar is not the polar angle). When measured in radian, Template:Mvar takes values from 0 to $2\pi \times \frac{\mathrm{LCM}(r,R)}{R}$ (where Template:Math is least common multiple).

Special cases include the hypocycloid with Template:Math and the ellipse with Template:Math and Template:Math. The eccentricity of the ellipse is

$e=\frac{2\sqrt{d/r}}{1+(d/r)}$

becoming 1 when $d=r$ (see Tusi couple).

The ellipse (drawn in red) may be expressed as a special case of the hypotrochoid, with Template:Math (Tusi couple); here Template:Math.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.

See also

• Cycloid
• Cyclogon
• Epicycloid
• Rosetta (orbit)
• Apsidal precession
• Spirograph

External links

• Template:MathWorld
• Flash Animation of Hypocycloid
• Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
• Interactive hypotrochoide animation
• Template:MacTutor