Hypotrochoid
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:<ref>Template:Cite book</ref>
- <math>\begin{align}
& x (\theta) = (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right) \\ & y (\theta) = (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right) \end{align}</math>
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 <math>2 \pi \times \tfrac{\operatorname{LCM}(r, R)}{R}</math> (where Template:Math is least common multiple).
Special cases include the hypocycloid with Template:Math and the ellipse with Template:Math and Template:Math.<ref>Template:Cite book</ref> The eccentricity of the ellipse is
- <math>e=\frac{2\sqrt{d/r}}{1+(d/r)}</math>
becoming 1 when <math>d=r</math> (see Tusi couple).
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.<ref>Template:Cite journal</ref>
See alsoEdit
ReferencesEdit
<references />
External linksEdit
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Hypotrochoid%7CHypotrochoid.html}} |title = Hypotrochoid |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}
- Flash Animation of Hypocycloid
- Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
- Interactive hypotrochoide animation
- Template:MacTutor