Everything about Hypocycloid totally explained
In
geometry, a
hypocycloid is a special
plane curve generated by the trace of a fixed point on a small
circle that rolls within a larger circle. It is comparable to the
cycloid but instead of the circle rolling along a line, it rolls within a circle.
If the smaller circle has radius
r, and the larger circle has radius
R =
kr, then the
parametric equations for the curve can be given by
»
If
k is an integer, then the curve is closed, and has
k cusps (for example, sharp corners, where the curve isn't
differentiable).
If
k is a
rational number, say
k =
p/
q expressed in simplest terms, then the curve has
p cusps.
If
k is an
irrational number, then the curve never closes, and fills the space within the larger circle except for a disk of radius
R −
r in the center of the larger circle.
Image:Hypocycloid-3.svg| k=3 - a deltoid
Image:Hypocycloid-4.svg| k=4 - an astroid
Image:Hypocycloid-5.svg| k=5
Image:Hypocycloid-6.svg| k=6
Image:Hypocycloid-2-1.svg| k=2.1
Image:Hypocycloid-3-8.svg| k=3.8
Image:Hypocycloid-5-5.svg| k=5.5
Image:Hypocycloid-7-2.svg| k=7.2
The hypocycloid is a special kind of
hypotrochoid, which are a particular kind of
roulette.
A hypocycloid with three cusps is known as a
deltoid.
A hypocycloid curve with four cusps is known as an
astroid.
Derived curves
The
evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while
the
involute of a hypocycloid is a reduced copy of itself.
(External Link
)
The
pedal of a hypocycloid with pole at the center of the hypocycloid is a
rose curve.
The
isoptic of a hypocycloid is a hypocycloid.
Hypocycloids in popular culture
Curves similar to hypocyloids can be drawn with the
Spirograph toy. Specifically, the Spirograph can draw
hypotrochoids and
epitrochoids.
The
Pittsburgh Steelers' logo includes three
astroids (hypocycloids of four
cusps). In his weekly NFL.com column
Tuesday Morning Quarterback, Gregg Easterbrook often refers to the Steelers as the Hypocycloids.
Portland, Oregon's flag features a hypocycloid; an
astroid.
The 2007 redesign of
The Price is Right's set features astroids on the three main doors and the turntable area
(External Link).
Further Information
Get more info on 'Hypocycloid'.
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