The Deltoid
Jared Kline
Stephen Butler
Fall 2007
Description
A deltoid is a type of Hypocycloid. Hypocycloids are plane curves traced
out by a point on a circle rolling along the inside of a larger circle.
In a specific type of Hypocycloid, the radius of the inner circle is 1/3 that of
the larger circle. This creates a hypocycloid of three cusps, or a deltoid.
History
The Deltoid, also known as the
Tricuspid, is just one case of
hypocycloid. It has three cusps
and looks very similar to the
Greek symbol delta. It was first
discovered in 1745 by Leonhard
Euler but was not greatly
studied until 1856 by Jakob
Steiner, therefore commonly
referred to as Steiner’s
Hypocycloid.
Parameterization
Large circle has radius = r
Small circle has radius = r/3
<BAC = T
<BDF = 3T
So the arc length between B and C = rT;
And the arc length between D and F = (r/3)t
Parameterization
To be able to parameterize the motion of the
circle we needed to make a couple more lines,
giving us triangles to find our needed angles.
Now by using some simple geometry, we are
able to find <GDF
<BAC = T
<BDG = T
<BDF = 3T
So, <GDF = 2T
Parameterization
Now that we have our angles, we can find the
coordinates of x and y in terms of (T)
X = AE + DG
X = (2r/3)cos(T) + (r/3)cos(2T)
Y = DE – FG
Y = (2r/3)sin(T) – (r/3)sin(2T)
Where 0
 T  2
Can you find a Deltoid?
Deltoids can be formed by 3 circles of equal radius
that are tangent to each other
Deltoids are common in places other than
piles of pipes. Many wheel designs are
based on the deltoid.
Works Cited
http://en.wikipedia.org/wiki/Deltoid_curve
http://mathworld.wolfram.com/Deltoid.html
http://www.xahlee.org/SpecialPlaneCurves
_dir/specialPlaneCurves.html
http://mathworld.wolfram.com/Deltoid.html
http://en.wikipedia.org/wiki/Deltoid_curve
http://www.daviddarling.info/encyclopedia/D/deltoid.html
http://www.2dcurves.com/roulette/rouletted.html