Jared Kline Stephen Butler Fall 2007 The Deltoid

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

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 _dir/specialPlaneCurves.html