By Nate Currier, Fall 2008 O-M-G! It’s AMAZING! O-M-D!! More like “oh my deltoid!”

Table of Contents A brief history The Hypocycloid Parametric Equations The Deltoid in ACTION! Deltoid Description The Deltoid in nature The Deltoid and Man Works Cited

A brief History The deltoid has no real discoverer. -The deltoid is a special case of a Cycloid; a three-cusped Hypocycloid. -Also called the tricuspid. -It was named the deltoid because of its resemblance to the Greek letter Delta. Despite this, Leonhard Euler was the first to claim credit for investigating the deltoid in Though, Jakob Steiner was the first to study the deltoid in depth in From this, the deltoid is often known as Steiner’s Hypocycloid. Leonhard Euler, Jakob Steiner,

The Hypocycloid To understand the deltoid, aka the tricuspid hypocycloid, we must first look to the hypocycloid. A hypocycloid is the trace of a point on a small circle drawn inside of a large circle. The small circle rolls along inside the circumference of the larger circle, and the trace of a point in the small circle will form the shape of the hypocycloid. The ratio of the radius of the inner circle to that of the outer circle ( a/b ) is what makes each Hypocycloid unique.

Parametric Equations The equation of the deltoid is obtained by setting n = a / b = 3 in the equation of the Hypocycloid: Where a is the radius of the large fixed circle and b is the radius of the small rolling circle, yielding the parametric equations. This yields the parametric equation:

The Deltoid in ACTION!

Deltoid Description Deltoid can be defined as the trace of a point on a circle, rolling inside another circle either 3 times or 1.5 times the radius of the original circle. The two sizes of rolling circles can be synchronized by a linkage: Let A be the center of the fixed circle. Let D be the center of the smaller circle. Let F be the tracing point. Let G be a point translated from A by the vector DF. G is the center of the larger rolling circle, which traces the same line as F. ADFG is a parallelogram with sides having constant lengths.

The Deltoid in Nature Yeah, that’s about as natural as it gets.

The Deltoid and Man Perhaps I should have said, the deltoid “in” man. Used in wheels and stuff.

Works Cited Weisstein, Eric W. “Deltoid.” Mathworld. Accessed 3 Dec, Lee, Xah, “A Visual Dictionary of Special Plane Curves” Accessed 4 Dec, Kimberling, Clark. “Jakob Steiner ( ) geometer” Accessed 4 Dec Qualls, Dustin. “The Deltoid”. Accessed 4 Dec 2008.

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