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 1754.

• Though, Jakob Steiner was the first to study the deltoid in depth in 1856.- From this, the deltoid is often known as Steiner’s Hypocycloid.

Leonhard Euler, 1701-1783

Jakob Steiner,1796-1863

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 EquationsThe 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!

Mydeltoid.gsp

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, 2008.

<http://mathworld.wolfram.com/Deltoid.html>.

• Lee, Xah, “A Visual Dictionary of Special Plane Curves” Accessed 4 Dec, 2008. <http://www.xahlee.org/SpecialPlaneCurves_dir/Deltoid_dir/deltoid.html>

• Kimberling , Clark. “Jakob Steiner (1796-1863) geometer” Accessed 4 Dec 2008. <http://faculty.evansville.edu/ck6/bstud/steiner.html>

• Qualls, Dustin. “The Deltoid”. Accessed 4 Dec 2008. <http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Fall98/DustinQ/deltoid1.htm>

THE END!!