Transcript Presentation

It’s AMAZING!
More like “oh my deltoid!”
O-M-G!
O-M-D!!
By Nate Currier,
Fall 2008
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 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
Used in wheels and stuff.
Perhaps I should have said, the deltoid “in” man.
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.ht
m>
THE END!!