The deltoid is a geometric curve traced by a point on a smaller circle rolling inside a larger circle with either three times or one and a half times the radius of the smaller circle. It can be defined parametrically by equations involving the radii of the circles. The deltoid has no single discoverer but was first investigated by Euler in 1754 and studied in depth by Steiner in 1856, and it arises in fields like complex eigenvalues and applications like the Kakeya needle problem.

1. DELTOID CURVE

2. DEFINATION
• In geometry, a deltoid, is the roulette created by a
point on the circumference of a circle as it rolls
without slipping along the inside of a circle with
three or one-and-a-half times its radius. a deltoid
can refer to any closed figure with three vertices
connected by curves that are concave to the
exterior, making the interior points a non-convex
set.

3. 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

4. •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.
The Hypocycloid

5. 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:

6. 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.

7. Deltoids arise in several fields of mathematics. For instance:
•The set of complex eigenvalues of unistochastic matrices of order three
forms a deltoid.
•A cross-section of the set of unistochastic matrices of order three forms a
deltoid.
•The set of possible traces of unitary matrices belonging to
the group SU(3) forms a deltoid.
•The intersection of two deltoids parametrizes a family of complex
Hadamard matrices of order six.
•The set of all Simson lines of given triangle, form an envelope in the
shape of a deltoid. This is known as the Steiner deltoid or Steiner's
hypocycloid after Jakob Steiner who described the shape and symmetry of
the curve in 1856.[3]
•The envelope of the area bisectors of a triangle is a deltoid (in the broader
sense defined above) with vertices at the midpoints of the medians. The
sides of the deltoid are arcs of hyperbolas that are asymptotic to the
triangle's sides.[4] [1]
•A deltoid was proposed as a solution to the Kakeya needle problem.
Application’s

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

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