Deltoid

Deltoid DELTOID

Balmoral Software

The deltoid is a three-cusped hypocycloid S with parametric equations

x(t) = 2cos(t) + cos(2t)
y(t) = 2sin(t) - sin(2t), 0 ≤ t < 2π

The curve is traversed in a counterclockwise direction, starting at the right cusp (3,0) at t = 0, reaching the next cusp at t = 2π/3 and then continuing symmetrically around the origin. Since S has two minimum-abscissa points

, it is non-convex by the multiple local extrema test. The width x height of its bounding rectangle is

Symmetry

Since

x(t + 2π/3) = cos(2π/3)x(t) - sin(2π/3)y(t)

and

y(t + 2π/3) = sin(2π/3)x(t) + cos(2π/3)y(t),

the conditions (R1) of Lemma R are satisfied and the deltoid is rotationally symmetric with period 2π/3.

Metrics

The perimeter of the deltoid is 16 and its area is 2π.

Boundary Circles & Circumellipse

Since the deltoid is rotationally symmetric with period 2π/3, its circumcircle and circumellipse are the same. The squared-distance function of S

x²(t) + y²(t) = 4cos(3t) + 5

ranges from 1 to 9, so its inradius is 1 and its circumradius is 3.

Inellipse

Using z = -1 in Lemma E,

d/dt [x(t) - z]y(t) = d/dt 4sin³(t)cos(t)

has a zero at t = ±π/3. The corresponding points on S are

Together with (z,0) = (-1,0), these points are all at a unit distance from the origin, so an axis-aligned ellipse tangent to S at (-1,0) is a unit circle by Lemma S. Therefore, the inellipse is the same as the incircle.

Convex Hull

The convex hull is an equilateral triangle created by connecting the maximum-radius points (3,0) and

by line segments of length

, as shown in blue in the left diagram below. The perimeter of the convex hull is

, which is about 3% shorter than that of the deltoid. The area of the convex hull is half that of its bounding rectangle:

and is about 86% more than the area of the deltoid.

Summary Table

Figure Parameters Perimeter Area Centroid

Incircle R = 1 6.283185 3.141593

Deltoid 16 6.283185

Convex hull 15.588457 11.691343

Circumcircle R = 3 18.849556 28.274334

Figure	Parameters	Perimeter	Area	Centroid
Incircle	R = 1	6.283185	3.141593
Deltoid		16	6.283185
Convex hull		15.588457	11.691343
Circumcircle	R = 3	18.849556	28.274334

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