The deltoid is a
three-cusped hypocycloid S with parametric equations
x(t) = 2cos(t) + cos(2t)
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
y(t) = 2sin(t) - sin(2t), 0 ≤ t < 2π
x(t + 2π/3) = cos(2π/3)x(t) - sin(2π/3)y(t)
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.
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.
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
(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.
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
and is about 86% more than the area of the deltoid.
|Incircle||R = 1||6.283185||3.141593|
|Circumcircle||R = 3||18.849556||28.274334|
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