Deltoid curve

More broadly, 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.

Equations

A hypocycloid can be represented (up to rotation and translation) by the following parametric equations

x = (b - a) \cos (t) + a \cos (\frac{b - a}{a} t)

y = (b - a) \sin (t) - a \sin (\frac{b - a}{a} t),

where a is the radius of the rolling circle, b is the radius of the circle within which the aforementioned circle is rolling and t ranges from zero to 6π. (In the illustration above b = 3a tracing the deltoid.)

In complex coordinates this becomes

z = 2 a e^{i t} + a e^{- 2 i t}

The variable t can be eliminated from these equations to give the Cartesian equation

(x^{2} + y^{2})^{2} + 18 a^{2} (x^{2} + y^{2}) - 27 a^{4} = 8 a (x^{3} - 3 x y^{2}),

so the deltoid is a plane algebraic curve of degree four. In polar coordinates this becomes

r^{4} + 18 a^{2} r^{2} - 27 a^{4} = 8 a r^{3} \cos 3 θ .

The curve has three singularities, cusps corresponding to $t = 0, \pm \frac{2 π}{3}$ . The parameterization above implies that the curve is rational which implies it has genus zero.

A line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency travels around the deltoid twice while each end travels around it once.

The dual curve of the deltoid is

x^{3} - x^{2} - (3 x + 1) y^{2} = 0,

which has a double point at the origin which can be made visible for plotting by an imaginary rotation y ↦ iy, giving the curve

x^{3} - x^{2} + (3 x + 1) y^{2} = 0

with a double point at the origin of the real plane.

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Equations

Area and perimeter

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