Deltoid curve

In geometry, a deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid of three cusps. In other words, it 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. It is named after the Greek letter delta which it resembles.

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. [1]

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

where a is the radius of the rolling circle, b is the radius of the circle within which the aforementioned circle is rolling. (In the illustration above b = 3a.)

In complex coordinates this becomes

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

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

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

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