Deltoid curve
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The red curve is a deltoid.
In geometry, a deltoid is a hypocycloid of three cusps.
A deltoid can be represented by the following parametric equations
The deltoid satisfies the cartesian equation
- (x2 + y2)2 + 18(x2 + y2) = 8x3 − 24y2x + 27
and is therefore a plane algebraic curve of degree four. It has three singularities, its three cusps, and is a curve of 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
- x3 − x2 − (3x + 1)y2 = 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
- x3 − x2 + (3x + 1)y2 = 0
with a double point at the origin of the real plane.
