Deltoid

DELTOID

Curve studied by Euler in 1745 and Steiner in 1856.
Delta: Greek letter reminder of the shape of the curve.
Other names: three-cusped hypocycloid, H₃, Steiner hypocycloid, or tricuspid curve.

Complex parametrization:

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Cartesian parametrization:

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Cartesian equation:

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Rational circular quartic.
Polar parametrization:

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Curvilinear abscissa: 1)

or 2)

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Cartesian tangential angle: 1)

or 2)

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Radius of curvature: 1)

or 2)

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Intrinsic equation 1 (1st form):

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Intrinsic equation 2 (1st form)):

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Polar equation:

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Length:

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Area:

(double of that of the inscribed circle).

The deltoid is the hypocycloid with three cusps (circle with radius a rolling inside a circle with radius 3a).

According to the double generation of hypocycloids, a point on a circle with radius 2a rolling inside a circle with radius 3a describes an isometric deltoid, but in the opposite directions and one of its diameters also envelopes an isometric deltoid.
Slowed down animation of the double generation

The deltoid is also the envelope of a chord (PQ) of the circle with centre O and radius a (circle inscribed in the deltoid), when P and Q describe this circle in opposite directions, one going twice as fast as the other one (generation of Cremona).
Above, the point n is linked to the point -2n modulo 30.

But the most elegant tangential generation is the one described opposite: the two tracing points P and Q describe the deltoid, and the line (PQ) stays tangent to the same deltoid!

Furthermore, the fact that the length PQ remains constant gives a (partial) answer to the problem of Kakeya: how to turn around a needle (of length 1) in the plane in such a way that it span an area as small as possible?
Here, the area spanned equals , and it was thought for a long that that this area was the smallest possible one. But in 1928, Besicovitch proved that the needle could be turned around while spanning an area as small as wanted. Nonetheless, in his method, the movement of the ends of the needle is realised by successive approximations, and cannot be made analytical like the deltoidian motion.
Remark: watch out! the needle slips on the deltoid during its movement!

Another mechanical construction of the tangential generation, dual to the previous one, stemming from the three-cusped epicycloid .

The envelope of the Simson lines (that pass by the projections on the three sides of a point on the circumscribed circle) of any triangle is a deltoid centred on the nine-point circle of the triangle, called the Steiner hypocycloid of the triangle.
See details on Wikipedia.

Like for any cycloidal curve, the evolute of the deltoid is a similar deltoid (with ratio 3):

Therefore, one of the involutes is a deltoid; the others are auto-parallel:

The pedals of the deltoid are the folia, and the negative pedal of an ellipse with respect to its principal summit gives a dilated deltoid (see Talbot curve).
Its caustic by reflection at infinity are the astroids.
Its orthoptic is its inscribed circle.
Its radial is a regular trifolium.

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But the most elegant tangential generation is the one described opposite: the two tracing points P and Q describe the deltoid, and the line (PQ) stays tangent to the same deltoid!
Furthermore, the fact that the length PQ remains constant gives a (partial) answer to the problem of Kakeya: how to turn around a needle (of length 1) in the plane in such a way that it span an area as small as possible? Here, the area spanned equals , and it was thought for a long that that this area was the smallest possible one. But in 1928, Besicovitch proved that the needle could be turned around while spanning an area as small as wanted. Nonetheless, in his method, the movement of the ends of the needle is realised by successive approximations, and cannot be made analytical like the deltoidian motion. Remark: watch out! the needle slips on the deltoid during its movement!
Another mechanical construction of the tangential generation, dual to the previous one, stemming from the three-cusped epicycloid .
The envelope of the Simson lines (that pass by the projections on the three sides of a point on the circumscribed circle) of any triangle is a deltoid centred on the nine-point circle of the triangle, called the Steiner hypocycloid of the triangle. See details on Wikipedia.

Like for any cycloidal curve, the evolute of the deltoid is a similar deltoid (with ratio 3):
Therefore, one of the involutes is a deltoid; the others are auto-parallel: