Quartic Curves > Deltoid Curve

A **deltoid curve**, also called *Steiner’s hypocycloid* or *tricuspoid curve*, is a three-cusped Hypocycloid. *Hypocycloids* are produced by tracing a fixed point on a small circle which rolls within a larger circle; The small circle rolls like a wheel inside the circumference of the bigger circle. The deltoid is produced when the radii of the circles has a ratio of 3:1. In other words, it is produced when a small circle of radius 1 rotates within a circle of radius 3. One other ratio gives a deltoid curve: 3:2.

More broadly, a deltoid can be any closed figure with three vertices, connected by curves that are concave to the exterior of the outer circle (which makes the interior points a non-convex set) [2].

The deltoid curve was first studied by the Swiss mathematicians Leonhard Euler (who discovered the curve while studying optics) and Jakob Steiner, who studied the curve in greater depth; Steiner identified it as the envelope of a family of the Simson lines of a triangle.

It’s thought that it’s called a “deltoid” because it resembled the Greek letter Delta Δ.

## Deltoid Curve Equations and Properties

The parametric equations for the deltoid curve are [1]:

- x =
*a*2cos(θ) +*a*cos(2θ) - y =
*a*2sin(θ) –*a*sin(2θ) - θ ≤ θ ≤ 2π

Where *a *is the rolling circle’s radius.

In complex coordinates this becomes z = 2ae^{it} + ae^{-2it} [2].

Eliminating t from the above equations gives the Cartesian equation:

- (x
^{2}+ y^{2})^{2}– 8x(x^{2}– 3y^{2}) + 18(x^{2}+ y^{2}) – 27 = 0.

The length of a deltoid curve is L = 16*a* and the area is A = 2π*a*^{2}. The area is exactly half of the unit circle [3].

The curve has a domain of -3/2 *a* ≤ x ≤ 3 *a* and a range of

The deltoid curve has three singularities, one at each cusp. These can be found at t = 0, ± 2/3.

## Properties: Tangent

The deltoid curve has some interesting properties [4].

Let:

- A = the center of the curve
- B = a cusp
- P = a point on the curve
- E, H = intersection of the curve and tangent at P.

- The line segment EH has constant length [E,H] = 4/3 * [A,B].
- The locus of midpoint D of the tangent segment EH is the inscribed circle.
- E,P,H are concurrent normals; the locus of these intersections is also the inscribed circle.
- If J is the intersection of another tangent that cuts the line segment EH at right angles, the locus traced by J is the inscribed circle.

## References

Deltoid curve illustration: Sam Derbyshire at the English Wikipedia,

[1] Sexton, E. (1999). The Deltoid Curve. Retrieved January 13, 2021 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/Sp99/Ed/Deltoid2.pdf

[2] Dayanithi, D. Combination of Cubic and Quartic Plane Curve. IOSR Journal of Mathematics (IOSR-JM)e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. – Apr. 2013), PP 43-53

[3] Gardner, M. (2020). The Unexpected Hanging and Other Mathematical Diversions

A Classic Collection of Puzzles and Games from Scientific American. American Mathematical Society.

[4] “Deltoid”. Retrieved January 13, 2021 from: http://virtualmathmuseum.org/docs/Deltoid.pdf

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