An **astroid curve** (also called a tetracuspid, cubocycloid, or paracycle) is a special case of a hypocycloid with four cusps.

The Cartesian equation is [1]

The parametric equation is

*Hypocycloids *are produced by tracing a fixed point on a small circle rolling inside the circumference of the larger circle. Cusps are meeting points for two branches of the curve with sharp corners and equal tangent lines.

## Equation for the Tangent Line

We can use implicit differentiation to find an equation of the tangent lineat a given point [2].

For example, to find the equation at (-3√3, 1):

Substituting in (-3√3, 1) gives

Finally, use the point-slope form of the equation for a line to get the solution

## History of the Astroid Curve

The discovery of the curve is attributed to Roemer in 1674 [3], although it was formally named the astroid curve centuries later — in 1836. During that time the curve took on many names including the “four cusp curve.” The modern word asteroid (as in a small rocky body orbiting the sun) is derived from the name [4]. Sometimes the two words are confused, with some authors referring to the curve as the *asteroid* instead of the *astroid*. The curve has an important application in magnetism, where it is called the *Stoner-Wohlfarth astroid*.

## References

Astroid graph created with Desmos.

[1] Tan, S. (2020). Handbook of Famous Plane Curves Using Mathematica.

[2] Stewart, J. Calculus, Early Transcendentals, p. 215, #30. Solution: https://sccollege.edu/Departments/MATH/Documents/Math%20180/03-05-030_Implicit_Differentiation.pdf

[3] O’Hanen, B. & Wisan, M. (2006). The Asteroid: Special Plane Curves/

Retrieved March 5, 2022 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/sp06/mattben/RealWebsite/AsteroidPaper.html

[4] Silva, L. & Yokoyama, B. Final Project – The Astroid. Retrieved March 5, 2021 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/sp07/breannelydia/Final.htm