The **bean curve**, named because it looks like a bean, is a quartic plane curve situated in the 1st and 4th quadrants with the origin at one end of the bean. It is defined by the implicit equation: x^{4} + x^{2}y^{2} + y^{4} = x(x

^{2} + y^{2}) [1].

There is one horizontal asymptote, at (2/3 a ± 2/3a) and two vertical asymptotes at (0, 0) and (a, 0).

## Area of the Bean Curve

The area of the bean curve is given by the integral [2]:

## Alternative Bean Curve

The curve defined above is the simplest type of bean-shaped curve. However, there is an alternative bean, described by the Cartesian equation (x^{2} + y^{2})^{2} = x^{3} + y^{3}, shown below.

This alternate curve can be described by the polar equation

r = a(sin^{3}θ + cos^{3}θ).

## Bean Curve History

The origins of the bean curve are somewhat of a mystery. One early mention of the curve was in Cundy and Rollett’s 1961 text Mathematical Models [3], where it had a one line entry as an “interesting quartic” with just the label “bean” and its Cartesian equation (p.72).

The first curve listed above is sometimes called the *egg curve*, presumably to differentiate it from the alternate bean-shaped curve. However, this can get a bit confusing as the alternate curve is sometimes called the “crooked egg curve” (even though it clearly looks like a lima bean!). None of this really matters, because the use of both curves seems to be mostly an academic pursuit, with very few (if any) real life applications.

## References

Bean curve graphed with Desmos.

[1] Knill, O. & Chi, A. (2003). Entry Curves. Retrieved December 31, 2021 from: https://abel.math.harvard.edu/~knill/sofia/data/curves.pdf

[2] Vaze, V. et al. (2005). Continuous space representations of human activity spaces. Arbeitsberichte Verkehrs- und Raumplanung.

[3] Cundy, H. & Rollett, A. (1951). Mathematical Models. Oxford University Press.