Here are three curves that have interesting names and interesting shapes.

## The fish curve

The fish curve has parameterization

*x*(*t*) = cos(*t*) – sin²(*t*)/√2

*y*(*t*) = cos(*t*) sin(*t*)

We can plot this curve in Mathematica with

ParametricPlot[ {Cos[t] - Sin[t]^2/Sqrt[2], Cos[t] Sin[t]}, {t, 0, 2 Pi}]

to get the following.

It’s interesting that the image has two sharp cusps: the parameterization consists of two smooth functions, but the resulting image is not smooth. That’s because the velocity of the curve is zero at 3π/4 and 7π/4.

I found the fish curve by poking around Mathematica’s `PlainCurveData`

function. I found the next two in the book Mathematical Models by H. Martyn Cundy and A. P. Rollett.

## The electric motor curve

Next is the “electric motor” curve. Cundy and Rollett’s book doesn’t provide illustrations for the plane curves it lists on pages 71 and 72, and I plotted the electric motor curve because I was curious why it was called that.

The curve has equation

*y*²(*y*² – 96) = *x*² (*x*² – 100)

and we can plot it in Mathematica with

ContourPlot[ y^2 (y^2 - 96) == x^2 (x^2 - 100), {x, -20, 20}, {y, -20, 20}

which gives the following.

Sure enough, it looks like the inside of an electric motor!

## The ampersand curve

I was also curious about the “ampersand curve” because of its name. The curve has equation

(*y*² – *x*²) (*x* – 1) (2*x* – 3) = 4 (*x*² + *y*² – 2*x*)²

and can be plotted in Mathematica with

ContourPlot[(y^2 - x^2) (x - 1) (2 x - 3) == 4 (x^2 + y^2 - 2 x)^2, {x, -0.5, 2}, {y, -1.5, 1.5}, PlotPoints -> 100]

The produces the following pleasant image.

Note that the code specifies `PlotPoints`

. Without this argument, the default plotting resolution produces a misleading plot, making it appear that the curve has three components.