Fitting the parameters of a Besace curve like the Meta logo

I saw a post yesterday saying that the Meta logo is a Besace curve.

Meta logo

A Besace curve has the implicit form

and the parametric form

$\begin{align*} x &= a\cos(t) - b \sin(t) \\ y &= -\sin(t) x\end{align*}$

where t ranges over [0, 2π].

So given a Besace curve, such as the Meta logo, how do you find the parameters a and b to fit the curve?

We can rewrite the parametric expression for x as a sine with a phase shift (see notes here)

$x = A \sin(t + \phi)$

where

$\begin{align*} A &= \sqrt{a^2 + b^2} \\ \phi &= -\arctan(a/b)\end{align*}$

Also, we can rewrite the parametric expression for y as

$\begin{align*} y &= A \sin(t) \sin(t + \phi) \\ &= \frac{A}{2} \left(\cos(\phi) - \cos(2t + \phi)\right) \\ \end{align*}$

Now the extreme values of x and y are easier to see. The maximum value of x is A and the minimum value is −A. The maximum value of y is A(cos(φ) + 1)/2 and the minimum value is A(cos(φ) − 1)/2.

We can simplify the cosine of an arctangent (see here) to find the height, i.e. the difference between the maximum and minimum y value, in terms of a and b.

$\begin{align*} \cos(\phi) &= \cos(-\arctan(a/b)) \\ &= \frac{1}{\sqrt{1 + (a/b)^2}} \\ &= \frac{b}{\sqrt{a^2 + b^2}} \end{align*}$

Then the height is given by

$\begin{align*} h &= \frac{A}{2}(\cos(\phi) + 1) - \frac{A}{2}(\cos(\phi) - 1) \\ &= A \cos(\phi) + A \\ &= b + \sqrt{a^2 + b^2} \end{align*}$

The width is given by

$w = 2A = 2\sqrt{a^2 + b^2}$

and so

and

$a = \pm \sqrt{\frac{w^2}{4} - b^2}$

Now the Meta logo is drawn with a thick line, and the line width isn’t constant. It’s a little fuzzy what the height and width of the middle of the curve are, but I estimated h = 120 and w = 200 from one image. This leads to b = 20 and a = 97.98.

The Mathematica code

ParametricPlot[{a Cos[t] + 
   b Sin[t], -Sin[t] ( a Cos[t] + b Sin[t])}, {t, 0, 2 Pi}, 
 PlotStyle -> Thickness[0.05]]

produces the following image.

Mathematica approximation of Meta logo

This is reminiscent of the Meta logo, but not a great match. I suspect the logo is not exactly a Besace curve. You could tinker with the a and b parameters and the aspect ratio to get a closer match. The logo may have been inspired by a Besace curve and then drawn by hand.