Becoming the parameters of a Besace curve just like the Meta emblem

I noticed a put up yesterday saying that the Meta emblem is a Besace curve.

A Besace curve has the implicit type

and the parametric type

the place t ranges over [0, 2π].

So given a Besace curve, such because the Meta emblem, how do you discover the parameters a and b to suit the curve?

We are able to rewrite the parametric expression for x as a sine with a section shift (see notes right here)

the place

Additionally, we are able to rewrite the parametric expression for y as

Now the intense values of x and y are simpler to see. The utmost worth of x is A and the minimal worth is −A. The utmost worth of y is A(cos(φ) + 1)/2 and the minimal worth is A(cos(φ) − 1)/2.

We are able to simplify the cosine of an artangent (see right here) to seek out the peak, i.e. the distinction between the utmost and minimal y worth, by way of a and b.

Then the peak is given by

The width is given by

and so

and

Now the Meta emblem is drawn with a thick line, and the road width isn’t fixed. It’s a bit of fuzzy what the peak and width of the center of the curve are, however I estimated h = 120 and w = 200 from one picture. This results in 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 next picture.

That is harking back to the Meta emblem, however not an important match. I think the brand shouldn’t be precisely a Besace curve. You can tinker with the a and b parameters and the side ratio to get a better match. The emblem could have been impressed by a Besace curve after which drawn by hand.