The Meta logo and fitting Besace curves

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

Meta logo

A Besace curve has the implicit form

(x^2 - by)^2 = a^2(x^2 - y^2)

and the parametric form

begin{align*} x &= acos(t) - b sin(t) \ y &= -sin(t) xend{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.

W#e can simplify the cosine of an artangent (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 = 2sqrt{a^2 + b^2}

and so

b = h - w/2

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.

The post The Meta logo and fitting Besace curves first appeared on John D. Cook.

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