The Meta logo and fitting Besace curves

I saw a post yesterday saying that the Meta logo is a Besace curve. A Besace curve has the implicit form and the parametric form 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) where Also, we can rewrite the parametric expression for y as 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. Then the height is given by The width is given by and so and 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…