That looks like some kind [two kinds, really] of formula in which the angle to a point on the curve is based on the inverse of a trigonometric function whose value swings between positive and negative without ever going to infinity [such as sine or cosine], applied to some multiplier of the local radius, and for the browner curve with a constant angle [looks like 45 degrees] added to the result.  That shouldn't be very hard to work out -- for a related routine that draws polynomial functions [y = ax^n + bx^(n-1) + .... + px^2 + qx + r], see my PolynomialFunction.lsp routine, available here.  But this one, being radial, would not involve X and Y coordinates, but instead (polar) functions measured from the origin as basepoint, in a (repeat) function stepping through small increments of radius as the distance arguments, and calculating the angle arguments for each based on the equation, and presumably used in a SPLINE, or perhaps a POLYLINE that you could spline-curve to smooth it out.

EDIT:  That link doesn't seem to work.  Try copy/pasting this:

https://forums.autodesk.com/t5/autocad-2013-2014-2015-2016-2017/modeling-a-mathematical-shape/m-p/6886507#M163554