this plot what created using a mathematical formula ,demostrated in mathlab.

thanks

 

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 http://cadtips.cadalyst.com/lisp-code-modules/draw-polynomial-functions" rel="nofollow - 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.

 

; for range (0-1.5>
(defun fPolar1 (u / r tau alfa R0)
;x = (* r (cos th))
;y = (* r (sin th))
 (setq r u)
 (setq tau 1.1) ; example
 (setq alfa 1.0) ; example
 (setq R0 0.1) ; example
 (setq th (* alfa (sin (/ (* pi (log (/ r R0))) (log tau))))) ; your expression
 (list (* r (cos th)) (* r (sin th))) ; return (X Y)
)

Then call:

 

(2Dplot fPolar1 0.001 1.5 0.001)

 

meaning - draw my function "fPolar1" from 0.001 to 1.5 (the radius), with ministeps of 0.001

 

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