Our curve is closed for some value of T near 3.307. Is this a Lemniscate? Wolfram gives a parametric form for the lemniscate as:
x = (cos t)/(1 + sin2 t)
y = (cos t)(sin t)/(1 + sin2 t)

We differentiate to find the tangent and then the derivative of its direction with respect to t.
dx/dt = –2(cos t)(1 + sin2 t)–2 (sin t)(cos t) – (sin t)/(1 + sin2 t) =(-2(cos t) – (sin t) – sin3 t)(1 + sin2 t)–2
dy/dt = (cos t)(sin t)/(1 + sin2 t)

This is too hard, but numerically: 

(define (sq x)(* x x))
(define (xf t) (/ (cos t) (+ 1 (sq (sin t)))))
(define (yf t) (/ (* (cos t)(sin t)) (+ 1 (sq (sin t)))))
(define (p t) (cons (xf t)(yf t)))
(define pi (* 4 (atan 1)))
(define dt .000001)
(define ((D f) t) (/ (- (f (+ t dt)) (f t)) dt))
(define xd (D xf))
(define yd (D yf))
(define (a t) (atan (/ (yd t) (xd t))))
(define ad (D a))
(define (cm t) (list (xf t) (ad t) (/ (xf t) (ad t)) (+ (sq (xd t)) (sq (yd t)))))


(cm 0)  => (1 2.999600567932248 0.33337772058409176 0.9999999999989169)
(cm .3) => (0.8786059087328341 2.636061980920701 0.3333024470183217 0.9196819031132135)
(cm .7) => (0.540518238412472 1.6215085905657567 0.33334281517674913 0.7067053298091556)
(cm 1)  => (0.31632264754418365 0.9489159250586354 0.33335160596513064 0.5854547720970557)
(cm 1.5) => (0.03545731062300114 0.10637113134759346 0.33333584191312005 0.5012540576558454)
(cm 2)  => (-0.22779826372270964 -0.6833838654962676 0.3333386625352714 0.5473988768989227)
It is not a lemniscate. da/dt = 3x but t is not the arc-length parameter as indicated by the last list element!! The curvature is not proportional to x.
Google excerpts a fragment of coyrightext suggesting that this document thinks that the lemniscate solves the bent beam problem.