A Lemniscate?

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 take p = (x, y) to be the vector position on the curve as a function of t or λ, the arc length. The curvature is the magnitude of d²p/dλ² = (d²p/dt²)/(dλ/dt)². (dλ/dt)² = (dx/dt)² + (dy/dt)².

We differentiate to find the tangent and then the derivative of its direction with respect to t.
Where r = 1/(1 + sin² t), dr/dt = ((2cos² t)(1 + sin² t)⁻²)

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 text suggesting that this document thinks that the lemniscate solves the bent beam problem. But Wikipedia says that the curvature of the lemniscate is proportional to the distance from the origin, not the x-axis as we have for this curve.