Such a transformation can be written down fairly straightforwardly, except that the requirement to keep the length constant introduces a somewhat implicit element: $$ x=\cos \theta \ , \ y=\sin \theta \cos t \ , \ z=a(t) \sin \theta \cos \theta \sin t $$ where $\theta \in [0,2\pi ]$, and $t\in [0,\pi /2]$ is the deformation parameter; at $t=0$, one has the circle, at $t=\pi /2$, one has the lemniscate. The function $a(t)$ varies monotonically between $a(0)=1$ and $a(\pi/2)=1.061395$; it is defined by the requirement of constant length at any given $t$, $$ 2\pi =\int_{0}^{2\pi } d\theta \sqrt{1+\sin^2 t(a^2 \cos^2 2\theta - \cos^2 \theta )} $$ and can be easily determined numerically.

Such a transformation can be written down fairly straightforwardly, except that the requirement to keep the length constant introduces a somewhat implicit element: $$ x=\cos \theta \ , \ y=\sin \theta \cos t \ , \ z=a(t) \sin \theta \cos \theta \sin t $$ where $\theta \in [0,2\pi ]$, and $t\in [0,\pi /2]$ is the deformation parameter; at $t=0$, one has the circle, at $t=\pi /2$, one has the lemniscate. The function $a(t)$ varies monotonically between $a(0)=1$ and $a(\pi/2)=1.061395$; it is defined by the requirement of constant length at any given $t$, $$ 2\pi =\int_{0}^{2\pi } d\theta \sqrt{1+(a^2 \cos^2 2\theta - \cos^2 \theta ) \sin^{2} t} $$ and can be easily determined numerically.