Implicit anisotropic similarity coordinates

Fix $0<d<1/2$ . For $\tau>0$ , $z\in\mathbb R$ , define $q$ as the unique root with $q>|z|^{1/d}$ of

q-z^2q^{1-2d}=\tau.

For $r\ge0$ , set $\eta=zq^{-d}$ , $X=r^2/(2q)$ , and $L=1-(1-2d)\eta^2$ . Then

\tau=q(1-\eta^2),\quad |\eta|<1,\quad L\ge2d, \qquad q\asymp\tau+|z|^{1/d}.

The constants of comparability depend only on $d$ .

Existence, uniqueness, and smoothness

The left side is zero at $q=|z|^{1/d}$ , tends to infinity, and has derivative $1-(1-2d)z^2q^{-2d}\ge2d$ on that branch. For $z=0$ the solution is $q=\tau$ . The implicit function theorem gives smooth dependence wherever $q>0$ .

The lower bound follows from $q\ge\max(\tau,|z|^{1/d})$ . If $q\le2\tau$ , the upper bound is immediate. Otherwise $q/2\le z^2q^{1-2d}$ , so $q\le2^{1/(2d)}|z|^{1/d}$ . At $\tau=0,z\ne0$ , the scale extends smoothly locally with $q=|z|^{1/d}$ and $L=2d$ ; the corner $(\tau,z)=(0,0)$ is excluded.