Let
Ω
\Omega
be a planar Jordan domain. We consider double-dome-like surfaces
Σ
\Sigma
defined by graphs of functions of
dist
(
⋅
,
∂
Ω
)
\operatorname {dist}(\cdot ,\partial \Omega )
over
Ω
\Omega
. The goal is to find the right conditions on the geometry of the base
Ω
\Omega
and the growth of the height so that
Σ
\Sigma
is a quasisphere or is quasisymmetric to
S
2
\mathbb {S}^2
. An internal uniform chord-arc condition on the constant distance sets to
∂
Ω
\partial \Omega
, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in
R
n
\mathbb {R}^n
, for any
n
≥
3
n\ge 3
.