 
Summary: A CHARACTERISATION OF PLANE QUASICONFORMAL MAPS USING
TRIANGLES
JAVIER ARAMAYONA AND PETER HA¨ISSINSKY
Abstract. We show that an injective continuous map between planar regions which distorts vertices
of equilateral triangles by a small amount is quasiconformal.
Accepted version
Quasiconformal maps have become an important class of homeomorphisms, for they arise in many
fields of mathematics, such as pde's, Teichm¨uller theory, hyperbolic geometry, complex dynamics
etc. Their involvement may be explained from the numerous characterisations of quasiconformality
involving different flavours, which generally amount to loosening characterisations of conformal maps.
Let C be a domain in the plane, and let us first define (z) = dist(z, C \ ). Let f : C
be an injective continuous map. For z and r (0, (z)), one may consider
Lf (z, r) = sup{f(z)  f(w), z  w = r} , and
f (z, r) = inf{f(z)  f(w), z  w = r} .
Let us set Hf (z, r) = Lf (z, r)/ f (z, r) and
Hf (z) = lim sup
r0
Hf (z, r) [1, ] .
The metric definition of F.W. Gehring asserts that f is Kquasiconformal if Hf is finite everywhere,
and if Hf K a.e. [3].
