91 research outputs found
Hyperbolic Lambert Quadrilaterals and Quasiconformal Mappings
We prove sharp bounds for the product and the sum of two hyperbolic distances
between the opposite sides of hyperbolic Lambert quadrilaterals in the unit
disk. Furthermore, we study the images of Lambert quadrilaterals under
quasiconformal mappings from the unit disk onto itself and obtain sharp results
in this case, too.Comment: 21 pages, 7 figure
Some remarks on the visual angle metric
We show that the visual angle metric and the triangular ratio metric are
comparable in convex domains. We also find the extremal points for the visual
angle metric in the half space and in the ball by use of a construction based
on hyperbolic geometry. Furthermore, we study distortion properties of
quasiconformal maps with respect to the triangular ratio metric and the visual
angle metric.Comment: 13 pages, 5 picture
On Quasi-inversions
Given a bounded domain strictly starlike with
respect to we define a quasi-inversion w.r.t. the boundary
We show that the quasi-inversion is bi-Lipschitz w.r.t. the
chordal metric if and only if every "tangent line" of is far away
from the origin. Moreover, the bi-Lipschitz constant tends to when
approaches the unit sphere in a suitable way. For the formulation
of our results we use the concept of the -tangent condition due to F.
W. Gehring and J. V\"ais\"al\"a (Acta Math. 1965). This condition is shown to
be equivalent to the bi-Lipschitz and quasiconformal extension property of what
we call the polar parametrization of . In addition, we show that
the polar parametrization, which is a mapping of the unit sphere onto is bi-Lipschitz if and only if satisfies the -tangent
condition.Comment: 22 pages; 5 figure
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