Home

Gegenflächen

Gegenflächen are a concept in geometry, primarily found in German-language texts, describing surfaces related to a given surface by a reversal of orientation or by a mirror-like transformation. The term literally means “opposite surfaces” and is used to distinguish these surfaces from the original orientation or embedding.

Definition and construction

For a smooth oriented surface S in three-dimensional space with a local parametrization X(u,v) and unit normal

Properties

The Gegenfläche shares the first fundamental form with the original surface, so distances and angles are preserved

Relation to boundaries and topology

If S has a boundary ∂S, the Gegenfläche often inherits the same boundary curve with opposite orientation,

See also

Fläche (surface), Orientierung (orientation), Gauss map, Minimal surfaces, Mirror symmetry.

N(u,v),
the
Gegenfläche
of
S
can
be
described
as
the
same
geometric
set
of
points
S
equipped
with
the
opposite
orientation,
i.e.,
using
the
normal
−N.
In
other
contexts,
Gegenflächen
may
refer
to
the
mirror
image
of
S
with
respect
to
a
plane
or
symmetry,
yielding
a
surface
with
the
same
boundary
and
the
same
intrinsic
metric
but
a
reversed
orientation.
locally.
The
second
fundamental
form
changes
sign
due
to
the
reversed
normal,
which
flips
the
orientation
of
curvature
measures.
Consequently,
Gaussian
curvature
K
remains
the
same
in
magnitude,
while
oriented
curvature
quantities
such
as
mean
curvature
H
change
sign
under
reversal
of
orientation.
consistent
with
the
reversal
of
orientation.
As
with
the
original
surface,
Gegenflächen
can
be
compact
or
non-compact,
and
they
may
play
a
role
in
discussions
of
duality
and
orientation-sensitive
theorems.