Normal Curves and Compression
The only normal people are the ones you don’t know very well.
— Alfred Adler
I have captured the signal, and am presently triangulating the vectors,
and compressing the data down, in order to express it as a function of my hand.
[Points.]
They’re over therrrrrrrrre!
— Prof. John Frink, “Wild Barts Can’t be Broken”,
The Simpsons
(1999)
13 Normal Curves and Compression
In the late 1920s, Helmuth Kneser
[
3
]
introduced the theory of
normal surfaces
to prove a certain
decomposition theorem about three-dimensional manifolds. Normal surfaces were further developed
by Wolfgang Haken
[
2
]
a few decades later to answer certain algorithmic questions about 3-manifolds.
Since Haken’s work, dozens of reﬁnements, extensions, and applications of normal surface theory have
been developed; a comprehensive survey is beyond the scope of this course (and the expertise of the
instructor!). In this lecture, I will describe some algorithmic results related to
normal curves
, which
are just like normal surfaces, only one dimension lower; I will return to (one of) Haken’s algorithmic
applications in the next lecture.
13.1 Normal Curves and Normal Coordinates
Fix a 2-manifold
M
, possibly with boundary. Let
T
be a
triangulation
of
M
: a cellularly embedded
graph in which every face has three sides. Tn particular, each boundary cycle of
M
is covered by a cycle
in
T
. A
simple cycle
in
M
is
the image
of a continuous injective map from
S
1
to
M
. A
simple arc
in
M
is the image of a continuous injective map
α
:
[
0,1
]
→
M
whose endpoints
α
(
0
)
and
α
(
1
)
lie on the
boundary of
M
. A
curve
is the union of a ﬁnite number of pairwise-disjoint simple cycles and arcs.
Two curves
γ
and
δ
are
isotopic
(relative to
∂
M
) if
γ
can be continuously deformed into
δ
without
moving any point on the boundary of
M
or introducing any intersections.
1
A curve is
normal
with
respect to
T
if every intersection with an edge of
T
is transverse, and the intersection of the curve with
any triangle is a ﬁnite set of
elementary segments
: simple paths whose endpoints lie on distinct sides of
the triangle. The endpoints of the elementary segments partition the edges of
T
into
ports
.
Nine elementary segments in a triangle.
Every triangle in
T
can contain three different types of elementary segments, each ‘cutting off’ one of
corners of the triangle. For any corner
x
of any triangle
A
, let
γ
(
A
,
x
)
denote the number of elementary
segments in
γ
∩
A
that separate
x
from the other two corners of
A
. Suppose
T
has
t
triangles. The vector
of 3
t
non-negative integers
γ
(
A
,
x
)
are the
normal coordinates
of
γ
. We denote the normal coordinate
vector of any normal curve
γ
by
⟨
γ
⟩
. Not every vector in
N
3
t
is a normal coordinate vector of a curve; for
1
More formally, an isotopy from
γ
to
δ
is a continuous function of the form
h
:
[
0,1
]
×
γ
→
M
, such that (1)
h
(
0,
γ
) =
γ