PA170 Digital Geometry
Lecture 07: Topological Characterization of Curves and Surfaces
Martin Maˇska (xmaska@fi.muni.cz)
Centre for Biomedical Image Analysis
Faculty of Informatics, Masaryk University, Brno
Autumn 2023
Recap: Common Subfields of Topology
Point Set Topology
It considers local properties of spaces, and is closely related to analysis
It generalizes the concept of continuity to define topological spaces, in which limits of
sequences can be considered
Combinatorial Topology
It is the oldest branch of topology, which dates back to Euler
It considers the global properties of spaces, built up from a network of vertices,
edges, and faces
Algebraic Topology
It also considers the global properties of spaces, and uses algebraic objects such as
groups or rings to answer topological questions
It converts topological problems into algebraic ones that are hopefully easier to solve
Differential Topology
It considers spaces with some kind of smoothness associated to each point (e.g., a
square and a circle are not differentiably equivalent to each other)
It is useful for studying properties of vector fields, such as magnetic or electric fields
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INTRODUCTION TO
COMBINATORIAL TOPOLOGY
Motivation: Combinatorial Topology
Combinatorial topology studies the topological properties of sets represented as
complexes of small parts
The topological properties are derived from these complexes
A Euclidean complex A simplicial complex
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Common Types of Complexes
Euclidean complexes consist of convex cells
Examples:
Simplicial complexes consist of simplices
All n-simplices (0 ≤ n ≤ 3):
Simplicial complexes are special cases of Euclidean complexes
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Euclidean Complexes: Definition
Let M ⊆ En be the union of a finite number of convex cells
A Euclidean complex is a partition S of M into a nonempty finite set of convex cells
with the following two properties:
EC1: If p is a cell of S and q is a side of p, q is a cell of S
EC2: The intersection of two cells of S is either empty or a side of both cells
A finite Euclidean complex that contains only triangles, line segments, and points is
called a triangulation
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CURVES
Simple Closed Curves: Definitions
A simple closed curve γ splits the plane into two open components. One component
is bounded and the other component is unbounded, with γ being the frontier between
these components
Parametric Definition
γ is a set of points {(x, y) : φ(t) = (x, y) ∧ a ≤ t ≤ b} where φ : [a, b] → R2 is a
continuous mapping the image of which is homeomorphic to a unit circle
Implicit Definition
γ is a set of points {(x, y) : f(x, y) = 0} satisfying an equation f(x, y) = 0
Topological Definition
γ is a one-dimensional continuum (a nonempty, compact, and topologically
connected subset of a topological space) in En
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Curves and Arcs: Terminology
A simple curve is a curve in which every point p has branching index 2
A simple arc is either a curve in which every point p has branching index 2 except for
its two endpoints, which have branching index 1, or a simple curve with one of its
points labeled as an endpoint
A regular point of a curve has branching index 2 and is not an endpoint
A branch point has branching index 3 or greater
A singular point is either an endpoint or a branch point
An elementary curve is the union of a finite number of simple arcs, each pair of which
have at most a finite number of points in common
Elementary curves can be approximated by polygonal chains
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Topological Characterization of Elementary Curves
An elementary curve γ can be partitioned into a one-dimensional geometric complex
S that consists of α1 simple arcs (1-cells), α0 endpoints (0-cells), and β0 components
The Euler characteristic χ(S) of S is defined as χ(S) = α0 − α1; χ(S) is preserved
for any partition of γ
The connectivity β1(S) of S is given as β1(S) = β0 − α0 + α1; β1(S) is equal to the
number of atomic cycles of S
Both the Euler characteristic and connectivity are topological invariants
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SURFACES
Manifolds
Let [S, G] be a topological space and p ∈ S
Any subset of S that contains an open superset of p is called a topological
neighborhood of p
[S, G] is called an n-manifold if every p ∈ S has a topological neighborhood in S,
which is homeomorphic to an open n-sphere (near each point resembles En)
An n-manifold is called hole-free iff it is compact (closed and bounded)
The surfaces of a ball and of a torus are examples of hole-free 2-manifolds
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Simple Closed Surfaces: Definitions
A simple closed surface σ splits E3 into two open components. One component is
bounded and the other component is unbounded, with σ being the frontier between
these components
Parametric Definition
σ is a set of points σ = {(x, y, z) : φ(s, t) = (x, y, z) ∧ a ≤ s, t ≤ b} where
φ : [a, b] × [a, b] → R3 is a continuous mapping the image of which is homeomorphic
to a unit sphere
Implicit Definition
σ is a set of points {(x, y, z) : f(x, y, z) = 0} satisfying an equation f(x, y, z) = 0
Topological Definition
σ is a hole-free surface (a hole-free 2-manifold)
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Surfaces with Frontiers: Definition
A surface S is called a surface with frontiers iff S is homeomorphic to a polyhedral
surface and can be partitioned into two nonempty subsets S◦ and ϑS such that every
p ∈ S◦ has a topological neighborhood in S, which is homeomorphic to an open disk,
and every p ∈ ϑS has a topological neighborhood in S, which is homeomorphic to
the union of the interior of a triangle and one of its sides (without endpoints) where p
is mapped onto that side
The points of S◦ are called interior points of S, and the points of ϑS are called frontier
points of S
The number of frontiers is a topological invariant
A surface with three frontiers A surface with one frontier
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Surfaces with Frontiers: Definition
A surface S is called a surface with frontiers iff S is homeomorphic to a polyhedral
surface and can be partitioned into two nonempty subsets S◦ and ϑS such that every
p ∈ S◦ has a topological neighborhood in S, which is homeomorphic to an open disk,
and every p ∈ ϑS has a topological neighborhood in S, which is homeomorphic to
the union of the interior of a triangle and one of its sides (without endpoints) where p
is mapped onto that side
The points of S◦ are called interior points of S, and the points of ϑS are called frontier
points of S
The number of frontiers is a topological invariant
A surface with three frontiers A surface with one frontier
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Topological Characterization of 2D Euclidean Complexes
Let S be a 2D Euclidean complex that consists of αi i-cells (0 ≤ i ≤ 2)
The Euler characteristic of S is defined as χ(S) = α0 − α1 + α2
If S is a simple polyhedron, χ(S) = 2 (Descartes & Euler)
In 1812, Lhuilier incorrectly derived the following formula:
α0 − α1 + α2 = 2(c − t + 1) + p
where c is the number of cavities, t is the number of tunnels, and p is the number of
polygons (“tunnel exits”) on the polyhedron faces
Tunnels and cavities What are tunnels?
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Betti Numbers
Betti numbers βi (0 ≤ i ≤ n) are topological invariants, which extend the polyhedral
formula to n-dimensional spaces (the Poincar´e formula):
χ(·) =
n
i=0
(−1)i
· βi
Informally, β0 is the number of connected components, β1 is the number of tunnels,
and β0 + β2 is the number of closed surfaces (so that there are β2 cavities)
Formally, βi is the rank of the i-th homology group of the particular topological space,
and can be algorithmically calculated
β0 = 1, β1 = 1059, β2 = 0
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The Orientability of Surfaces
An oriented triangle is a triangle with a direction on its frontier (e.g., clockwise or
counterclockwise), which is called the orientation of that triangle
Two triangles are called coherently oriented if they induce opposite orientations on
their common side
A triangulation of a surface is called orientable iff it is possible to orient all of the
triangles in such a way that every pair of triangles with a common side is coherently
oriented; otherwise it is called nonorientable
All triangulations of the same surface are either orientable or nonorientable
A surface is called orientable iff it has an orientable triangulation
The orientability of a surface is a topological invariant
A M¨obius strip
A starting configuration The configuration after one loop
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The Genus of Orientable Surfaces
Let S be an orientable surface. The genus g(S) of S is the number of handles of S
It can be shown that χ(S) = 2 − 2g(S)
The genus is a topological invariant
Genus 0 Genus 1 Genus 2 Genus 3
Source: MathWorld, Wikipedia
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Take-Home Messages
A simple closed curve is defined as a one-dimensional continuum
The topology of elementary curves can be characterized by the Euler characteristic
and connectivity
A simple closed surface is defined as a hole-free 2-manifold
The topology of surfaces is uniquely determined by the number of frontiers,
orientability, and Euler characteristic
Betti numbers can correctly describe tunnels and cavities in surfaces
A M¨obius strip is a popular example of a nonorientable surface
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