Download 1. Topology

Shape Modeling
Lecture 12
Basics of Topology
1. Topology
Topology is a branch of mathematics that deals with connectivity among geometric
elements.
Topology derives from concept of topological space, which is based on the theory of
“neighborhoods”.
(a)
(b)
(c)
(d)
Figure 1. Homeomorphic figures. (1D)
In Figure 1 the figures shown in (a), (b), (c), (d) are, within each set, topologically
equivalent, or homeomorphic, to one another. If one figure can be stretch to make another,
then one can consider them to be homeomorphic.
Figure 2. Homeomorphic figures (2D)
In Figure 2 the figures are homeomorphic within each set, but not homeomorphic between
sets.
“Homeomorphism” definition. X and Y are assumed to be sets.
Definition 1. (Surjection)
Given the mapping f: X Y , if f(X) = Y holds, that is, for an arbitrary point y in Y, there
exist a point x in X such that f(x) = y, then f is called a surjection or onto-mapping.
Definition 2. (Injection)
Given the mapping f: X Y , if x x holds that f(x)  f(x), then f is called an in injection
or one-to-one mapping.
Definition 3. (Bijection)
If the mapping f: X Y is, at the same time, a surjection and an injection, it is called a
bijection.
Definition 4. (Homeomorphism)
Let X and Y be subsets of Euclidian space En. If mapping f: X Y is a a bijection, and if f
and its inverse mapping g: YX are both continuous, then f is called homeomorphism.
Definition 5. (Homeomorphic)
Let X and Y be subsets of Euclidian space En. If there exists between X and Y
a
homeomorphism f: X Y, then X and Y are homeomorphic.
2. Euler Formulas
In 18 century mathematician, Leonard Euler, discovered a formula showing the relationship
among the numbers of vertices, edges (line segments) and faces of polyhedra. This formula
was later further generalized by H. Poincare.
2.1 Figures of Points and Line Segments
A geometric figure consisting of points and line segments is called a graph. The set of
points one passes when setting out from some point on the graph and arriving at another
point is called a route. A close route is called cycle.
Figures with No Cycle
Figures which has no cycle and consists of connected series of points and line segments is
called a tree graph. The conditions of a tree graph are that:
1. it must be connected; and
2. it must contain no cycle.
Figure 3. A Tree graph
Figure 3 shows a tree graph. A point from which only a single line segment originates is called a
terminal point. The tree graph has the following properties.
Theorem (Tree graph)
1. There always exists one and only one route connecting two arbitrary points in a tree graph.
2. If any line segment other than a lower end line segment is removed from the tree graph, the
tree graph will be divided into two.
To construct a tree graph, one must initially set down a point. A tree graph can be produced by
repeating the procedure of connecting a line segment to a pre-existing arbitrary point, then attaching
to other end of the line segment a point.
In a tree graph the number of points is large than of line segments by one. Denoting by v and e the
number of points and line segments, respectively, this is expressed as
v-e= 1
This is called Euler formula for tree graphs. v – e is called Euler characteristic. It means “that the
Euler characteristic of tree graph is one”.
A figure consisting of two or more tree graphs is called a forest. The Euler formula for k tree graph
is given by:
v–e=k
Figure 4. A forest.
An Italian mathematician E. Betti proposed a method of classifying geometric figures using what is
known Betti numbers.
The 0-Betti number is the number of parts into which a given figure is divided, i.e it is the number of
connected components. The 0-Betti number for forest Figure 4 is 3. Thus the 0-Betti number is
equal to Euler characteristic of a forest. We denote the 0-Betti number by 0.
Figures with Cycles
We can begin by making a cycle in a tree graph. As Figure 5 shows, this is done by connecting two
points with a line segment. Thus, the Euler characteristic of a graph containing a single cycle is:
v – e = 0.
Because the Euler characteristic is reduced by one for each cycle formed, for a graph with c cycles,
it is:
v – e = 1- c.
This formula was that of a single graph with c cycles. The Euler formula for a forest which has the
0-Betti number 0 and c cycles is given by:
v – e = 0- c.

We now introduce the 1-Betti number, which we denote by 1. This is the number of independent
cycles.
Figure 5. Making cycles from a tree graph.
c on the right-hand side of equation  can be replaced by 1 , or:
v – e = 0- 1.
2.1 Figures of Points, Line Segments and Faces
Figures on an Open Surface
Let us take some graph containing a cycle and made of points and line segments, and assume that a
surface has been stretched across that cycle, bordered by points and line segments. To distinguish it
from the entire surface, we shall call this surface portion covering the cycle a face. Denoting by f
the number of faces, we then have f = c. So, equation  can be written as:
v – e + f = 0.

The left-hand side is called the Euler characteristic of a surface figure.
Note that the face does not have to be planar and can be a curved surface as well. The figures for e
can include curves as well. The only requirement is that they must be homeomorphic to a line
segment. Also a figure counted under “f” must be homeomorphic to a triangular face. Figure 6
presents a disc and an annulus, partitioned to be applied to the Euler formula.
Figure 6. Partition of disc surface and annulus surface.
Let us first take a disc and draw on it a circle c1 (see Figure 7). Next, draw another smaller circle, c2,
inside c1. Repeating this procedure an infinite number of times will eventually reduce the circle to a
single point. The circle c1 , in this case is said to be homologous to 0.
Figure 7. Circles homologous to 0 and 1-Betti number.
1-Betti number is the number of independent circles that are not homologous to 0. In (c) it is
possible to draw tree circles that cannot be reduced to a point. However, setting down two circles
automatically determines the third, so there are only two independent circles, hence 1 = 2. We see
that, with regard to discs with holes, 1-Betti number equals the number of holes.
Writing down the right-hand side of equation  in the general form, we get
v–e +
f = 0 - 1 + 2.

For a hole-less figure on an open surface, the right-hand side yields:
0 -  1 +  2 = 1 - 0 +
0 = 1.
For a geometric figure which consists of points, line, segments and faces, drawn on a surface, and
which is gomeomorphic to a disc, the Euler characteristic equals 1.
Figures on Closed Surfaces
Consider stretching another face onto the back of this surface, thus creating a closed surface. Such a
figure on a closed surface would posses an Euler characteristic of 2, or:
v–e +
f =2.
This is a well-known formula for simple polyhedra, that is , polyhedra that are homeomorphic to a
spherical surface. In fact, formula  applies to figures on a closed surface as well. That is, for a
simple polyhedron, the right-hand side of  is:
0- 1 + 2 = 1 - 0
+ 1=2
To apply equation  to varios figures on closed surfaces, the count for e must include figures
homeomorphic to a line segment, while f must include those homeomorphic to a (open) triangular
face. The surfaces of a sphere cylinder, cone, etc, can be partitioned as shown in Figure 8.
Figure 8. Partitioning of geometric figures.
A holed polyhedron can be made by connecting two U-shaped polyhedra, as in Figure 9.
Figure 9. Connecting two U-shaped polyhedra.
At the connections, 24 vertices, 24 edges and 4 faces disapper. Denoting by v, e and f the
nimber of vertices, line segments and faces, respectively, of one of the original polyhedra, the Euler
formula after connection is as follows:
(2v -24) – (2e - 24) + (2f – 4) = 2(v – e + f) – 4 = 22 - 4 = 0
Thus the Euler formula for a polyhedron with a single penetrating hole is given by:
v – e + f = 0.
The number of penetrating holes a polyhedron has ordinarily equals its genus.
The Euler characteristic of a polyhedron with genus g is:
v – e + f = 2 – 2g.
A connected surface is called a shell (see Figure 10). If we denote by s the number of shells, we
have a widely used a practical Euler formula:
v – e + f = 2(s – g).
Figure 10. Solid with two shells.
Euler-Poincare formula.
Given a polyhedron, divide it into simplices. Denoting its r-Betti numbers by r (r = 0,1,2, …, n) and
the number of r-simplices by r (r = 0,1,2, …, n), the following relation holds:
n

r 0
(-1)rr =
n

(-1)r r

r 0
Figure 11. Simplices.
3. Simplices, Complexes and Cells
A point, line segment (including the end points), triangle and solid tetrahedron are
respectively called a 0-, 1-, 2-, and 3-simplices(see Figure 11). The end points, for instance,
of a 1-simplex 1 are called the 0-faces of 1.
Simplices can be thought of as the basic figures in their respective dimension. For example,
the polygon shown in Figure 12 can be thought of as a combination of five triangles. The
reverse process is called simplicial decomposition.
Figure 12. Combining simplices to construct a figure.
A geometric figure which is produced by combining simplices, and for which the
intersection of two arbitrary simplices forms a face simplex is called a complex (see Figure
13).
Figure 13. Complexes and non-complexes
By extending simplices, we obtain what is known as cells. An n-cell is a convex open set:
that is, 0-, 1-, 2-and 3-cells are, respectively, points, open line segments, the interior of
convex polygons and the interiors of convex polyhedra (see Figure 14).
Figure 14. Cells.
4. Connectivity Number of Complexes
Definition 6. (Connectivity number of 1-complex)
Given a 1-complex C, the maximum number of line segments that can be removed without changing
the number of connected components (i.e., its 0-Betti number 0) is called the connectivity number
of C.
Consider Figure 15. Here the number of connected complexes, or connected components, is 0 = 3.
The line segments referred to in the definition above must be open, that is, they exclude the end
points. For instance removal of line segment v1v2 will isolate point v1, thus increasing the nimber of
connected components by one. One may, however, remove a line segment from a cycle without
changing the number of connected components. There are three cycles in the figure shown, so as
many as three line segments can be removed without affecting the number of components: for
example, v3v4, v8v9 and v13v14. Thus, the connectivity number of a 1-complex is equal to the number
of cycles present, or its 1-Betti number.
Figure 15. Connectivity number of 1-complex.
Definition 7. (Connectivity number of 2-complex)
Given some curved surface f, we obtain by triangulation a connected 2-complex C. Suppose that the
removal of a 1-complex of connectivity number q will leave C connected as one. The largest such q
is called the connectivity number of the curved surface F.
Figure 16. Determining the connectivity number of a 2-complex.
Examine the connectivety number of a cube (see Figure 16). Here, removing any closed polygonal
line will end up splitting the cube into two parts. For instance, if we remove the closed polygonal
line ABCDA, the top surface will become separated. Because the cube surface will always be
divided by removing a closed polygonal line, which is a 1-complex of connectivity number 1, we
know that the cube surface has a connectivity number of 0.
5. Manifolds
Lines, line segments and closed polygonal lines are one-dimensional figures. In general, an ndimensional figure is one in which n independent variables are needed to specify the position of a
point. For instance, cubes and spheres are examples of three-dimensional figures.
By imposing a certain constraint onto these figures, we arrive at the concept of a manifold.
Figure 17. Manifolds and non-manifolds
Definition 8. (Manifolds)
A 1-manifold is a figure which is homeomorphic to a line, line segment or closed polygonal line. A
2-manifold is a figure such that the neighborhood of any point is homeomorphic to a single plane or
half-plane.
The two figures on the right in Figure 17 (a) are not manifold, since the junction points do not
possess the properties of a line segment.
6. Practical Euler Formula
1. Face-loop
The Euler-Poincare formula, equation , involves a count of simplex for each dimension.
In applying this formula, however, it is not always necessary to triangulate the faces of a polyhedron.
For instance, the diagonal in the face of Figure 18 can be removed without affecting the formula;
this is because the numbers of line segments and faces both decrease by one. Therefore, such a face
i.e., one homeomorphic to a triangle, can be counted as one face for f count in the Euler formula.
Figure 18. Face that does not require triangulation.
This does not apply, however, for face such as shown in Figure 19. That is, based on a simple count
(i.e., without triangulating, the faces), we obtain for the Euler characteristic:
v – e + f = 16 – 24 + 11 = 3.
Figure 19. Polyhedron with a multiply-connected face.
The error here that the count for f must only involve faces that are homeomorphic to a triangular
face, there is a face with a hole.
Let us therefore triangulate this (see Figure 20(b). We then have for the Euler characteristic:
v – e + f = 16 -32 + 18 = 2,
which indeed satisfies the Euler formula.
Figure 20. Removal of line segments from a triangulated multiply-connected face.
Let us now remove from (b) a single line segment to arrive at the figure in (c). The Euler formula
still stands because, along with the removal line segments, the number of faces also decreases by
one.
We can conclude that the Euler formula also applies to a polyhedron containing multiply-connected
faces. With each face being counted as one, and with additional “virtual edges” being counted as
many as the number of holes.
We now introduce an element called the loop or a face-loop. A loop is a cycle of vertices that form
the boundary of face. If we denote by lf the total number of face-loops of an arbitrary polyhedron,
the total number of holes is given by (lf - f). We shall insert lf – f edges. Since this edge insertion
does not affect the right hand side of  it can be written as :
v – (e + lf - f) + f = (s – g),
where we do not have to take into account the “virtual edges” any more.
References
[1] F. Yamaguchi, Computer-Aided Geometric Design, Springer, 2002
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