Topology Topology Concepts Geometric Modeler
What is Topology
This paper presents the general topological concepts that are supported by CATIA V5. After defining the topology, the basic entities (cell, domain, body) are precisely
described. Then, non-manifold topologies are introduced and illustrated. A summarized
chart allows the reader to visualize the links between those different concepts. 本文介绍的一般拓扑的概念？是由CATIA V5的支持。
; Topology for Geometric Design
; The Basic Topological Objects
; The Manifold and Non Manifold Concepts
o Dividing a Body into Domains
; Relations between the Topological Objects
; In Short
Topology for Geometric Design
Topology allows to represent objects, by detailing their boundaries and the connections
between their different parts. This figure shows an example of the topological
description of a simple shell object.
Fig 1: A Topological description of a shell object
; The shell object is made of one
topological 2D entity called a face
; The face F is the limitation of the
surface S by four connected 1D
boundaries called edges (E1, E2,
; Each edge (E1, for example) is a
limitation of a geometric curve (C),
lying on the surface. It is bounded
by two vertices (V1, V2).
; The edges are connected by their
vertices to bound the face.
"Regular" objects are called manifold. Objects presenting "hairs" or "scales", are called
non-manifold. The use of non manifold topology is very useful to simplify the
representation of objects: in an early stage of design, a thin stiffener of a solid object
may be represented as a 2D element ("scale").
Fig 2: An Example of a non-manifold object
In this object, a stiffener has been
modelized as a 2D topological element (the
face F). The edge E1 is an external
boundary of the face F, but is also
immersed into a face of the 3D object V:
this is a non-manifold configuration.
The object B without the face F is manifold.
See The Manifold and Non Manifold Concepts for a detailed presentation of these concepts.
CGM uses the technology called "cell complexes" (see the paper of Rossignac for
instance), which allows to:
; Handle multidimensional concepts in an unified way
; Represent all manifold and non-manifold objects.
Basic Topological Objects
The topology manages three types of entities:
; The cell: most basic topological entity.
; The domain: set of connected cells grouped to define boundaries.
; The body: the "concrete" object to modelize. 该拓扑管理三种类型的实体，
We detail here these entities.
A cell is a connected limitation of an underlying geometry. There are four types of cells according to the dimension of the space in which they lie.
Space Dimension Cell Type Associated geometry
0 Vertex Point
1 Edge Curve
2 Face Surface
3 Volume 3D Space
Cells of upper dimensions are bounded by cells of lower dimensions: a volume is the limitation of the 3D space by faces, a face is the limitation of a surface by edges and an edge is the limitation of a curve by vertices.
Fig 3: Examples of cells
A domain is a set of cells of dimension n connected by cells of dimension n-1. A domain can possibly contain only one cell.
Domains are useful to manipulate altogether the boundaries of a cell of upper dimension. If a face, for instance, is bounded by four connected edges, all those edges are conveniently grouped into a domain. Like cells, domains bear specific names according to what they actually contain.
A ... is a set of ... bounding ...
loop edges connected by vertices a face
vertex in face one vertex a face
lump volumes connected by faces the 3D space shell faces connected by edges the 3D space or a volume wire edges connected by vertices the 3D Space
vertex in volume one vertex the 3D Space or a volume
Lumps, shells, wires and vertices in volume are boundaries of 3D entities. Loops and
vertices in faces are boundaries of 2D entities. No domain is associated to the
boundaries of edges (1D entities): vertices directly bounds edges, because such domain
does not bring any added value to the model.
Fig 4: Examples of domains
; A loop is
a set of
g a face
; A shell is
a set of
; A wire is
a set of
; A vertex
in face is
C: this is
Domains can define outer, inner, or immersed frontiers: vertex in face or vertex in volume are typical immersed boundaries. Notice that loops (resp. shell) can also be immersed into a face (resp. volume), but this type of domain is always called a loop (resp. shell) and not a "edge in face" (resp. "face in volume"). Reading the different definitions of the domains, you can see that two faces (resp. two volumes) cannot be connected only by a vertex (resp. by an edge or a vertex). In this case, it will be necessary to have two shells (resp. two lumps). Domains define manifold components inside non-manifold objects.
Fig 5: Domains define manifold components inside non manifold objects
face F in
; The cubes C3
edge E in
a lump is
a set of
lump is a
A body is a set of domains non necessarily connected (with non common boundary of
any dimension). Bodies must satisfy the following properties:
1. Any cell bounding a cell in a body also belongs to the body.
2. The intersection of the underlying geometry for any two cells in a body is also the
underlying geometry for a cell ( and this cell must belong to the body, following
the property 1). In other words, "no intersection of the underlying geometries
without having a cell representing the intersection".
Fig 6: The intersection of the geometry of two cells is the geometry of a cell of the same
Let F1 be a
face of the
body B. The
F1, has also
to belong to
the body B.
If faces F1
) are cells of
the body B,
edge E, lying
of S1 and
S2, has to
exist and is
also a cell of
The body only references domains, even if there is only one cell in the domain. See the
example of the following section: the body contains only one volume, but it contains it
through the lump domain.
This example shows the breaking up into cells and domains of a body representing a
cuboid with a cavity. In order to keep things clear, some relations have not been
Fig 7: Decomposition of a body into domains and cells
The body is composed of a Lump
The Volume has two
boundaries: an inner and an
Each Shell is made of six Faces.
Each Face is bounded by a Loop.
Each Loop owns 4
Edges and each Edge is bounded by two
Notice that each edge is used by
two faces and each vertex is also
The Manifold and Non Manifold Concepts
CGM allows you to create and use manifold and non-manifold bodies. Mathematically speaking, a N-manifold object is a set of points which neighborhood is represented by a N-dimensional bowl. Take a lump domain (resp. shell, loop). If for each point of this domain, there exists a neighborhood of the domain equivalent to only one piece of a
sphere (resp. disk, segment), the lump (resp. shell, loop) is 3D (resp. 2D, 1D ) -manifold. Otherwise, it is non manifold.
The following figures shows examples of manifold and non manifold objects. The place where there are non manifold are highlighted. The bodies can be:
; singular: if there exists cells of dimension n that are only connected by cells of
dimension n-2. (B4, B5, C4, C5)
; heterogeneous: mixing of domains of different dimensions in the same body
; general: cells of dimension n with more than 2 connections with other cells of
same dimension (A2, B6)
manifold non manifold