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TopologyConcept

By Jerry Daniels,2014-09-15 11:35
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Geometric Modeler Topology Topology Concepts What is Topology Technical Article Abstract 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..

    Topology Topology Concepts Geometric Modeler

    What is Topology

Abstract

    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。

    定义的拓扑结构后?基本实体;单元格?网域?体?精确地描述。然后?非流形拓扑结构进行了介绍和说明。汇总图表?让读者直观了解这些不同的概念之间的联系。

    ; Topology for Geometric Design

    ; The Basic Topological Objects

    o Cell

    o Domain

    o Body

    o Example

    ; The Manifold and Non Manifold Concepts

    o Definition

    o Dividing a Body into Domains

    ; Relations between the Topological Objects

    ; In Short

    ; References

    拓扑几何设计

    基本拓扑对象

    O单元

    O

    Ø

    O例子

    歧管和非流形的概念

    O定义

    Ø划分体到域里

    拓扑对象之间的关系

    在短

    参考文献

    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.

    拓扑允许代表对象?详细介绍他们的界限和他们的不同部分之间的连接。此图显示了一个简单的shell对象

    的拓扑描述的例子

    Fig 1: A Topological description of a shell object

    一个shell对象的拓扑描述

    ; The shell object is made of one

    topological 2D entity called a face

    (F).

    ; The face F is the limitation of the

    surface S by four connected 1D

    boundaries called edges (E1, E2,

    E3, E4).

    ; 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.

    ; •Shell对象是一个拓扑的二维实体称为一

    个面;F?的。

     面对F是表面S的四个连接1D称为边缘

    界限;E1E2E3E4?的限制。

    每个边缘;E1?例如?是趴在表面上的几

    何曲线;c?的限制。它是由两个顶点V1

    V2?界。

    边缘连接必然会面对自己的顶点。

    "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").

    被称为普通对象多方面的。提出的鳞片的对象?被称为非流形。非流形拓扑的使用是非常有用的?

    以简化对象的代表性,在设计的早期阶段?加强筋的薄了坚实的对象可能是作为一个2D元素;规模?的

    代表。

    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.

    [Top]

    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,

    单元,最基本的拓扑实体。

    域,设置分组来定义边界连接细胞。

    体,具体的对象modelize

    We detail here these entities.

    [Top]

    Cell

    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.

    一个单元是一个基本的几何的连接限制。

    根据维度的空间?有四种类型的单元

    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

    [Top]

    Domain

    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

    edges

    connect

    ed by

    vertices

    boundin

    g a face

    ; A shell is

    a set of

    faces

    connect

    ed by

    edges

    boundin

    g a

    volume

    ; A wire is

    a set of

    edges

    connect

    ed by

    vertices

    in the

    3D

     space

    ; A vertex

    in face is

    immerse

    d into

    the face.

    In the

    case of

    the

    figure, it

    represen

    ts the

    connecti

    on

    between

    the face

    F and

    the cone

    C: this is

    a non

    manifold

    configur

    ation.

    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

    ; The

    cubes C1

    and C2

    have the

    face F in

    common

    .They

    can be

    grouped

    in the

    same

    lump.

    ; The cubes C3

    and C4

    have

    only the

    edge E in

    common

    : they

    must be

    put into

    different

    lumps,

    because

    a lump is

    a set of

    volumes

    connecte

    d by

    faces.

    Each

    lump is a

    manifold

    compon

    ent of

    the

    non-man

    ifold

    global

    object.

    [Top]

    Body

    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

    body

    Property 1:

    Let F1 be a

    face of the

    body B. The

    edge E,

    boundary of

    F1, has also

    to belong to

    the body B.

    Property 2:

    If faces F1

    and F2

    (lying on

    surfaces F1

    and F2

    respectively

    ) are cells of

    the body B,

    then the

    edge E, lying

    on the

    intersection

    of S1 and

    S2, has to

    exist and is

    also a cell of

    B.

    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.

    [Top] Example

    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

    displayed.

    Fig 7: Decomposition of a body into domains and cells

    The body is composed of a Lump

    made of

    one

    Volume.

    The Volume has two

    shell

    boundaries: an inner and an

    outer Shell.

    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

     Vertices.

    Notice that each edge is used by

    two faces and each vertex is also

    referred three

    times.

    [Top]

    The Manifold and Non Manifold Concepts

Definition

    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

    (C6).

    ; general: cells of dimension n with more than 2 connections with other cells of

    same dimension (A2, B6)

     manifold non manifold

    1D_

2D_

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