内容发布更新时间 : 2024/12/23 9:31:32星期一 下面是文章的全部内容请认真阅读。

OPEN CASCADE NOTES

A BLOG ABOUT THE OPEN SOURCE 3D MODELING KERNEL: NOTES FROM ITS FORMER DEVELOPER AND PROJECT MANAGER

Topology and Geometry in Open CASCADE. Part 1

Let us consider these fundamental concepts of Open CASCADE. From time to time questions about them pop up on the forum, so I hope this post will be useful for many readers. Understanding these concepts is important if you have to work with most modeling algorithms and especially if you want to develop you own. You might also want to re-read various chapters from the Modeling Data User's Guide to refresh you memory.

Abstract topology

Open CASCADE topology is designed with reference to the STEP standard ISO-10303-42. Perhaps reading some its concepts would be helpful (I myself did this once in 1997).

The structure is an oriented one-way graph, where parents refer to their

children, and there are no back references. Abstract structure is implemented as C++ classes from the TopoDS package. Here is an inheritance diagram taken from the Doxygen-generated documentation.

TopoDS_Shape is manipulated by value and contains 3 fields – location,

1

orientation and a myTShape handle (of the TopoDS_TShape type) – see the diagram below (this and other contain only most important fields):

myTShape and Location are used to share data between various shapes and thus save huge amounts of memory. For example, an edge belonging to two faces has equal Locations and myTShape fields but different Orientations (Forward in context of one face and Reversed in one of the other).

Connection with Geometry

Now let's consider how this abstract topology structure is connected with geometry.

This is done by inheriting abstract topology classes from the TopoDS package by those implementing a boundary representation model (from the BRep

package). Only 3 types of topological objects have geometric representations – vertex, edge, and face (see the diagram below).

2

Let's consider them more closely.

To be continued...

Topology and Geometry in Open CASCADE. Part 2

Continued...

Vertex

Vertex has a primary geometric representation which is a point in 3D space encoded as gp_Pnt. It may have other representations but they are virtually never used in practice.

Another important attribute of a vertex is its tolerance which indicates possible inaccuracy of its placement. Geometrical meaning of the vertex

tolerance T is a sphere with a radius T centered in vertex’s point. This sphere must encompass curves ends of all edges connected at that point.

3

Tolerances

Let me elaborate a little bit on tolerances.

Unlike some other geometric libraries which have some global precision (e.g. as 1/1000th of the model bounding box dimension), Open CASCADE treats tolerances as local properties. As indicated in the diagram in Part1, tolerance is a field of any vertex, edge, or face instance. This approach helps to describe the model with higher accuracy in general. Look at the picture below:

Individual tolerances allow to specify local inaccuracies while leaving the rest of the model well defined. If you have to build your shape bottom up, the best approach is to specify the smallest valid tolerance. By default, it is Precision::Confusion() which is 1e-07.

In general some modeling algorithms are quite sensitive to tolerances,

especially if they have to work with a single value to be specified by the user. For instance, importing a model from another 3D geometric kernel (via IGES or STEP formats, or directly from native format of other kernels or CAD systems), involves a global precision specified in a file header. The importer tries to be robust regardless of that value, no matter how coarse it is (e.g. it

4

could that big so that some tiny faces are fully encompassed by its boundaries what would even violate the standard).

Another example is Sewing (stitching topologically disconnected faces into a shell). Gaps between faces can be quite different across the model. Specifying too small tolerance would leave too many disconnected faces, specifying too big upfront would connect too distant faces (maybe even implied too be disconnected by user intent, like tiny holes).

Coming back to vertices, let me mention about their orientation field. It does not have a direct geometric meaning, but by convention vertex with

TopAbs_FORWARD orientation must match an edge’s end corresponding to the smaller parameter of its curve. Respectively a vertex with

TopAbs_REVERSED – to the curve’s end with greater parameter. For

instance, if you have an edge lying on a circular arc of radius 1 on plane Z=0 starting at point (1, 0, 0) and moving anti-clockwise (if to look in the direction opposite to Z axis) then its forward vertex will have a point (1, 0, 0) and reversed one – (0, 1, 0):

Building a vertex bottom-up

BRep_Builder is the tool that actually performs low-level operations.

gp_Pnt aPoint (100., 200., 300.); BRep_Builder aBuilder; TopoDS_Vertex aVertex;

aBuilder.MakeVertex (aVertex, aPoint, Precision::Confusion()); aVertex.Orientation (TopAbs_REVERSED);

5