内容发布更新时间 : 2024/5/10 14:58:42星期一 下面是文章的全部内容请认真阅读。

There is a convenience class BRepBuilderAPI_MakeVertex which eventually uses BRep_Builder inside. So, if you have to construct topology bottom up make sure you are familiar with BRep_Builder.

Accessing vertex info

BRep_Tool is the tool that provides access to geometry of the topological entities. Most its methods are static.

Standard_Real aTolerance = BRep_Tool::Tolerance (aVertex); gp_Pnt aPoint = BRep_Tool::Pnt (aVertex);

To be continued...

Topology and Geometry in Open CASCADE. Part 3

Continued...

OK, let's continue to eat our elephant bit by bit. The next bit is edge. Hope you won't get difficulties with it.

Edge

Edge is a topological entity that corresponds to 1D object – a curve. It may designate a face boundary (e.g. one of the twelve edges of a box) or just a ‘floating' edge not belonging to a face (imagine an initial contour before

constructing a prism or a sweep). Face edges can be shared by two (or more) faces (e.g. in a stamp model they represent connection lines between faces) or can only belong to one face (in a stamp model these are boundary edges). I'm sure you saw all of these types – in default viewer, in wire frame mode, they are displayed in red, yellow and green respectively.

6

Edge contains several geometric representations (refer to the diagram in Part1):

- Curve C(t) in 3D space, encoded as Geom_Curve. This is considered as a primary representation;

- Curve(s) P(t) in parametric 2D space of a surface underlying each face the edge belongs to. These are often called pcurves and are encoded as Geom2d_Curve;

- Polygonal representation as an array of points in 3D, encoded as Poly_Polygon3D;

- Polygonal representation as an array of indexes in array of points of face triangulation, encoded as Poly_PlygonOnTriangulation.

The latter two are tessellation analogues of exact representations with the help of former two.

These representations can be retrieved using already mentioned BRep_Tool, for instance:

Standard_Real aFirst, aLast, aPFirst, aPLast;

Handle(Geom_Curve) aCurve3d = BRep_Tool::Curve (anEdge, aFirst, aLast); Handle(Geom2d_Curve) aPCurve = BRep_Tool::CurveOnSurface (anEdge, aFace, aPFirst, aPLast);

The edge must have pcurves on all surfaces, the only exception is planes where pcurves can be computed on the fly.

7

The edge curves must be coherent, i.e. go in one direction. Thus, a point on

the edge can be computed using any representation - as C(t), t from [first, last]; S1 (P1x (u), P1y(u)), u from [first1, last1], where Pi – pcurve in parametric space of surface Si

Edge flags

Edge has two special flags:

- \and last = last_i, i.e. all geometric representations are within the same range; - \= S1(P1x(t), P1y(t)), i.e. any point along the edge corresponds to the same parameter on any of its curves.

Many algorithms assume that they are both set, therefore it is recommended that you ensure that these conditions are respected and flags are set.

Tolerance

Edge's tolerance is a maximum deviation between its 3D curve and any other representation. Thus, its geometric meaning is a radius of a pipe that goes along its 3D curve and encompass curves restored from all representations.

Special edge types

There are two kinds of edges that are distinct from others. These are:

- seam edge – one which is shared by the same face twice (i.e. has 2 pcurves on the same surface)

8

- degenerated edge – one which lies on a surface singularity that corresponds to a single point in 3D space.

The sphere contains both of these types. Seam-edge lies on pcurves

corresponding to surface U iso-lines with parameters 0 and 2*PI. Degenerated edges lies on North and South poles and correspond to V iso-lines with parameters –PI/2 and PI/2.

Other examples - torus, cylinder, and cone. Torus has two seam-edges – corresponding to its parametric space boundaries; cylinder has a seam-edge. Degenerated edge represents on a cone apex.

To check if the edge is either seam or degenerated, use BRep_Tool::IsClosed(), and BRep_Tool::Degenerated().

Edge orientation

Forward edge orientation means that its logical direction matches direction of its curve(s). Reversed orientation means that logical direction is opposite to curve's direction. Therefore, seam-edge always has 2 orientations within a face – one reversed and one forward.

To be continued...

Topology and Geometry in Open CASCADE. Part 4

Continued...

9

Though the next topology type after the edge considered in the previous post is a wire, let's jump to the face which is the last topology entity that binds with geometry. We'll consider a wire as well as the rest of topology types in the future posts. Frankly, I was not originally going to speak of them but several folks on the blog asked to.

Face

A face is a topology entity that describes a boundary unit of the 3D body. A face is described with its underlying surface and one or more wires. For instance a solid cylinder consists of 3 faces – bottom and top, and lateral. Each of respective underlying surfaces is infinite (Geom_Plane and

Geom_CylindricalSurface), while each face bounds its surface with a wire – two of them consists of a single edge lying on Geom_Circle and the lateral face consists of 4 edges – 2 are shared with top and bottom faces, and remaining two represent a seam edge (see previous post on edges), i.e. the face contains it twice in its wire (with different orientations).

Surface

Let's briefly recall what a surface is. If you had a course of mathematical analysis in your higher school you likely remember this by heart, if not you might want to read some articles to educate yourself. First part of this one on wikipedia give simple examples of parametric surfaces.

A surface maps its 2D parametric space {U,V} into a 3D space object (though still two-dimensional). Compare it with molding when a planar steel sheet is transformed into something curved. Parametric space can be bounded or unbounded, or (un)bounded in one direction only. For instance, a plane's parametric space is unbounded, while NURBS is bounded, and a cylindrical surface is bounded in U (from 0 to 2*PI) and is unbounded in V (-infinity, + infinity).

Geom_Surface::Value() returns a 3D point (X, Y, Z) from a point in a

parametric space (U, V). For instance any location on Earth is described by latitude (V) and longitude (U), but can be viewed as 3D point in the Universe (if Earth center had some origin defined).

Let's recall that an edge must have both a 3D curve and a pcurve in its face surface space. Open CASCADE requires that face boundaries (wires) represent closed contours in 3D and 2D space. Therefore a face cylinder's lateral face is described as we considered above.

Not all modeling kernels impose such restrictions. I remember a cylinder coming from Unigraphics (via STEP) which lateral face was described with 2 wires, each consisting of a circular edge. We had a few discussions with my colleagues from the Data Exchange team on how to recognize such cases and

10