Open CASCADE notes

Continued...

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 how to convert them into Open CASCADE valid wire. As a result, Shape Healing's ShapeFix_Face has been extended with FixMissingSeam().

Orientation
Face orientation shows how face normal is aligned with its surface normal. If orientation is TopAbs_FORWARD then normals match, if TopAbs_REVERSED then they are opposite to each other.
Face normal shows where body material is – it lies ‘behind' the face. In a correct solid body all face normals go ‘outward' (see below):



Material on the face is defined by orientation of its edges. The side is defined by a cross product of a surface (not face!) normal and edge derivative. Edge derivative equals its 3D curve derivative if edge is forward and is opposite, if reversed. Perhaps easier understanding can be got if to consider edge pcurves: if an edge is forward then material is on the left, if reversed, material is on the right. This may sound complicated but once you get it, it will be simpler ;-). We'll speak more of orientation in a dedicated chapter.



Tolerance
Geometric meaning of a face tolerance is a thickness of a pie surrounding a surface.



A face tolerance is used by modeling algorithms significantly more rarely than edge's or vertex' and is often retained at default level (Precision::Confusion()).
By convention, Open CASCADE requires that the following condition is respected:
Face tolerance <= Edge tolerance <= Vertex tolerance, where an Edge lies on a Face, and a Vertex – on an Edge.

Triangulation
In addition to underlying surface, a face may contains a tessellated representation (composed of triangles). It is computed, for example, when a face is displayed in shading mode. Visualization algorithms internally call BRepMesh::Mesh() which calculates and adds triangulation for every face.

Additional location
Unlike an edge or a vertex, a face has an additional location (TopLoc_Location) which is a member field of BRep_TFace. So, do not forget to take it into account when using underlying surface or triangulation. Stephane has recently pointed this out on a forum.

Creating a face bottom-up and accessing its data
Like in the case of a vertex and an edge, BRep_Builder and BRep_Tool is what you need.
BRep_Builder aBuilder;
TopoDS_Face aFace;
aBuilder.MakeFace (aFace, aSurface, Precision::Confusion());
...

TopLoc_Location aLocation;
Handle(Geom_Surface) aSurf = BRep_Tool::Surface (aFace, aLocation);
gp_Pnt aPnt = aSurf->Value (aU, aV).Transformed (aLocation.Transformation());
//or to transform a surface at once
//Handle(Geom_Surface) aSurf = BRep_Tool::Surface (aFace);
//gp_Pnt aPnt = aSurf->Value (aU, aV);

Handle(Poly_Triangulation) aTri = BRep_Tool::Triangulation (aFace, aLocation);
aPnt = aTri->Nodes.Value (i).Transformed (aLocation.Transformation());

Hope this was not too boring ;-). Just bookmark it and get back when you need it!

To be continued...

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 wireframe mode, they are displayed in red, yellow and green respectively.



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.

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:
- "same range" (BRep_Tool::SameRange()), which is true when first = first_i and last = last_i, i.e. all geometric representations are within the same range;
- "same parameter" (BRep_Tool::SameParameter()), which is true when C(t) = 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)
- 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...

P.S. As usual, many thanks to those who voted and sent comments. Is this series helpful ?

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.



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 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);

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...

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, 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).



Let's consider them more closely.

To be continued...