Boundary representation

A brep model with composite surface areas

Boundary representation , in German boundary surface model , ( B-rep or Brep ) is a form of representation of a surface or volume model in which objects are described by their bounding surfaces. The term is made up of the English words boundary for limitation, edge and representation for representation .

application

Visualization

Boundary representation models are preferably used in the visualization of 3D computer graphics and in CAD programs, as they can be processed quickly using algorithms.

Solid modeling

Volume models can also be described with the boundary representation model. Since a body is only represented by its bounding surfaces, one speaks of indirect modeling (as opposed to direct modeling with Constructive Solid Geometry , which is used to construct with bodies). The user or an intelligent test algorithm must ensure that it is a closed envelope.

Object creation

The definition of instances can be done with a node-edge-surface graph ( vef graph , engl. Vertex, edge, face). The geometry is determined by the coordinates of the points. The topology, i.e. the relationships between the points, describe the edges and surfaces. Edges refer to points and surfaces to edges.

example

A vef graph is set up for a tetrahedron as an example. The information is stored using a relational database model . The description of the object can be done in many ways. The following three lists are defined here:

The node list, which contains the coordinates of the points,
the edge list, which references two points for each edge, and
the surface list, which has a closed sequence of edges for each surface.

In order to achieve uniqueness, the direction of rotation of the edge sequence is defined with the definition that the surface is z. B. left of it, firmly. So it is possible in 2D to describe holes with opposite directions of rotation. In 3D, the surface normal is determined by the three-finger rule , which in turn can be used to determine the "front". It should be noted that it is not the order of the points but the edges that is evaluated.

Tetrahedron defined by four points

Node list
Node number	x	y	z
1	2	−2	0
2	−2	2	0
3	2	2	4th
4th	−2	−2	4th

Edge list
Edge number	Node number 1	Node number 2
1	1	2
2	2	3
3	1	3
4th	1	4th
5	2	4th
6th	3	4th

Area list
Area number	Edge sequence (edge number 1, edge number 2, ...)
1	1 2 3
2	3 6 4
3	2 5 6
4th	1 4 5

If you want to describe a volume model, you need a fourth table that lists all the surrounding areas. As mentioned above , the user must ensure that the partial areas completely delimit the volume and that no gaps remain. The entry in the "Orientation" column determines whether the normal vector of the first surface specified in the list of boundary surfaces points away from the volume or into the volume. As in 2D for surfaces, holes can be modeled in this way.

Volume list
Volume number	orientation	Boundary areas (area number 1, area number 2, ...)
1	1	1 2 3 4

literature

Christoph Martin Hoffmann: Geometric & Solid Modeling . Morgan Kaufmann Publishers, San Mateo, California 1989, ISBN 1-55860-067-1 .
Martii Mäntylä: An Introduction to Solid Modeling . Computer Science Press, Rockville, Maryland 1988, ISBN 0-88175-108-1 .