Computer Aided Engineering Design

SOLID MODELING 263

protruding flat face with 10 vertices, 15 edges, 7 faces and no holes with a shell value 1. Even though
the Euler-Poincaré formula gives 10 – 15 + 7 – (7 – 7) –2 (1 – 0) = 0, it is not a valid solid because
of the protruding face that has zero thickness and thus no interior within itself.

8.8.3 Euler-Poincaré Operators

Given a polyhedron model, we may want to edit it by adding or deleting edges, vertices, faces and
genus to create a new polyhedron using Euler operators. The operators are designed such that the
Euler-Poincaré formula in Eq. (8.3) is always satisfied for the intermediate results. Two groups of
Euler operators are put to use, the Make and Kill groups for adding and deleting, respectively. Euler
operators are written as MxyzorKxyz for the Make and Kill groups, respectively, where x,y and z
represent a vertex, edge, face, loop, shell or genus. For instance, MEV implies making (or adding) an
edge and a vertex while KEV means killing or deleting an edge and a vertex. Euler operators form a
complete set of modeling primitives in that any polyhedron satisfying Euler-Poincaré relation can be
constructed using a finite sequence of operators. Euler operators, thus, are significant from the
viewpoint of constructing B-rep solid models. The make group table shows some operators of the
Make group used to add elements in the existing polyhedral topology and one for the Make-Kill
group used to add and delete some elements at the same time.

Operator Implication VEFL SGChange in Euler-
Poincaré formula

MEV Make an edge and a vertex +1 +1 0

MFE Make a face and an edge + 1 +1 +1 0

MSFV Make a shell, a face and a vertex +1 +1 +1 +1 0

MSG Make a shell and a genus +1 +1 0

MEKL Make an edge, Kill and loop +1 –1 0

Note that the operations above are designed such that they do not cause any change in the Euler-
Poincaré relation as shown in the rightmost column. MEV implies adding an edge and a vertex. A
face and an edge are added via MFE.When adding a face, a loop also gets added which causes
no change in the expression (L –F) of Eq. (8.3). MSGmakes a shell with a hole and MEKLmakes
an edge and kills a loop. MEKLoperation is commonly employed when connecting the outer loop
with the inner one through an auxiliary edge as suggested in Figure 8.18(b).
The Kill group of Euler operators performs the deletion operations, and exchanging M with K
in the Make group table yields the operators of the Kill group given below. With these operations,
we can reduce, for instance, a cube to its non-existence. Otherwise, we may partially delete
the entities of an existing polyhedron and then use the Make group operators to reconstruct a new
one.

Operator Implication VEFLS G Change in Euler-

Poincaré formula

KEV Kill an edge and a vertex –1 –1 0

KFE Kill a face and an edge – 1 –1 –1 0

KSFV Kill a shell, a face and a vertex –1 –1 –1 –1 0

KSG Kill a shell and a genus –1 –1 0

KEKL Kill an edge, Make and loop –1 +1 0