Computer Aided Engineering Design

262 COMPUTER AIDED ENGINEERING DESIGN

vertices,E the number of edges, F the number of faces, G the number of holes (or genus) penetrating
the solid, S the number of shells and L as the total number of outer and inner loops, the Euler-Poincaré
formula is given as

V – E + F – (L – F) – 2(S – G) = 0 (8.3)

Here, a shell is an internal void of a solid bounded by a closed connected surface that can have its own
genus value. The solid itself is counted as a shell. Euler-Poincaré formula is employed to test the
topological validity of a solid, that is, if the right hand side of Eq. (8.3) is non-zero, the solid is an
invalid solid. However, the vice-versa is not true, that is, a zero value of the formula does not
necessarily mean that the solid is valid.

Example 8.1.Verify the Euler-Poincaré formula for the solids shown in Figure 8.19.

Figure 8.19 (a) A cube, (b) cube with a partial void, (c) cube with penetrating void, (d) half section of the
cube with three orthogonally through voids and (e) cube with a rectangular hanging face.

(a) (b) (c)

(d) (e)

Second

void

First through

void

Third void

A cube has 8 vertices, 12 edges, 6 faces and therefore 6 loops with a shell value 1. Euler-Poincaré
formula results in 8 – 12 + 6 – (6 – 6) – 2 (1 – 0) = 0. A cube with a partial void in Figure 8.19 (b)
has 16 vertices, 24 edges, 11 faces (6 outer and 5 inner) and 12 loops (one inner loop on top surface)
for which the formula gives 16 – 24 + 11 – (12 – 11) – 2(1 – 0) = 0. For a through void in Figure
8.19(c), the solid has 16 vertices, 24 edges, 10 faces, 12 loops (2 inner loops on the top and bottom
surfaces, respectively) and one hole for which 16 – 24 + 10 – (12 – 10) –2(1 – 1) = 0. For a solid
in Figure 8.19 (d) which is a cube with three voids orthogonal to each other, there are 40 vertices,
72 edges, 30 faces, 36 loops, 1 shell and, say, x voids. The Euler-Poincaré result for this solid is
40 – 72 + 30 – (36 – 30) – 2(1 – x) = 0 implying that x = 5, that is, the number of voids is five which
seems counter intuitive and can be explained. The first void is a through hole as shown in the figure,
and in an orthogonal direction, there are two voids (as opposed to one) as shown. Likewise, in the
third direction, there are two voids making a total of 5. The solid in Figure 8.19(e) is a cube with a