The Euler-Poincaré Formula

The Euler-Poincaré formula is a formula describing the relationship of the number of vertices, the number of edges and the number of faces. It has been generalized to include potholes and holes that penetrate the solid. To state the Euler-Poincaré formula, we need the following definitions:

• V: the number of vertices
• E: the number of edges
• F: the number of faces
• G: the number of holes that penetrate the solid, usually referred to as genus in topology
• S: the number of shells. A shell is an internal void of a solid. A shell is bounded by a 2-manifold surface, which can have its own genus value. Note that the solid itself is counted as a shell. Therefore, the value for S is at least 1.
• L: the number of loops, all outer and inner loops of faces are counted.

Examples

• A cube has eight vertices (V = 8), 12 edges (E = 12) and six faces (F = 6), no holes and one shell (S=1); but L = F since each face has one outer loop. Therefore, we have V-E+F-(L-F)-2(S-G) = 8-12+6-(6-6)-2(1-0) = 0
• The following solid has 16 vertices, 24 edges, 11 faces, no holes, 1 shell and 12 loops (11 faces + one inner loop on the top face). Therefore, V-E+F-(L-F)-2(S-G) = 16-24+11-(12-11)-2(1-0)=0
  

• The following solid has 16 vertices, 24 edges, 10 faces, 1 hole (i.e., genus is 1), 1 shell and 12 loops (10 faces + 2 inner loops on top and bottom faces). Therefore, V-E+F-(L-F)-2(S-G) = 16-24+10-(12-10)-2(1-1)=0
  

• The following solid has a penetrating hole and an internal cubic chamber as shown by the right cut-away figure. It has 24 vertices, 12*3 (cubes) = 36 edges, 6*3 (cubes) - 2 (top and bottom openings) = 16 faces, 1 hole (i.e., genus is 1), 2 shells and 18 loops (16 faces + 2 inner loops on top and bottom faces). Therefore, V-E+F-(L-F)-2(S-G) = 24-36+16-(18-16)-2(2-1)=0
  

• The following solid has two penetrating holes and no internal chamber as shown by the right cut-away figure. It has 24 vertices, 36 edges, 14 faces, 2 hole (i.e., genus is 2), 1 shells and 18 loops (14 faces + 4). Therefore, V-E+F-(L-F)-2(S-G) = 24-36+14-(18-14)-2(1-2)=0
  

Part of the information recorded in a B-rep is topological (i.e., adjacency relations). Invalid solids may be generated if the representation is not carefully constructed. One way of checking this topological invalidity is to use the Euler-Poincaré formula. If its value is not zero, we are sure something must be wrong in the representation. However, this is only a one-side test. More precisely, a zero value of the Euler-Poincaré formula does not mean the solid is valid.

The figure above has a box and an additional sheet which is simply a rectangle. This object has 10 vertices, 15 edges, 7 faces, 1 shell and no hole. Its loop number is equal to the number of faces. The value of the Euler-Poincaré formula is zero as shown below,

V-E+F-(L-F)-2(S-G) = 10-15+7-(7-7)-2(1-0)=0 but this is not a valid solid! Therefore, if the value of Euler-Poincaré formula is non-zero, the representation is definitely not a valid solid. However, the value of the Euler-Poincaré formula being zero does not guarantee the representation would yield a valid solid.