Geometry and topology; Euler check of solid

Geometry and topology

Geometry is the metric information of the model: actual coordinates of points, equations of curves and surfaces, lengths, angles, radii. Topology is the connectivity information: how vertices, edges and faces are joined, and which side of a face is material. It does not change when the shape is stretched.

Euler-Poincare formula

For a valid solid with holes (genus) and inner loops:

$$V - E + F - R = 2\,(S - H)$$

where $V$ = vertices, $E$ = edges, $F$ = faces, $R$ = rings (inner loops on faces), $S$ = shells, $H$ = through-holes (genus).

Numerical

Given: $V = 16$, $E = 24$, $F = 10$, one shell $S=1$, one through-hole $H=1$.

$$\begin{aligned} 16 - 24 + 10 - R &= 2(1-1) = 0 \\ 2 - R &= 0 \\ R &= 2 \end{aligned}$$

The two rings are the square hole boundaries on the top face and on the bottom face. Each of these faces has an outer loop and one inner loop, so $R = 2$, which agrees with the physical block. The formula is satisfied, so the topology is valid.

Answer: $R = 2$ inner loops (top and bottom faces); $V-E+F-R = 0 = 2(S-H)$, so the solid is valid.