Solution
ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2015/geometric-modelling/3/b/solution
Past exam of the computer science course of the University of Oxford 2015 Geometric modelling 3 b Solution by 
Codex 0 2026-10-31
Codex 0 2026-10-31For an ordinary closed manifold boundary representation, three requirements are:
- Valid embedded faces. Each face is planar and bounded by simple closed edge loops with nonzero area; distinct face interiors do not cross, and faces meet only in their declared common edges or vertices of a polyhedron. A quadrilateral with four noncoplanar vertices of a polyhedron is not one planar face; a bow-tie loop does not bound a simple face. Holes, when allowed, need separate correctly oriented loops.
- Closed manifold incidence. Exactly two faces meet at each edge, and the incident faces around each vertex of a polyhedron form one cyclic fan. A single missing face leaves edges incident with only one face. Three faces sharing an edge, or two closed shells touching only at one vertex of a polyhedron, violate the local surface condition. The second example shows why the edge count alone is insufficient.
- Consistent orientation. Adjacent oriented faces traverse their common edge in opposite directions, giving compatible outward normal vectors. Reversing just one face of a tetrahedron breaks this condition. For a cavity, the material's outward normal vector points into the cavity, so shell orientation must also respect which region is solid.
Together these support the answerFor a connected orientable closed boundary of genus , the Euler characteristic provides an additional check, , when faces are disk-like. The familiar value applies to a sphere-like shell; it must not be imposed blindly on every polyhedron. Correct counts alone do not exclude intersections or invalid local incidence.
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