This lecture describes some of the basic building blocks for digitally encoding geometry using meshes and mesh data structures, namely simplicial complexes, adjacency matrices, and the halfedge data structure:

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What’s the boundary of a line? (Not including endpoints) I’m having a hard time figuring out from the definition

The boundary is only defined on pure simplicial complexes. The line without its endpoints is not a pure simplicial complex (because it’s not a simplicial complex), so its boundary is not defined according to our definition. If we add in the endpoints to make our line into a pure 1-complex, then its boundary is just the two endpoints, since they’re both proper faces of exactly one simplex (the edge between them)