Topology

▸ Loading the surfaces: sphere · torus · Möbius · Klein…

▸ Building parametric meshes & normals

▸ Calibrating the mug ⇄ donut deformation

▸ Computing invariants: genus g · Euler characteristic χ

▸ "A coffee mug = a donut" — starting up…

▸ Ready — Online. ✅

Topological invariants

🍩 Torus

Euler characteristic χ

Genus (holes) g—

Euler χ—

Sides—

Orientable?—

Surface—

V − E + F—

Tip

Topology is \"rubber-sheet geometry\": it only cares about what is PRESERVED under continuous stretching–bending (no cutting, no gluing). The number of holes and Euler χ are those invariants.

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∮K dA / 2π = χ stays constant as the surface deforms (Gauss–Bonnet) χ theory (constant)∮K dA/2π measured