The Minimal Number of Triangles Needed to Span a Polygon Embedded in ℝd

Abstract

Given a closed polygon P having n edges, embedded in R d , we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having P as its boundary which is immersed in R d and whose interior is disjoint from P . The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in R 3 there exists an embedded orientable triangulated PL surface having at most 7n 2 triangles, whose boundary is a subdivision of P . We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 1 2 n 2 − O(n) triangles. We also exhibit families of polygons in R 3 for which Ω(n 2 ) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n 2 ) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between 1/2 and 7.

Cuts for 3-D magnetic scalar potentials: Visualizing unintuitive surfaces arising from trivial knots

“…For more studies of related problems, see works of Hass et al, e.g. [13,14,15,16]. The problem of having the least number of triangles to span a polygon embedded in R n is studied in [15].…”

“…[13,14,15,16]. The problem of having the least number of triangles to span a polygon embedded in R n is studied in [15]. It brings us to the context of finite element meshes, where we can also consider different mesh-dependent optimality criteria for cuts.…”

Area Inequalities for Embedded Disks Spanning Unknotted Curves

“…In the direction of PL combinatorial bounds, Hass and Lagarias [17] showed that there is a (qualitative) combinatorial analogue of the isoperimetric inequality (1) above in R 3 .…”

“…embedded polygons in R 3 . For piecewise linear curves, one can consider combinatorial analogues of isoperimetric inequalities, in which the "length" is replaced by the number of edges n in a polygon, and the "area" is the number of triangles in an embedded triangulated surface having the polygon as boundary, as in [17]. We establish the following PL analogue of Theorem 1.1 above.…”