On the non-degeneracy property of the longest-edge trisection of triangles

摘要

The longest-edge (LE) trisection of a triangle t is obtained by joining the two equally spaced points of the longest-edge of t with the opposite vertex. In this paper we prove that for any given triangle t with smallest interior angle τ>0, if the minimum interior angle of the three triangles obtained by the LE-trisection of t into three new triangles is denoted by τ1, then τ1⩾τ/c1, where c1=π/3arctan(3/5)≈3.1403. Moreover, we show empirical evidence on the non-degeneracy property of the triangular meshes obtained by iterative application of the LE-trisection of triangles. If τn denotes the minimum angle of the triangles obtained after n iterative applications of the LE-trisection, then τn>τ/c where c is a positive constant independent of n. An experimental estimate of c≈6.7052025350 is provided.