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

Abstract

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 τ 1 , then τ 1 ⩾ τ / c 1 , where c 1 = π / 3 arctan ( 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 τ n denotes the minimum angle of the triangles obtained after n iterative applications of the LE-trisection, then τ n > τ / c where c is a positive constant independent of n. An experimental estimate of c ≈ 6.7052025350 is provided.