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Abstract:The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bisection is iteratively applied to an initial triangle, then minimum angle of the resulting triangles is greater or equal than a half of the minimum angle of the initial angle. The novelty of the proofs is the use of an hyperbolic metric in a shape space for triangles. MSC:51M09, 65M50, 65N50

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We introduce a new triangle transformation, the shortest-edge (SE) duplication, as a natural way of mesh derefinement suitable to those meshes obtained by iterative application of longest-edge bisection refinement. Metric properties of the SE duplication of a triangle in the region of normalised triangles endowed with the Poincare hyperbolic metric are studied. The self-improvement of this transformation is easily proven, as well as the minimum angle condition. We give a lower bound for the maximum of the smallest angles of the triangles produced by the iterative SE duplication α=π6. This bound does not depend on the shape of the initial triangle.