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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A lower bound on the angles of triangles constructed by bisecting the longest side


Authors: Ivo G. Rosenberg and Frank Stenger
Journal: Math. Comp. 29 (1975), 390-395
MSC: Primary 50B15; Secondary 65M10
MathSciNet review: 0375068
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Abstract: Let $ \Delta {A^1}{A^2}{A^3}$ be a triangle with vertices at $ {A^1},{A^2}$ and $ {A^3}$. The process of "bisecting $ \Delta {A^1}{A^2}{A^3}$" is defined as follows. We first locate the longest edge, $ {A^i}{A^{i + 1}}$ of $ \Delta {A^1}{A^2}{A^3}$ where $ {A^{i + 3}} = {A^i}$, set $ D = ({A^i} + {A^{i + 1}})/2$, and then define two new triangles, $ \Delta {A^i}D{A^{i + 2}}$ and $ \Delta D{A^{i + 1}}{A^{i + 2}}$.

Let $ {\Delta _{00}}$ be a given triangle, with smallest interior angle $ \alpha > 0$. Bisect $ {\Delta _{00}}$ into two new triangles, $ {\Delta _{1i}},i = 1,2$. Next, bisect each triangle $ {\Delta _{1i}}$, to form four new triangles $ {\Delta _{2i}},i = 1,2,3,4$, and so on, to form an infinite sequence T of triangles. It is shown that if $ \Delta \in T$, and $ \theta $ is any interior angle of $ \Delta $, then $ \theta \geqslant \alpha /2$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0375068-5
PII: S 0025-5718(1975)0375068-5
Keywords: Triangulation, bisection, finite element method
Article copyright: © Copyright 1975 American Mathematical Society