AI 中文总结
研究平面封闭定向多边形角切方案，证明连续极限曲线存在并导出其定向面积公式，区分初始轮廓几何形状与细化规则贡献，还探讨常数几何意义，给出相关应用及\(\pi\)和\(\log 2\)的级数。
AI 中文摘要
我们考虑平面上封闭定向多边形的角切方案。在每一步中，每个顶点被相邻边上的两个点取代；当前多边形的所有顶点使用相同的切割系数，但该系数可能随步骤变化。对于此类方案，我们证明了连续极限曲线的存在性，并导出其定向面积的显式公式。主要结果区分了两种贡献：初始轮廓的几何形状通过其定向面积和由相邻边构建的行列式局部和进入，而对细化规则的整个依赖包含在一个数值常数中。我们还讨论了该常数的几何意义，即标准角度中产生的面积。作为应用，我们考虑平稳德拉姆曲线、柴金方案、与阿基米德抛物线求积法的联系，以及导致圆和双曲线弧的特殊系数序列；这些给出了\(\pi\)和\(\log 2\)的相应级数。
英文摘要
We consider corner-cutting schemes for closed oriented polygons in the plane. At each step every vertex is replaced by two points on the adjacent edges; the same cutting coefficient is used at all vertices of the current polygon, but this coefficient may vary from step to step. For such schemes we prove the existence of a continuous limit curve and derive an explicit formula for its oriented area. The main result separates two contributions: the geometry of the initial contour enters through its oriented area and a local sum of determinants built from neighboring edges, while the entire dependence on the refinement rule is contained in a single numerical constant. We also discuss the geometric meaning of this constant as an area arising in the standard angle. As applications we consider stationary de Rham curves, the Chaikin scheme, the connection with Archimedes' quadrature of the parabola, and special sequences of coefficients leading to a circle and to an arc of a hyperbola; these give corresponding series for $π$ and $\log 2$.
CommentsEnglish version followed by the Russian version; 24 pages in English and 25 pages in Russian, 4 figures in each version
