The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants interpolating boundary points is assumed. However, the study of approximants, which additionally interpolate corresponding tangent directions and curvatures at the boundary of an arc, is also considered. Several low-degree polynomial approximants are studied in detail. When several solutions fulfilling the interpolation conditions exist, the optimal one is characterized, and a numerical algorithm for its construction is suggested. Theoretical results are demonstrated with several numerical examples and a comparison with general (i.e. non-one-sided) approximants are provided.

翻译：本文研究了圆弧的最优单侧参数多项式逼近问题。更精确地说，逼近曲线必须完全位于圆弧所在圆的内侧或外侧。研究假设逼近曲线自然满足插值圆弧边界点的约束条件，同时进一步探讨了在边界处额外插值对应切线方向与曲率的逼近曲线。详细分析了若干低次多项式逼近情况。当存在多个满足插值条件的解时，给出了最优解的特征描述，并提出了构造最优解的数值算法。通过多个数值实例验证理论结果，并与一般（即非单侧）逼近方法进行了比较。