关于卵形弧的扭转

On the deformation of an oval arc

证明了下列定理：定理设ＡＢ是一正则卵形弧，其全曲率小于π２，将ＡＢ扭转成为ＡＢ，ＡＢ上任一点Ｍ在ＡＢ上的对应点为Ｍ．则Ｍ到直线ＡＢ的距离不小于Ｍ到直线ＡＢ的距离．还给出了这定理的一些应用．

The following theorem is proved. Theorem Let AB be a regular oval arc with arc length s and curvature k(s), A *B * a carve of the same length referred to the same parameter s such that its curvature k *(s) satisfies|k *(s)|≤|k(s)|Let D(s),D *(s) be points on AB and A *B * respectively, d the distance from D(s) to AB and d * the distance from D *(s) to A *B * .If ∫ AB |k(s)|ds<π2 ,then d≥d * where the equality holds when and only when AB and A *B * are congruent. Some applications of this theorem have been given.