We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal curves, the discrete normals are built up through a Gram–Schmidt procedure applied to consecutive oriented segments, and they naturally live in the projective space associated with the Gauss hyper-sphere. By using sequences of inscribed polygonals with infinitesimal modulus, a relaxed notion of total variation of the jth normal to a generic curve is then introduced. For smooth curves satisfying the Jordan system, in fact, our relaxed notion agrees with the length of the smooth jth normal. Correspondingly, a good notion of weak jth normal of irregular curves with finite relaxed energy is introduced, and it turns out to be the strong limit of any sequence of approximating polygonals. The length of our weak normal agrees with the corresponding relaxed energy, for which a related integral-geometric formula is also obtained. We then discuss a wider class of smooth curves for which the weak normal is strictly related to the classical one, outside the inflection points. Finally, starting from the first variation of the length of the weak jth normal, a natural notion of curvature measure is also analyzed.

我们处理了在高维欧氏空间中定义的各种不规则曲线的弱法线的强健概念。关于多边形曲线，离散法线是通过将Gram–Schmidt程序应用于连续的定向线段而建立的，它们自然地生活在与高斯超球面相关的投影空间中。通过使用具有无限小模数的内接多边形序列，然后引入了第j条法向于一般曲线的总变化的松弛概念。实际上，对于满足约旦系统的平滑曲线，我们的松弛概念与平滑第j法线的长度一致。相应地，弱j的一个好概念引入具有有限松弛能量的不规则曲线的法线，它是逼近多边形的任何序列的强极限。我们的弱法线的长度与相应的松弛能量相符，为此，还获得了相关的积分几何公式。然后，我们讨论一类较宽的平滑曲线，在拐点之外，弱法线与经典曲线严格相关。最后，从弱第j个法线的长度的第一个变化开始，还分析了曲率测度的自然概念。