For the spherical unit speed nonlightlike curve in pseudo-hyperbolic space and de Sitter space [Formula: see text] and a given point P, we can define naturally the pedal curve of [Formula: see text] relative to the pedal point P. When the pseudo-sphere dual curve germs are nonsingular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the pseudo-sphere dual curve germs are nonsingular. Furthermore, we obtain the extension results in dualities, which has wide influence on the open and closed string field theory and string dynamics in physics, and can be used to better solve the dynamics of trajectory particle condensation process.

中文翻译：

---

伪双曲空间和de Sitter空间中踏板曲线的奇异性和对偶性

对于伪双曲空间和德西特空间[公式：见文]和给定点P中的球面单位速度非光曲线，我们可以自然地定义[公式：见文]相对于踏板点P的踏板曲线。当伪球双曲线细菌是非奇异的，这种踏板曲线的奇异类型仅取决于踏板点的位置。在本文中，我们给出了当伪球双曲线细菌是非奇异时奇异点和踏板点位置的完整范式列表。此外，我们得到了对偶性的推广结果，这对物理学中的开闭弦场理论和弦动力学有广泛的影响，可以更好地解决轨迹粒子凝聚过程的动力学问题。