In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and swallowtails. In order to characterize these singularities, two geometric invariants are discovered. At last, an example is given to demonstrate the main results.

In this research article, we prove the existence and uniqueness of common best proximity points for a given class of discontinuous mappings. For that, we have introduced the notions of neutrosophic proximally compatible mappings, neutrosophic proximally reciprocal and weak reciprocal mappings, and R-proximally weak reciprocal commuting mappings of type I and type II. We have given examples to validate our findings.

In this paper, we have established some basic inequalities for the submanifolds of conformal Kenmotsu manifolds. As an application, we have also derived the same inequalities for the $\theta$-slant submanifolds of conformal Kenmotsu manifolds.

In optics, light rays emitted from a light source form a wavefront (orthotomic) upon reflection by a mirror. The mirror is referred to as an antiorthotomic of the orthotomic. The investigation of the relationship between orthotomics and antiorthotomics constitutes an interesting problem in physics. However, the study becomes ambiguous when the orthotomic exhibits singular points. In this paper, we define generalized antiorthotomics of (n, m)-cusp curves in the Euclidean plane by using the singular curve theory. We demonstrate that the singular points of the generalized antiorthotomic sweep out the evolute of the (n, m)-cusp curve. We also investigate the behavior and singular characteristics of the antiorthotomic of the (n, m)-cusp curve. Moreover, we define parallels for (n, m)-cusp curves and reveal the relationship between parallels and generalized antiorthotomics. Finally, repeated antiorthotomics are studied, which is useful for identifying the characteristics of (n, m)-cusp curves.