Topological photonics offers enhanced control over electromagnetic fields by providing a platform for robust trapping and guiding topological states of light. The topological rainbow can separate and distribute different wavelengths of topological photonic states into different positions, but related topological devices have not yet been fully explored. In this work, topological rainbows are realized by inserting a sandwiched gradient structure in the topological waveguide to realize the topological edge states. A robust one‐way slow‐light coupling state is realized to broaden the frequency range of the formed topological rainbow with the robust transmission. Thus, multiple topological rainbows are realized by combining the topological property with immunity to backscattering and coupling states of the slow light. Accordingly, light can spread, separate, and thus be trapped at different positions in different directions. This …

In this paper, we address a new version of Darboux frame using a common tangent vector
field to a surface along a curve and call this frame the rotation minimizing Darboux frame (RMDF).
Then, we give the parametric equation due to the RMDF frame for a sweeping surface and show that the
parametric curves on this surface are curvature lines. Consequently, necessary and sucient conditions
for sweeping surfaces to be developable ruled surfaces are derived. Also, we analyze the conditions
when the resulting developable surface is a cylinder, cone or tangential surface. We also provide some
examples to illustrate the main results.

In this paper using the Blaschke approach we generalized the Bertrand curves to spacelike
ruled and developable surfaces. It is proved that, every spacelike ruled surface have a Bertrand oset
if and only if an equation should be fulfilled among their dual integral invariants. Consequently, some
new relationships and theorems for the developability of the Bertrand osets of spacelike ruled surfaces
are outlined.

In this work, we introduce a time-like ruled surface in one-parameter hyperbolic dual spherical motions. This provides the ability
to derive some formulae of surface theory into line spaces. Then, a time-like Plücker conoid associated with the motion has been
obtained, and its kinematic geometry is researched in detail. Consequently, a characterization for a time-like ruled surface to be a
constant Disteli-axis is derived and investigated in detail. At last, we have discussed some special cases which lead to some special
time-like ruled surfaces such as the time-like helicoids, Lorentzian sphere, and time-like cone.

In this paper, the relationships among the instantaneous invariants of a one-parameter Lorentzian spatial movement and the
local invariants of the axodes are studied. New proofs for Euler–Savary, and Disteli formulae are given which demonstrate
the elegance and efficiency of the E. Study map in Lorentzian spatial kinematics. Consequently, two spacelike line congruences
are introduced and their spatial equivalent are examined in detail.