In the field of differential geometry, a recent study conducted by Hande Nur DALKILIÇ and Yusuf YAYLI from Ankara University focuses on the intricate connections between pseudo-evolute curves and caustic surfaces. Their research, titled “Pseudo-evolute curves and caustic surfaces,” delves into the types of developable surfaces, specifically examining conical, cylindrical, and tangent surfaces that arise from space curves.
Developable surfaces are fundamental constructs in differential geometry, with tangent surfaces receiving significant attention in prior studies. While existing literature, including research by Hoffmann et al. in 2022, has made strides in defining parameters for caustic surfaces across different types, the investigation into how to derive specific caustic developable surfaces—namely rectifying, osculating, and normal surfaces—remains incomplete. This study aims to close those gaps by adjusting light sources and base curves and clarifying the relationships between pseudo-evolute curves and these caustic surfaces.
The authors clarify several core concepts essential to their work. These include the Frenet frame, which consists of unit tangent, normal, and binormal vectors for space curves, and the Darboux frame, which incorporates unit tangent, auxiliary, and normal vectors for surfaces along curves. They also define expressions for pseudo-evolute curves related to curvature and torsion for space curves, as well as geodesic and normal curvature for surface-borne curves. Additionally, they explain flat approximation surfaces, where the normal vector field aligns with the base surface, and normal approximation surfaces, where the normal vector field is orthogonal to the base surface.
One of the key findings involves rectifying caustic surfaces. When the light source is positioned in the negative unit binormal direction of a unit-speed space curve along the mirror surface”s tangent plane, and the resulting reflected vector aligns with the unit binormal, this configuration leads to a rectifying caustic developable surface. Notably, the striction curve of this caustic corresponds to the pseudo-evolute curve of the base curve, allowing the caustic to be expressed as the tangent surface of the pseudo-evolute.
Another significant aspect of their findings pertains to osculating caustic surfaces. For a unit-speed surface-borne curve, placing the light source in the direction of the negative auxiliary vector (from the Darboux frame) along the mirror”s tangent plane results in a reflected vector that follows the auxiliary vector. This setup produces an osculating caustic developable surface, which also serves as the flat approximation surface of the base surface. Here, the striction curve corresponds to the base curve”s pseudo-evolute on the surface.
Moreover, the researchers investigate normal caustic surfaces. When the reflected vector aligns with the unit normal direction of the surface-borne base curve along the mirror”s tangent plane, it creates a normal caustic developable surface, which also acts as the normal approximation surface of the base surface. Similar to the previous findings, the striction curve for this case is the pseudo-evolute of the base curve on the surface.
To validate their findings, the study confirms that if the base curve exhibits zero geodesic curvature—meaning the Frenet and Darboux frames coincide—the striction curve of the osculating caustic aligns with the pseudo-evolute of the space curve. Likewise, a zero normal curvature condition yields the same alignment for the normal caustic. The authors utilize specific examples, such as helicoids, to calculate and visualize these curves and surfaces, thereby reinforcing their conclusions.
The full text of the study “Pseudo-evolute curves and caustic surfaces” is available for open access at this link.
