Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces...

Area of Science:

• Differential Geometry
• Geometric Analysis
• Topology

Background:

• Developable surfaces are crucial in geometry and have familiar constructions like the tangent and rectifying developables.
• The tangent developable relates the curve's curvature (κ) to its geodesic curvature (κg) as |κg| = κ.
• The rectifying developable is characterized by vanishing geodesic curvature (κg = 0).

Purpose of the Study:

• To introduce and characterize a generalized family of developable surfaces constructed from a space curve.
• To demonstrate that geodesic curvature can be precisely controlled on these new surfaces.
• To extend the understanding of developable surfaces beyond the classical tangent and rectifying constructions.

Main Methods:

• The study involves the theoretical construction of developable surfaces based on a given space curve.
• A key method is the parameterization of a family of surfaces using an arbitrary function k defined on the curve.
• Mathematical analysis is used to determine the geodesic curvature of the curve relative to these newly constructed surfaces.

Main Results:

• A novel family of developable surfaces is shown to be generatable from any space curve.
• For each surface in this family, the geodesic curvature (κg) of the curve is precisely equal to the function k, where |k| ≤ κ.
• This provides a continuous range of geodesic curvatures, bridging the gap between the tangent and rectifying developables.

Conclusions:

• The findings reveal a broader class of developable surfaces than previously known.
• This generalized framework allows for the explicit control of a curve's geodesic curvature relative to its developable surface.
• The research offers new perspectives on the relationship between curves and their associated developable surfaces in differential geometry.