Cartan ribbonization and a topological inspection

Area of Science:

• Differential Geometry
• Computational Geometry
• Topology

Background:

• Approximating complex surfaces is crucial in various scientific fields.
• Existing methods may lack intrinsic flatness or topological simplicity.

Purpose of the Study:

• To develop a novel method for surface ribbonization using Cartan ribbons.
• To enable simplified topological analysis of surfaces through ribbon organization.

Main Methods:

• Introduced the concept of Cartan ribbons.
• Developed a rolling-based method for surface approximation.
• Ensured geodesic curvature alignment between surface and development curves using Darboux frames.
• Utilized closed contact centre curves for closed ribbon generation.

Main Results:

• Achieved intrinsically flat ribbon approximations of surfaces.
• Obtained closed Cartan ribbons with zero total curvature integral.
• Demonstrated simplified topological inspection based on ribbon edge arrangement.
• Successfully ribbonized a torus and an ellipsoid.

Conclusions:

• Cartan ribbons offer an effective method for surface approximation and topological analysis.
• The rolling-based approach ensures intrinsic flatness and simplifies complex surface studies.
• The method provides a new perspective on surface topology through ribbon organization.