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Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique

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Smooth and Discrete Cone-Nets.

Martin Kilian¹, Christian Müller¹, Jonas Tervooren¹

¹Institute for Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Vienna, Austria.

Results in Mathematics

|April 3, 2023

Summary

Cone-nets, a type of conjugate net, are explored in both smooth and discrete differential geometry. Their projective invariance and transformation theory reveal connections to known surface classes, including tractrix surfaces.

Keywords:

Cone-nets combescure transformation conjugate nets discrete nets

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Area of Science:

Differential Geometry
Geometric Analysis
Surface Theory

Background:

Cone-nets are defined as conjugate nets on surfaces with tangential contact along parameter curves.
These networks exhibit projective invariance and are linked to specific transformations.
Understanding these transformations is key to classifying various surface types.

Purpose of the Study:

To investigate the properties of cone-net transformation theory.
To demonstrate how established surface classes fit within the cone-net framework.
To present both smooth and discretely defined cone-nets and their associated concepts.

Main Methods:

Analysis of projective invariance and characteristic transformations of conjugate curve networks.
Development of a consistent discrete setting for cone-nets, mirroring smooth concepts.
Focus on tractrix surfaces as principal cone-nets with constant geodesic curvature.

Main Results:

The study establishes a unified framework for analyzing cone-nets.
Several known surface classes are shown to be instances of cone-nets.
Smooth and discrete tractrix surfaces are characterized within this framework.

Conclusions:

Cone-nets provide a powerful lens for understanding surface geometry.
The framework unifies smooth and discrete differential geometry concepts for cone-nets.
Tractrix surfaces represent a significant subclass of principal cone-nets.