Untitled Research Article

Title: Untitled Research Article

Abstract: This research article explores the development of a curve on a plane, specifically focusing on the intersection of a plane with a polytope. The main result is that slice curves always develop to simple curves, i.e., they do not self-intersect. The authors achieve this by generalizing Cauchy's arm lemma, allowing for nonconvex "openings" of a planar convex chain. This generalization is crucial in proving that the hand cannot enter the forbidden disk D(a), ensuring that the curve develops without self-intersection.

Research Question: How do slice curves develop on a plane without self-intersection?

Methodology: The authors employ a generalization of Cauchy's arm lemma, which permits opening of the angles beyond π. This generalization is used to prove that slice curves always develop to simple curves, preventing self-intersection.

Results: The main result of this paper is that slice curves always develop to simple curves, i.e., they do not self-intersect. This is achieved by proving that the hand cannot enter the forbidden disk D(a) in any reconfiguration of the chain, ensuring that the curve develops without self-intersection.

Implications: This research has significant implications for the field of geometry, particularly in the study of curve development and self-intersection. The generalization of Cauchy's arm lemma used in this study can be applied to other areas of mathematics and may lead to further advancements in the field.

Conclusion: In conclusion, the authors have successfully generalized Cauchy's arm lemma to permit nonconvex "openings" of a planar convex chain. This generalization has been used to prove that slice curves always develop to simple curves, preventing self-intersection. This research has significant implications for the field of geometry and may lead to further advancements in the study of curve development and self-intersection.

Link to Article: https://arxiv.org/abs/0006035v4 Authors: arXiv ID: 0006035v4