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Simplifying intersections of surfaces.
1DEPARTMENT OF MATHEMATICS, FLORIDA STATE UNIVERSITY, TALLAHASSEE.
Summary
This study explores geometric properties of intersecting arcs within disk boundaries. It demonstrates that the union arc either lies on a disk boundary or is formed by two disks meeting at the intersection arc.
Area of Science:
- Topology
- Geometric Analysis
- Differential Geometry
Background:
- Investigates the topological and geometric properties of arcs and disks.
- Builds upon existing research in piecewise linear topology.
Purpose of the Study:
- To analyze the relationship between the union and intersection of two arcs (mu, v) within disk boundaries (M, N).
- To establish conditions under which the union arc (alpha) or the constituent arcs lie on disk boundaries.
Main Methods:
- The study employs geometric constructions and topological arguments.
- It analyzes the configuration of two disks (M, N) and their boundary arcs (mu, v).
Main Results:
- It is proven that either the union arc (alpha) lies on a disk boundary, or the original disks (M, N) can be replaced by new disks (M', N') whose boundaries meet only at the intersection arc (beta).
Conclusions:
- The findings provide a fundamental construction for understanding the boundaries of intersecting disks.
- This work facilitates the exploration of new problems in geometric topology, extending beyond the piecewise linear category.