# Shadows of a Closed Curve

## TL;DR

The paper proves that a topologically embedded circle in Euclidean space can have at most two simple-path shadows in linearly independent directions, extending previous results from three dimensions.

## Contribution

It generalizes a prior result by showing the maximum number of simple-path shadows for embedded circles in arbitrary dimensions.

## Key findings

- Maximum of two simple-path shadows in linearly independent directions
- Extension of previous 3D result to higher dimensions
- Topological proof using degree of maps on a circle

## Abstract

A shadow of a geometric object $A$ in a given direction $v$ is the orthogonal projection of $A$ on the hyperplane orthogonal to $v$. We show that any topological embedding of a circle into Euclidean $d$-space can have at most two shadows that are simple paths in linearly independent directions. The proof is topological and uses an analog of basic properties of degree of maps on a circle to relations on a circle. This extends a previous result which dealt with the case $d=3$.