[Paper Review] Bodies invisible from one point

Alexander Plakhov, Vera Roshchina|arXiv (Cornell University)|Dec 28, 2011

Advanced Mathematical Theories and Applications6 references3 citations

TL;DR

This paper constructs a connected three-dimensional body with a mirror surface that is invisible from a single point in the framework of geometrical optics. Using confocal ellipses and hyperbolas in a rotational construction, the authors prove that billiard trajectories from a focal point emerge along the same ray after reflections, making the body undetectable from that viewpoint.

ABSTRACT

We show that there exist bodies with mirror surface invisible from a point in the framework of geometrical optics. In particular, we provide an example of a connected three-dimensional body invisible from one point.

Motivation & Objective

• To demonstrate the existence of bodies invisible from a single point under geometrical optics assumptions.
• To extend previous results on invisibility in one direction to point-based invisibility in higher dimensions.
• To provide a constructive proof using billiard dynamics and conic section properties.
• To show that such invisible bodies can be connected in three or more dimensions.
• To establish a direct geometric and dynamical mechanism for ray redirection that preserves the initial ray direction.

Proposed method

• Construct a 2D body from the union of two arcs of confocal ellipses and hyperbolas, with foci at $F_1$ and $F_2$.
• Use the focal properties of ellipses ($|F_1C| + |F_2C| = \text{const}$) and hyperbolas ($|F_1D| - |F_2D| = \text{const}$) to control trajectory behavior.
• Apply Lemma 1, which characterizes angle bisectors in triangles via the algebraic condition $(a_1 + b_1)(a_2 - b_2) = f^2$, to verify reflection symmetry.
• Prove that after two reflections, the outgoing ray from $F_1$ lies along the same ray as the initial incoming ray, ensuring invisibility.
• Extend the 2D construction to 3D by rotating the body around the axis $F_1F_2$, preserving rotational symmetry and invisibility.
• Verify that all trajectories from $F_1$ emerge along the same initial ray after reflections, due to the geometric constraints and symmetry.

Experimental results

Research questions

• RQ1 Can a connected three-dimensional body be invisible from a single point under geometrical optics?
• RQ2 What geometric and dynamical conditions ensure that all rays from a point emerge along the same initial direction after reflections?
• RQ3 How can confocal conic sections be used to construct a body that redirects all incident rays from a focal point to re-emerge along the same ray?
• RQ4 What role do the focal properties of ellipses and hyperbolas play in achieving invisibility from a point?
• RQ5 Is it possible to generalize 2D invisibility constructions to higher dimensions while preserving connectivity and invisibility?

Key findings

• A connected three-dimensional body with a mirror surface can be constructed that is invisible from a single point in geometrical optics.
• The construction relies on confocal ellipses and hyperbolas in a 2D cross-section, with the body formed by rotating this shape about the line joining the foci.
• All billiard trajectories emanating from one focus $F_1$ reflect in such a way that the initial and final segments lie on the same ray, making the body undetectable from that point.
• The key geometric condition ensuring this behavior is the identity $(|F_1C| + |F_2C|)(|F_1D| - |F_2D|) = |F_1F_2|^2$, derived from the focal properties of conic sections.
• The proof uses a lemma that characterizes angle bisectors in triangles via the algebraic relation $(a_1 + b_1)(a_2 - b_2) = f^2$, which confirms that the reflection path preserves the ray direction.
• The construction is defined by three independent parameters: scale, and two angles ($\measuredangle HF_1F_2$ and $\measuredangle BF_1F_2$), with two constraints from the confocal and intersection conditions (1) and (2).