Equal Aperture Angles Curve for some Convex Sets in the Plane

For given convex set $K$ in the plane (not necessarily bounded), we can construct the curve $C$ for which the visibility (aperture) angle of this set has the same, prescribed value. We give the implicit formula for $C$, discuss some issues concerning practical computations of $C$ and bring several simple examples, when the equal visibility angle curves can be effectively computed explicitly. We conclude with some remarks about possible directions for further research in this area. Extensive use of Open Source software (Sage, Pylab, IPython,…) is a key feature of this article.

💡 Research Summary

The paper investigates the geometric locus of points from which a given convex set K in the plane is seen under a constant aperture (visibility) angle α. The visibility angle φ(X,K) of a point X∉K is defined as the angle of the smallest cone with apex X that contains K. For a prescribed angle α (0 ≤ α < π) the set C(α,K) = {X ∉ K | φ(X,K)=α} is called the “constant visibility‑angle curve”.

The authors first treat a simple wedge W with interior angle β. Depending on the relation between α and β, C(α,W) is either empty (α<β), the wedge itself rotated by π (α=β), or a new wedge opposite to W with angle 2α−β (α>β). This illustrates that C can be a two‑dimensional region, a one‑dimensional curve, or even degenerate.

Next, they consider the unbounded strictly convex set Q = {(x,y) ∈ ℝ² | y ≥ x²}, i.e., the parabola y = x² together with its interior. For an exterior point A = (x₁,y₁) the two tangent lines to Q have slopes

k₁ = 2(x₁ + √(x₁² − y₁)), k₂ = 2(x₁ − √(x₁² − y₁)).

The condition that the angle between these tangents equals α translates into

(k₁−k₂)/(1 + k₁k₂) = tan α = K.

When k₁k₂ ≠ −1, substitution yields an explicit quadratic relation for the curve:

yₚ(x) = −