The Voronoi Diagram of Rotating Rays With applications to Floodlight Illumination

We introduce the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are rays, and the distance function is the counterclockwise angular distance between a point and a ray-site. This novel Voronoi diagram is motivated by illumination and coverage problems, where a domain has to be covered by floodlights (wedges) of uniform angle, and the goal is to find the minimum angle necessary to cover the domain. We study the diagram in the plane, and we present structural properties, combinatorial complexity bounds, and a construction algorithm. If the rays are induced by a convex polygon, we show how to construct the ray Voronoi diagram within this polygon in linear time. Using this information, we can find in optimal linear time the Brocard angle, the minimum angle required to illuminate a convex polygon with floodlights of uniform angle. This last algorithm improves upon previous results, settling an interesting open problem

Brocardove točke i Brocardov kut

Brocardova točka trokuta jedna je od osobitih točaka trokuta, a definira se kao točka $\Omega$ unutar trokuta ABC za koju duzine $\bar{PA}, \bar{PB}$ i $\bar{PC}$ zatvaraju sukladne kutove sa stranicama $\bar{AB}, \bar{BC}$ i $\bar{CA}$, redom. Pripadne sukladne kutove nazivamo Brocardov kut trokuta. Na analogan način, promjenom orijentacije trokuta, definira se i druga Brocardova točka. Pojam Brocardove točke trokuta moguće je generalizirati i na poligon te tada govorimo o Brocardovoj točki i Brocardovom kutu poligona. Za razliku od trokuta koji uvijek ima Brocardovu točku i to jedinstvenu, za općenite poligone s $n \geq 4$, točka i kut s odgovarajućim svojstvom mogu, ali ne moraju postojati. U prvom dijelu rada izloženi su osnovni, dobro poznati rezultati o Brocardovoj točki i Brocardovom kutu trokuta: dokaz postojanja pomoću konstrukcije, ocjena veličine Brocardova kuta i formula $ctg \omega = ctg \alpha + ctg \beta + ctg \gamma$, gdje su $\alpha, \beta, \gamma$ kutovi trokuta. U drugom dijelu prikazani su neki rezultati o Brocardovoj točki i Brocardovom kutu općenitog poligona. Postojanje Brocardove točke istražuje se pomoću niza poligona dobivenih tzv. Brocardovom transformacijom koji konvergira prema nepraznom skupu, a taj se ili sastoji od jedne točke ili je segment. Dokazan je kriterij postojanja Brocardove točke pomoću sličnosti promatranog poligona s bilo kojom njegovom Brocardovom transformacijom. U slučaju postojanja Brocardove točke pokazano je i svojstvo stabilnosti, što znači da sve Brocardove transformacije poligona imaju zajedničku Brocardovu točku. Jedan od glavnih navedenih i dokazanih rezultata je veličina Brocardovog kuta $\omega$ za $n$-poligon koja glasi $\omega \leq \frac{\pi}{2} - \frac{\pi}{n}$, pri čemu jednakost vrijedi ako i samo ako je poligon pravilan. Tek je nedavno uočeno da ta ocjena jednostavno slijedi iz znatno općenitijeg rezultata Dmitrieva i Dynkina iz 1945. godine. Na kraju rada navedeni su primjeri nepravilnih poligona koji imaju Brocardovu točku.Brocard point is one of the many special points of a triangle. It is defined as a point $\Omega$ inside a triangle ABC such that line segments $\bar{PA}, \bar{PB}$ and $\bar{PC}$ form congruent angles with sides $\bar{AB}, \bar{BC}$ and $\bar{CA}$, respectively. Corresponding congruent angles are called Brocard angle. By reversing the order of vertices we obtain second Brocard point. It is possible to generalize the concept of the Brocard point to n-polygons. As opposed to the case of a triangle, where a unique Brocard point always exists, for n-polygons where $n \geq 4$ a point and an angle with corresponding properties may exist, but not necessarily. In the first chapter we explain the basic, well-known results on the Brocard point and Brocard angle of a triangle: proofs of existence by construction, the upper bound for Brocard angle and the formula $ctg \omega = ctg \alpha + ctg \beta + ctg \gamma$; where $\alpha, \beta, \gamma$ are angles of a triangle. In the second chapter we present some results about the Brocard point and the Brocard angle of a general polygon. The existence of the Brocard point is investigated using a sequence of the polygons obtained by the so called Brocard transformation, that converges to a nonempty set, which is either a one point set or a line segment. A Criterion for the existence of the Brocard point is proven, expressed in terms of similarity between the given polygon and any of its Brocard transforms. In the case of existence of the Brocard point the stability property is also shown, meaning that all the Brocard transforms have the same Brocard point. One of the main results proven for $n$-polygons which have the Brocard point is the estimate of the Brocard angle $\omega$, showing that $\omega \leq \frac{\pi}{2} - \frac{\pi}{n}$ where equality holds only for a regular polygon. It was recently noticed that this estimate easily follows from a much more general result by Dmitriev and Dynkin (1945.). This chapter ends with example of irregular polygons for which the Brocard points exist

Brocardove točke i Brocardov kut

Brocardova točka trokuta jedna je od osobitih točaka trokuta, a definira se kao točka $\Omega$ unutar trokuta ABC za koju duzine $\bar{PA}, \bar{PB}$ i $\bar{PC}$ zatvaraju sukladne kutove sa stranicama $\bar{AB}, \bar{BC}$ i $\bar{CA}$, redom. Pripadne sukladne kutove nazivamo Brocardov kut trokuta. Na analogan način, promjenom orijentacije trokuta, definira se i druga Brocardova točka. Pojam Brocardove točke trokuta moguće je generalizirati i na poligon te tada govorimo o Brocardovoj točki i Brocardovom kutu poligona. Za razliku od trokuta koji uvijek ima Brocardovu točku i to jedinstvenu, za općenite poligone s $n \geq 4$, točka i kut s odgovarajućim svojstvom mogu, ali ne moraju postojati. U prvom dijelu rada izloženi su osnovni, dobro poznati rezultati o Brocardovoj točki i Brocardovom kutu trokuta: dokaz postojanja pomoću konstrukcije, ocjena veličine Brocardova kuta i formula $ctg \omega = ctg \alpha + ctg \beta + ctg \gamma$, gdje su $\alpha, \beta, \gamma$ kutovi trokuta. U drugom dijelu prikazani su neki rezultati o Brocardovoj točki i Brocardovom kutu općenitog poligona. Postojanje Brocardove točke istražuje se pomoću niza poligona dobivenih tzv. Brocardovom transformacijom koji konvergira prema nepraznom skupu, a taj se ili sastoji od jedne točke ili je segment. Dokazan je kriterij postojanja Brocardove točke pomoću sličnosti promatranog poligona s bilo kojom njegovom Brocardovom transformacijom. U slučaju postojanja Brocardove točke pokazano je i svojstvo stabilnosti, što znači da sve Brocardove transformacije poligona imaju zajedničku Brocardovu točku. Jedan od glavnih navedenih i dokazanih rezultata je veličina Brocardovog kuta $\omega$ za $n$-poligon koja glasi $\omega \leq \frac{\pi}{2} - \frac{\pi}{n}$, pri čemu jednakost vrijedi ako i samo ako je poligon pravilan. Tek je nedavno uočeno da ta ocjena jednostavno slijedi iz znatno općenitijeg rezultata Dmitrieva i Dynkina iz 1945. godine. Na kraju rada navedeni su primjeri nepravilnih poligona koji imaju Brocardovu točku.Brocard point is one of the many special points of a triangle. It is defined as a point $\Omega$ inside a triangle ABC such that line segments $\bar{PA}, \bar{PB}$ and $\bar{PC}$ form congruent angles with sides $\bar{AB}, \bar{BC}$ and $\bar{CA}$, respectively. Corresponding congruent angles are called Brocard angle. By reversing the order of vertices we obtain second Brocard point. It is possible to generalize the concept of the Brocard point to n-polygons. As opposed to the case of a triangle, where a unique Brocard point always exists, for n-polygons where $n \geq 4$ a point and an angle with corresponding properties may exist, but not necessarily. In the first chapter we explain the basic, well-known results on the Brocard point and Brocard angle of a triangle: proofs of existence by construction, the upper bound for Brocard angle and the formula $ctg \omega = ctg \alpha + ctg \beta + ctg \gamma$; where $\alpha, \beta, \gamma$ are angles of a triangle. In the second chapter we present some results about the Brocard point and the Brocard angle of a general polygon. The existence of the Brocard point is investigated using a sequence of the polygons obtained by the so called Brocard transformation, that converges to a nonempty set, which is either a one point set or a line segment. A Criterion for the existence of the Brocard point is proven, expressed in terms of similarity between the given polygon and any of its Brocard transforms. In the case of existence of the Brocard point the stability property is also shown, meaning that all the Brocard transforms have the same Brocard point. One of the main results proven for $n$-polygons which have the Brocard point is the estimate of the Brocard angle $\omega$, showing that $\omega \leq \frac{\pi}{2} - \frac{\pi}{n}$ where equality holds only for a regular polygon. It was recently noticed that this estimate easily follows from a much more general result by Dmitriev and Dynkin (1945.). This chapter ends with example of irregular polygons for which the Brocard points exist