Efficient algorithms for beacon routing in polygons

In a paper by Biro et al. [7], a novel twist on guarding in art galleries is introduced. A beacon is a fixed point with an attraction pull that can move points within the polygon. Points move greedily to monotonically decrease their Euclidean distance to the beacon by moving straight towards the beacon or sliding on the edges of the polygon. The beacon attracts a point if the point eventually reaches the beacon. Unlike most variations of the art gallery problem, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. For a given point in the polygon, the inverse attraction region is the set of beacon locations that can attract the point. We first study the characteristics of beacon attraction. We consider the quality of a "successful" beacon attraction and provide an upper bound of $\sqrt{2}$ on the ratio between the length of the beacon trajectory and the length of the geodesic distance in a simple polygon. In addition, we provide an example of a polygon with holes in which this ratio is unbounded. Next we consider the problem of computing the shortest beacon watchtower in a polygonal terrain and present an $O(n \log n)$ time algorithm to solve this problem. In doing this, we introduce $O(n \log n)$ time algorithms to compute the beacon kernel and the inverse beacon kernel in a monotone polygon. We also prove that $\Omega(n \log n)$ time is a lower bound for computing the beacon kernel of a monotone polygon. Finally, we study the inverse attraction region of a point in a simple polygon. We present algorithms to efficiently compute the inverse attraction region of a point for simple, monotone, and terrain polygons with respective time complexities $O(n^2)$, $O(n \log n)$ and $O(n)$. We show that the inverse attraction region of a point in a simple polygon has linear complexity and the problem of computing the inverse attraction region has a lower bound of $\Omega(n \log n)$ in monotone polygons and consequently in simple polygons.

"Fit Cradle of the Reviving Art": An inquiry into the the Evolving Material Reality of the Murals in the Camposanto of Pisa from the Perspective of their British Reception from the 17th to the 19th century

The Camposanto of Pisa is an extraordinarily complex and evocative monument, which has captured the imagination of pilgrims, both religious and secular, for centuries. The late Medieval and early Renaissance wall paintings that line the perimeter of the portico surrounding a vast inner courtyard, are unparalleled in early Italian art, not only for their striking variety of composition and narrative complexity, but also for the sheer grandeur of their proportion. However, the passage of time has scarred the structure of the Camposanto and inflicted terrible damage on its wall paintings. This thesis explores the material reality of the Camposanto as experienced over three centuries through the eyes of British travelers. In order to situate the Camposanto mural cycle within an historical and cultural context, the first chapter provides an overview of the construction and decoration of the monument. Notably, Giorgio Vasari (1511-1574), the Italian Humanist often recognized as the father of art history, included numerous descriptions of the Camposanto murals in his highly influential text Vite de' più eccellenti pittori, scultori, ed architettori. Accordingly, the second chapter provides an analysis of Vasari’s descriptions and reflects upon the influence that the Renaissance author may have had upon the subsequent British reception of the Camposanto murals. The third chapter utilizes three centuries of travel writing in order to investigate the aesthetic impact of the Camposanto mural cycle upon British tourists from the seventeenth through to the nineteenth century.