Spaces of Geodesic Triangulations of Surfaces

We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any $$n>0$$ n > 0 , we show that there exists a space of geodesic triangulations of a polygon with a triangulation, whose n-th homotopy group is not trivial.

The Perceived Beauty of Convex Polygon Tilings: The Influence of Regularity, Curvature, and Density

Polygon tilings in natural and man-made objects show great variety. Unlike previous studies that have mainly focused on their classification and production methods, this study aimed at exploring factors that may contribute to the perceived beauty of convex polygon tilings. We analyze the dimensions of regularity, curvature, and density, as well as individual differences. Triangle tilings and hexagon tilings were tested in Experiment 1 and 2, respectively. The results showed that the perceived beauty of convex polygon tilings can be enhanced by higher levels of regularity and nonobvious local curvature. Surprisingly, the effect of density appeared to be different, with the dense triangle tilings and the less dense hexagon tilings scoring higher than the reverse. We discuss a possible explanation based on trypophobia caused by different types of polygons, as well as the observers’ personality trait of agreeableness.

An extremum problem for the power moment of a convex polygon contained in a disc

In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.

WILAYAH JELAJAH DAN AKTIVITAS HARIAN ELANG JAWA (Nisaetus bartelsi STRESEMANN, 1924) DI BUKIT MAYANA KECAMATAN KADUGEDE, KABUPATEN KUNINGAN

Abstrak : Elang jawa (Nisaetus bartelsi) merupakan jenis raptor dengan penyebaran terbatas (endemik). Elang jawa dilindungi oleh Peraturan Menteri Lingkungan Hidup dan Kehutanan Republik Indonesia Nomor P.92 tahun 2018 dan termasuk kategori endangered menurut IUCN Redlist (2017). Tujuan penelitian ini untuk mengetahui wilayah jelajah elang jawa dan aktivitas hariannya. Penelitian dilaksanakan di Bukit Mayana Kecamatan Kadugede, Kabupaten Kuningan selama 4 bulan dari Desember 2017 sampai Maret 2018. Metode yang digunakan yaitu Cooperative Method dan Focal Animal Sampling. Analisis wilayah jelajah menggunakan Minimum Convex Polygon dan Kernel Density Estimation dan analisis deskriptif untuk aktivitas harian. Berdasarkan hasil penelitian dan analisis didapatkan wilayah jelajah elang jawa dengan MCP individu Maya dan Yana masing-masing seluas 79.8 ha dan 189.9 serta KDE seluas 118.6 ha dan 180.6 ha. Aktivitas harian elang jawa di bukit Mayana yang paling banyak terlihat adalah terbang dengan presentase 79% sedangkan ketika tengger hanya sebesar 21%. Intensitas terbang paling tinggi mulai dari jam 09:00-11:30.Kata Kunci: Aktivitas harian, Wilayah jelajah, kernel density estimation, , Nisaetus bartelsi

Route-Planning Method for Plant Protection Rotor Drones in Convex Polygon Regions

Aiming at the problem of low operating efficiency due to the poor endurance of plant protection rotor drones and the small volume of pesticide carried, this paper proposes a route-planning algorithm for convex polygon regions based on the initial heading angle. First, a series of coordinate conversion methods ranging from the Earth coordinate system to the local plane coordinate system are studied. Second, in the local plane coordinate system, a route generation method based on subregion is proposed; therefore, multiple routes can be generated with different initial heading angles. Lastly, the optimal route and the best initial heading angle can be obtained after the comparison according to the three evaluation criteria: number of turns, route distance, and pesticide waste rate. The simulation results show that, compared with the common grid method, the route generation method based on subregion reduces the route distance and pesticide waste rate by 2.27% and 13.75%, respectively. Furthermore, it also shows that, compared with the route generated by the initial heading angle of 0°, the optimal route reduces the number of turns, route distance, and pesticide waste rate by 60%, 17.65%, and 38.18%, respectively. The route was optimized in three aspects and reached the best overall result using this method, which in turn proved its feasibility.

Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon

A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process