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14 pages, 300 KB

Open AccessFeature PaperArticle

Translative Covering a Square with Isosceles Right Triangles

by Janusz Januszewski and Łukasz Zielonka

Geometry 2026, 3(2), 8; https://doi.org/10.3390/geometry3020008 - 13 Apr 2026

Viewed by 805

A translative covering the rectangle

a \times b

with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of

a \times b

) is considered. It is shown that any collection of equal triangles homothetic to T [...] Read more.

A translative covering the rectangle

a \times b

with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of

a \times b

) is considered. It is shown that any collection of equal triangles homothetic to T with the total area at least 2 permits a translative covering of

1 \times 1

; this bound is tight. It is also demonstrated that any collection of positive homothetic copies of T with the total area at least 3 permits a translative covering of

1 \times 1

. Moreover, it is proven that if

a \geq \frac{5 + \sqrt{33}}{4} b

, then any collection of triangles homothetic to T with the total area at least

\frac{1}{2} {(a + b)}^{2}

permits a translative covering of

a \times b

; this bound is tight. Full article

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15 pages, 4618 KB

Open AccessArticle

HTCRL: A Range-Free Location Algorithm Based on Homothetic Triangle Cyclic Refinement in Wireless Sensor Networks

by Dan Zhang, Zhiyi Fang, Hongyu Sun and Jie Cao

Information 2017, 8(2), 40; https://doi.org/10.3390/info8020040 - 27 Mar 2017

Cited by 3 | Viewed by 5421

Abstract

Wireless sensor networks (WSN) have become a significant technology in recent years. They can be widely used in many applications. WSNs consist of a large number of sensor nodes and each of them is energy-constrained and low-power dissipation. Most of the sensor nodes are tiny sensors with small memories and do not acquire their own locations. This means determining the locations of the unknown sensor nodes is one of the key issues in WSN. In this paper, an improved APIT algorithm HTCRL (Homothetic Triangle Cyclic Refinement Location) is proposed, which is based on the principle of the homothetic triangle. It adopts perpendicular median surface cutting to narrow down target area in order to decrease the average localization error rate. It reduces the probability of misjudgment by adding the conditions of judgment. It can get a relatively high accuracy compared with the typical APIT algorithm without any additional hardware equipment or increasing the communication overhead. Full article

(This article belongs to the Special Issue Sensor Networks for Emergent Technologies)

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