## 1 Introduction  
Wireless sensor networks help us remotely explore unknown fields or dangerous areas such as military areas, hazard environment sensing, and high-temperature industrial monitoring. These tiny devices can monitor the information of their locations and deliver the collected data to the data center via the wireless communication technique. These sensing techniques can reduce the exploring workforce and prevent humans from stepping into dangerous regions.  
Numerous studies have focused on detecting the boundary area of the intangible continuous object. Some of these studies organize the deployed sensors into multiple clusters [1, 2]. The methods in these studies provide rough boundary results of the continuous object. Their main objective is to track the boundary of a mobile object. Similar proposed studies trace the boundary of a mobile object by using grid architecture [3, 4]. These methods divided the deployed region into multiple grids. The sensor in each grid reports whether the continuous object exists or not. The grid size determines the precision of the detected boundary. For refining the detected results, each grid is further divided into multiple smaller grids [4] to improve the detection precision.  
In addition, many boundary detection studies used planarization algorithms to figure out the boundary of the continuous object [5, 6, 7, 8, 9, 10, 11]. The sensors in these studies are stationary after being deployed. Their proposed methods treat the sensors as vertices on the planer graph. They create a coarse boundary region to detect the boundary of the continuous objects. The sensors that are covered by the continuous object and close to the boundary line are the inner boundary nodes, and the sensors, which are near the boundary line but uncovered by the continuous object, are outer boundary nodes. All these inner and outer boundary nodes are involved in building the boundary faces. Then, these methods applied the planarization algorithms, such as the Gabriel Graph (GG), Relative Neighborhood Graph (RNG), and k-Localized Delaunay Graph (\(LDel^{k}\)), to create the boundary faces. Considering the communication latency caused by delivering the positions of the sensors, [PERSON] et al. proposed the farthest-first routing technique to lower the latency [12].  
Generally, these planarization algorithms assume that the sensors' communication range is infinity. The assumption that every sensor can directly communicate with others without considering the physical distance is unrealistic. While considering the limited communication range, the network may determine that the sensors cannot return their positions to the controller. In this scenario, the planarization algorithms will lose their functionality.  
Therefore, some studies proposed using mobile sensors to refine the detected boundary of the continuous object [13; 14; 15]. These methods use a set of mobile sensors traveling within the deploying area to refine the boundary that the stationary sensors have roughly figured out [13; 14], or use all sensors to explore all areas of the continuous object. When the continuous object has a long boundary line, these methods may lose their timeliness.  
The above methods using stationary sensors to detect the boundary without considering the sensors' limited communication range are impractical. Therefore, this paper assumes that the sensors have a limited communication range and gives them the mobile ability to refine the boundary precision. The proposed methods applied the convex hull and the Delaunay triangulation methods to estimate the initial area of the continuous object. Next, the sensors' moving algorithm is adapted to refine the contour of the intangible continuous object (ICO).