Search Results (88)

A geodesic or geodetic line on a sphere is called the orthodrome. Research has shown that the orthodrome can be defined in a large number of ways. This article provides an overview of various definitions of the orthodrome. We recall the definitions of the orthodrome according to the greats of geodesy, such as Bessel and Helmert. We derive the equation of the orthodrome in the geographic coordinate system and in the Cartesian spatial coordinate system. A geodesic on a surface is a curve for which the geodetic curvature is zero at every point. Equivalent expressions of this statement are that at every point of this curve, the principal normal vector is collinear with the normal to the surface, i.e., it is a curve whose binormal at every point is perpendicular to the normal to the surface, and that it is a curve whose osculation plane contains the normal to the surface at every point. In this case, the well-known Clairaut equation of the geodesic in geodesy appears naturally. It is found that this equation can be written in several different forms. Although differential equations for geodesics can be found in the literature, they are solved in this article, first, by taking the sphere as a special case of any surface, and then as a special case of a surface of rotation. At the end of this article, we apply calculus of variations to determine the equation of the orthodrome on the sphere, first in the Bessel way, and then by applying the Euler–Lagrange equation. Overall, this paper elaborates a dozen different approaches to orthodrome definitions. Full article

This strategic document outlines Bulgaria’s roadmap for modernizing its cartographic sector from 2025 to 2035, addressing the outdated geospatial infrastructure, lack of standardized digital practices, lack of coordinated digital infrastructure, outdated standards, and fragmented data management systems. The strategy was developed in accordance with the national methodology for strategic planning and through preliminary consultations with key stakeholders, including research institutions, business organizations, and public institutions. It aims to build a human-centered, data-driven geospatial framework aligned with global standards such as ISO 19100 and the EU INSPIRE Directive. Core components include: (1) modernization of the national geodetic system, (2) adoption of remote sensing and AI technologies, (3) development of interactive, web-based geospatial platforms, and (4) implementation of quality assurance and certification standards. A SWOT analysis highlights key strengths—such as existing institutional expertise—and critical challenges, including outdated legislation and insufficient coordination. The strategy emphasizes the need for innovation, regulatory reform, inter-institutional collaboration, and sustained investment. It ultimately positions Bulgarian cartography as a strategic contributor to national sustainable development and digital transformation. Full article

This paper proposes a systematic ground experimental method to address the insufficient initial pointing accuracy of optical terminals in free space optical communication (FSO). By utilizing a multi-coordinate system transformation model combined with geodetic coordinates obtained from a Global Navigation Satellite System (GNSS), the elevation and azimuth angles of the optical terminal are calculated to achieve initial pointing. High-precision horizontal installation and true north direction calibration are accomplished using a GNSS dual-antenna system and a digital inclinometer to suppress mechanical installation errors. Furthermore, an iterative stellar calibration method is proposed, leveraging ephemeris to precompute stellar positions and optimize correction values through multiple observations, significantly improving pointing accuracy. In a 104.68 km span experiment conducted in the Qinghai Lake area, the azimuth and elevation angle errors of the optical terminal were reduced to −0.0293° and −0.0068°, respectively, with the uncertainty region narrowed to 0.0586°. These results validate the effectiveness of the proposed method in high-precision rapid link establishment, providing technical support for the engineering application of satellite-to-ground laser communication. Full article

Modern surveying is increasingly focused on fast data acquisition and processing using lightweight, low-cost equipment, particularly for the continuous monitoring of structures and infrastructures. This study investigates the use of LiDAR and RGB sensors embedded in Apple and Android smartphones, paired with an innovative device, the viDoc RTK Rover, for centimeter-level surveying. Three case studies were selected, each characterized by different materials, functional uses, and environmental contexts. The methodology centers on evaluating final accuracy during both the data acquisition and processing phases. Coordinates of target points were obtained directly via the viDoc device and indirectly through dense point clouds. Validation was conducted using a geodetic GNSS receiver. Results demonstrate that, in most cases, the system achieves accuracy comparable to traditional surveying methods. The findings confirm that these emerging tools offer a reliable and efficient solution for rapid 3D surveys with centimeter-level accuracy. Full article

This study evaluates coordinate consistency in the static Korean Geodetic Datum 2002 (KGD2002) by comparing GNSS station positions derived independently from GAMIT/GLOBK and Bernese software. Using a nationwide network of approximately 3000 unified geodetic control points (UGCPs), we analyze horizontal coordinate differences (ΔN, ΔE) to identify regional patterns and potential systematic biases. The results indicate that both solutions are closely aligned with the official KGD2002 coordinates, generally within a few millimeters to sub-centimeter levels. However, small regional discrepancies are evident; for example, some provinces exhibit consistent mean northward or southward offsets on the order of 0.1–0.3 cm, and greater dispersions—up to 2 cm—are observed in peripheral regions such as Jeollanam. Notably, the Bernese solution demonstrates slightly tighter agreement, with lower standard deviations compared to GAMIT/GLOBK. The application of two distinct processing strategies within a unified static reference frame is a novel aspect of this study, revealing subtle differences attributable to network geometry, environmental factors, and software modeling approaches. The findings also underscore the limitations of KGD2002’s static nature, particularly its fixed epoch and lack of motion modeling. In response to these issues, this study discusses the rationale for transitioning to a dynamic geodetic reference frame, such as ITRF2020, to improve compatibility with international systems and account for ongoing crustal motions. Overall, the results provide a foundation for the future modernization of Korea’s spatial reference infrastructure and highlight the importance of adopting time-dependent datums in geodetic applications. Full article

This study examines the influence of gravity on different height systems by integrating Global Navigation Satellite Systems (GNSS), leveling, and gravimetric measurements. Although the theoretical influence of gravity on height systems is well known, empirical studies that quantify these effects along steep terrain are rare—particularly within the Croatian reference systems. Geometric leveling, recognized for its precision in geodesy, was employed alongside gravimetric data to analyze the relationship between gravity variations and height differences. The research was conducted along Sljeme Road on Mt. Medvednica, Croatia, where altitude-dependent gravity effects were systematically investigated along an elevation profile with a height difference of about 650 m. GNSS measurements provided positional coordinates referenced to the Croatian Terrestrial Reference System 1996 (HTRS96) (EPSG:4888), while leveling and gravimetric data were analyzed within the Croatian Height Reference System 1971 (HVRS71) (EPSG:5610) and Croatian Gravimetric Reference System 2003 (HGRS03), respectively. The results demonstrate that differences between geometric and normal–orthometric heights become more pronounced at higher elevations but remain at the millimeter level. Notably, the impact of gravity is evident in normal and orthometric heights, with differences from geometric heights reaching up to 3.7 cm at the highest points. Additionally, a comparison between normal and orthometric heights reveals that at the beginning of the leveling line, the difference is around 4 mm. However, as the elevation increases, this difference grows, reaching over 1 cm at the end of the leveling line. The study also confirms the theoretical correlation between the geoid–quasigeoid height difference and terrain elevation, with increasing differences observed at higher altitudes. To examine the consistency of different height determination methods, two approaches were applied: one based on adjustment within the geopotential system, and the other involving direct adjustment in the desired height system, with specific height corrections applied. The results confirmed that the height differences between the two methods were 0, to the tenth of a millimeter, indicating that both methods provided identical results. These findings contribute to a deeper understanding of geodetic height systems and the role of gravity in height determination. Full article

Traditional 3D radar can only detect the planar characteristic information of a target. Thus, it cannot describe its spatial geometric characteristics, which is critical for accurate vehicle classification. To overcome these limitations, this paper investigates elevation features using 4D millimeter-wave radar data and presents a maximum likelihood estimation (MLE)-based vehicle classification method. The elevation data collected by 4D radar from a real road scenario are applied for further analysis. By establishing radar coordinate systems and geodetic coordinate systems, the distribution feature of vehicles’ elevation is analyzed by spatial geometric transformation referring to the radar installation parameters, and a Gaussian-based probability distribution model is subsequently proposed. Further, the data-driven parameter optimization on likelihood probabilities of different vehicle samples is performed using a large-scale elevation dataset, and an MLE-based vehicle classification method is presented for identifying small and large vehicles. The experimental results show that there are significant differences in elevation distribution from two different vehicle types, where large vehicles exhibit a wider range of left-skewed distribution in different cross-sections, while small vehicles are more concentrated with a right-skewed distribution. The Gaussian-based MLE method achieves an accuracy of 92%, precision of 87% and recall of 98%, demonstrating excellent performance for traffic monitoring and related applications. Full article

Spherical buoys serve as water surface markers, and their location information can help unmanned surface vessels (USVs) identify navigation channel boundaries, avoid dangerous areas, and improve navigation accuracy. However, due to the presence of disturbances such as reflections, water obstruction, and changes in illumination for spherical buoys on the water surface, using binocular vision for positioning encounters difficulties in matching. To address this, this paper proposes a monocular vision-based localization method for spherical buoys using elliptical fitting. First, the edges of the spherical buoy are extracted through image preprocessing. Then, to address the issue of pseudo-edge points introduced by reflections that reduce the accuracy of elliptical fitting, a multi-step method for eliminating pseudo-edge points is proposed. This effectively filters out pseudo-edge points and obtains accurate elliptical parameters. Finally, based on these elliptical parameters, a monocular vision ranging model is established to solve the relative position between the USV and the buoy. The USV’s position from satellite observation is then fused with the relative position calculated using the method proposed in this paper to estimate the coordinates of the buoy in the geodetic coordinate system. Simulation experiments analyzed the impact of pixel noise, camera height, focal length, and rotation angle on localization accuracy. The results show that within a range of 40 m in width and 80 m in length, the coordinates calculated by this method have an average absolute error of less than 1.2 m; field experiments on actual ships show that the average absolute error remains stable within 2.57 m. This method addresses the positioning issues caused by disturbances such as reflections, water obstruction, and changes in illumination, achieving a positioning accuracy comparable to that of general satellite positioning. Full article

Long linear constructions, particularly high-speed railway lines, place very high demands on the accuracy of the fundamental geodetic controls on which they are based. In general, national reference coordinate systems possess deformations (relative inconsistencies) due to historical factors—primarily the low accuracy of the measurement methods that were used at the time of their development. The reference coordinate system S-JTSK used in the Czech Republic comprises local deformations that are determined using a modern space geodesy method—the Global Navigation Satellite Systems (GNSS). An analysis of the local deformations revealed that, in the Czech Republic, 99% of the length corrections from the coordinate system deformations are in the interval of −1.4 to +1.4 cm/km with a standard deviation of up to 1.4 cm/km. The extreme value in this regard is the length correction from the deformation of +2.5 cm/km and the standard deviation of 3.6 cm/km. These values exceed the required accuracy of 1 cm/km, and the length corrections from the deformations, together with the length corrections from the used projection, must be taken into account when undertaking surveying work on high-speed railway lines in the Czech Republic (also with the height correction for lengths). The magnitude of the scatter also indicates the distortion of the S-JTSK projection conformality due to the deformations. Full article

The integration of Global Navigation Satellite Systems (GNSS) has revolutionized geodetic positioning, with techniques like Precise Point Positioning with Ambiguity Resolution (PPP-AR) offering highly accurate results with reduced convergence times. The full deployment of the BeiDou Navigation Satellite System-3 (BDS-3) has spurred interest in assessing its standalone and combined performance with GPS in PPP-AR applications. This study evaluates the performance of BDS-3-based PPP-AR across diverse geographical regions considering space weather conditions (SWCs) for the first time. GNSS data from six International GNSS Service (IGS) stations located in the Asia–Pacific, Europe, Africa, and the Americas were processed for 15 consecutive days. The three scenarios (BDS-3 only, GPS only, and BDS-3 + GPS) were analyzed using the open-source raPPPid v2.3 software developed in 2023. The estimated coordinates were statistically compared to the IGS-derived coordinates to assess accuracy. Results demonstrate that BDS-3 PPP-AR can independently deliver reliable positioning for many applications and that the accuracy of BDS-3-based PPP-AR is relatively low in the Americas. However, combining BDS-3 with GPS significantly enhances horizontal and vertical accuracies, especially in the Americas, achieving improvements of up to 86% and 82%, respectively. These findings highlight the potential of BDS-3 for complementing GPS for precise geodetic applications. Full article

The BeiDou global navigation satellite system (BDS-3) has been widely applied in various geodetic applications since its full operation. However, the estimated station coordinates using BDS-3 are less precise compared to GPS results. It contains systematic errors caused by scale bias with respect to International GNSS Service (IGS) 2020 frame and Inter-System Translation Parameters (ISTPs). In order to improve the consistency of BDS-3-derived station coordinates with respect to IGS20 products, we firstly estimated the satellite antenna Phase Center Offsets (PCOs) for BDS-3 Medium Earth Orbit (MEO) constellation, and then estimated station-specific ISTPs from GPS to BDS-3 systems. The results indicate that the PCO-Z estimates show large differences among satellites from different manufacturers and orbit planes. The estimated BDS-3 satellite PCOs exhibit a systematic bias of −9.3 cm in the Z-direction compared to ground calibrations. The maximum mean station-specific ISTPs can reach up to 3 mm, highlighting significant variability and the need for refinement in positioning. When using the estimated PCOs instead of igs20.atx values, the estimated scale bias with respect to the IGS20 frame is reduced from 0.38 ppb to −0.12 ppb, indicating that the refined BDS-3 satellite PCOs are well compatible with IGS20. Regarding the Up component that is correlated with the scale factor, the station coordinate differences with respect to the IGS20 frame is reduced from 7.0 mm to 6.2 mm in terms of the root mean square (RMS), which is improved by 11.4%. Considering the additional ISTP corrections, a further improvement of 17% was obtained in station coordinates. The RMS of station coordinate differences with respect to the IGS20 frame is 2.3 mm, 2.7 mm, and 5.2 mm for the North, East, and Up components, respectively. Full article

Carrier-phase measurements are essential in precise Global Navigation Satellite System (GNSS) positioning applications. The quality of those observations, as well as the final positioning result, is influenced by an extensive list of GNSS error sources, one of which is the receiver antenna phase center (PC) model. It has been well established that the antenna PC exhibits variability depending on the frequency, direction, and intensity of the incoming GNSS signal. To mitigate the corresponding range errors, phase center corrections (PCCs) are determined through a specialized procedure known as receiver antenna calibration and subsequently applied in data processing. In 2023, the Laboratory for Measurements and Measuring Technique (LMMT) of the Faculty of Geodesy, University of Zagreb, Croatia, initiated the development of a new robotic GNSS receiver antenna calibration system. The system implements absolute field calibration and PCC modeling through triple-difference (TD) carrier-phase observations and spherical harmonics (SH) expansion. This study presents and documents dual-frequency (L1 and L2) Global Positioning System (GPS) calibration results for several distinct receiver antennas. Furthermore, the main goals of this contribution are to evaluate the accuracy of dual-frequency GPS calibration results on the pattern level with respect to independent calibrations obtained from Geo++ GmbH and to extensively experimentally validate LMMT calibration results in the spatial (coordinate) domain, i.e., to investigate how the application of LMMT PPC models reflects on geodetic-grade GNSS positioning. Our experimental research results showed a submillimeter calibration accuracy, i.e., 0.36 mm for GPS L1 and 0.54 mm for the GPS L2 frequency. Furthermore, our field results confirmed that the application of LMMT PCC models significantly increases baseline accuracy and GNSS network solution accuracy when compared to type-mean PCC models of the International GNSS Service (IGS). Full article

Selecting an appropriate bra product has long been a challenge for adolescent girls, whose breasts are rapidly growing. This challenge arises due to the absence of a rational benchmark for breast measurement. Traditional benchmarks are based on ergonomic principles, using reference points located on the human skeleton. However, the breast lacks a bone structure for support, leading to highly variable measurement results. In this study, the jugular notch and the xiphoid process were selected as breast measurement points according to academic principles. Their accuracy, stability, and deviation were experimentally investigated through tactile assessment of adolescent girls at different stages of development. Based on the experimental results, a novel breast measurement coordinate system was established, and its accuracy was verified by conversion calculations using a geodetic coordinate system. The results indicated that the numerical calculation of the breast shape was more accurate using the newly established breast measurement coordinate system. Full article

Vertical displacements are traditionally measured with precise levelling, which is inherently time consuming. Rapid or even real-time height determination can be achieved by the Global Navigation Satellite System (GNSS). Nevertheless, the accuracy of real-time GNSS positioning is limited, and the deployment of a network of continuously operating GNSS receivers is not cost effective unless low-cost GNSS receivers are considered. In this study, we examined the use of geodetic-grade and low-cost GNSS receivers for static and real-time GNSS levelling, respectively. The results of static GNSS levelling were processed in four different software programs or services. The largest differences for ellipsoidal/normal heights reached 0.054 m/0.055 m, 0.046 m/0.047 m, and 0.058 m/0.058 m for points WRO1, BM_ROOF, and BM_CP, respectively. In addition, the values depended on the software used and the location of the point. However, the multistage experiment was designed to analyze various strategies for GNSS data processing and to define a method for detecting vertical displacement in a time series of receiver coordinates. The developed method combined time differentiation of coordinates estimated for a single GNSS receiver using the Precise Point Positioning (PPP) technique and Butterworth filtering. It demonstrated the capability of real-time detection of six out of eight displacements in the range between 20 and 55 mm at the three-sigma level. The study showed the potential of low-cost GNSS receivers for real-time displacement detection, thereby suggesting their applicability to structural health monitoring, positioning, or early warning systems. Full article

This paper shows the results obtained by using MATLAB/Simulink and X-Plane as co-simulation tools for the comparison of control schemes for takeoff and landing maneuvers of a quadrotor. Two control schemes based on nested saturations are compared to ensure the convergence of $\theta$ and $\varphi$ angles to the equilibrium point, each with its own specific characteristics in its design and tuning procedure. Furthermore, in both proposals, a Generalized Proportional Integral (GPI) control is used for the height part, while a feedforward PID control is used for the $\psi$ angle. The control schemes are proposed from a local geodetic coordinate system East, North, Up (ENU). Feedback data for the control schemes are obtained from X-Plane via User Datagram Protocol (UDP)-based interface; they are used in MATLAB/Simulink for the calculation of the control actions; the control actions are then entered into a transformation matrix that converts the actions into rotor angular velocities, which are sent to X-Plane. Several numerical simulations are presented to demonstrate the effectiveness and robustness of the proposed schemes, considering the presence of disturbances mainly due to wind speed. Finally, different performance indices are used to evaluate the schemes based on error; in this way, the use of X-Plane as a Model-in-Loop (MIL) environment is validated, which helps to identify errors or problems of the proposed controllers before their coding and physical implementation. Full article