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Overview of GNSS
Published in Basudeb Bhatta, Global Navigation Satellite Systems, 2021
A reference point (also known as control point in surveying) is a location used to determine (or express) the location or position of another one, by giving the relative position. To determine the location of one object we need to refer to reference points or locations of other objects. For example, to define the location of a glass, one might say ‘on the table’; hence the table is a reference point which has a location. In this example, however, the position of the glass is not very precise. One may ask ‘where on the table?’ Therefore, for precise determination of position, we need to adopt a geometrical approach like, ‘30 cm from left edge of the table, 20 cm from front edge of the table, and on top of the table’; and thus, it defines the exact position.
Coordinate Systems and Vector Algebra
Published in Ahmad Shahid Khan, Saurabh Kumar Mukerji, Electromagnetic Fields, 2020
Ahmad Shahid Khan, Saurabh Kumar Mukerji
Geometry is a branch of mathematics that deals with the measurement, property, and relations of lines, angles, surfaces, and volumes. A practical device or part thereof in which the field distribution is to be studied for its behavior requires representation of its shape and size in such a way that all its physical aspects fully conform to the geometry. The system that deals with geometrical aspects of an object is referred to as a coordinate system. The necessity to enhance these limits has led to the development of number of tools. A coordinate system helps us to visualize relative positions of independent points or those belong to a line, a surface or a volume. Thus, while introducing the coordinate systems, it is presumed that a reader fully understands the meanings of point, line, surface, and volume, and also is aware of normal, tangent, and geometrical properties. Furthermore, as and when any point on, or a segment in terms of line, area, or volume of, the system is to be represented (or identified), it should fit in well in the coordinate system selected. The following subsections describe the types of coordinate systems.
Polar coordinates
Published in W. Bolton, Mathematics for Engineering, 2012
The Cartesian coordinate system enables the position of a point on a plane to be specified by stating its x and y coordinates, i.e. the displacement of the point relative to two perpendicular axes. There is another way of specifying the position of a point on a plane and that is to use polar coordinates. A reference axis is specified as emanating from some fixed point O, often referred to as the pole. A line is then drawn from O to the point being specified. The location of the point is then specified by the angle θ that line makes with the reference axis and the distance along that line to the point, i.e. the radius r of a circle with its centre at O and which would pass through the point concerned (Figure 16.1). The polar coordinates are then (r, θ). The angle θ is a positive angle when measured anticlockwise from the reference axis and negative when measured in a clockwise direction. The angles can be specified in either degrees or radians. This chapter is a consideration of the polar coordinate method of specifying points on a plane and the relationship between polar and Cartesian coordinates.
Direct 3D coordinate transformation based on the affine invariance of barycentric coordinates
Published in Journal of Spatial Science, 2021
Whether in the two-dimensional space or the three-dimensional space, to describe the position of a material point, it is an essential prerequisite to establish a corresponding coordinate system, such as a Cartesian coordinate system, a polar coordinate system or a spherical coordinate system, and it is also indispensable to specify the origin of the coordinate system and mutually orthogonal unit vectors. Unlike traditional practices, the barycentric coordinates can locate the position of a point through the existing points (also referred to as reference points) rather than the origin, which are referred to as local coordinates as well (Vince 2006). The barycentric coordinates were proposed by the German mathematician (Möbius 1827) and have been successfully applied to many fields, such as 2D datum transformation in geodetic networks (Ansari et al. 2018), texture mapping in computer graphics (Hormann and Sukumar 2017), PnP (Perspective-n-Points) in computer vision (Lepetit et al. 2009), LiDAR point cloud filtering (Gézero and Antunes 2018), unmixing of hyperspectral remote sensing images (Honeine and Richard 2012), and real-time path planning for unmanned aerial vehicles (Zollars et al. 2018).
An indoor spatial accessible area generation approach considering distance constraints
Published in Annals of GIS, 2020
Lina Yang, Hongru Bi, Xiaojing Yao, Wei Chen
Geometric model uses the coordinate information in Euclidean space to express the position of spatial objects. Based on the coordinates, the distances among objects can be calculated accurately. But the relationship among them, such as connectivity and adjacency, cannot be represented by geometric model (Lee et al. 2014; Teo and Cho 2016).