This chapter discusses coordinate geometry of three dimensions. Mutually perpendicular straight lines intersecting at the origin are called the coordinate axes, and they determine three mutually perpendicular planes called the coordinate planes. The position of any point in space is determined uniquely by...
Cartesian coordinate systems have two great virtues: (1) they are uniform, with the local geometry identical at all points, and (2) they are orthogonal; in two dimensions, the lines of constant x are perpendicular to the lines of constant y. In three dimensions, the points of constant x ...
Analytic Geometry> There are several different ways to define a bipolar coordinate system. Two of the most common are Grubert’s bipolar system and the bipolar system of two concentric circles. Grubert’s Bipolar System Thebipolar coordinate system, created by Grubert in 1859 [1], is an unus...
The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been...
The face centered cubic grid is the densest possible packing in three dimensions [20]. The shape of the cells in an FCC grid is rhombic dodecahedron, which is a space filling polyhedron described by 12 faces, 24 edges, and 14 vertices. The FCC grid can be seen as the union of four di...
More re- cently, GoMatch [67] and MeshLoc [40] removed the need to store descriptors entirely by matching against the scene geometry. None of the aforementioned strategies reduce the mapping time – on the converse, often they incur additional post-proces...
the spaces of interest need not simply be the physical space of the real world. One can imagine the temperature-pressure-density space of thermodynamics or many of the other spaces where the dimensions are physical variables. One of the most important of these spaces for mechanics is phase spac...
Coordinate measuring machines are used to check workpieces, for example as part of quality assurance, or to determine the geometry of a workpiece completely as part of what is known as “reverse engineering”. Moreover, multifarious further application possibilities are conceivable. ...
Interestingly, subsets of MUC1- or PSA-secreting cells expressed a breast stem cell-like phenotype (MUC1/CK19) or secreted FGF-2 as factor relevant for the growth of DTC, respectively. A significant fraction of cancer patients harbor viable and biologically active tumor cells in their blood ...
Figure 7. Initial position of the two geometries used for the simulation of molecular adsorption on graphene. As referred to in the text, (a) Geometry 1, (b) top view of the selected part of (a), with the graphene substrate in light blue; the closest carbon atom of the molecule to ...