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General introduction
Published in Adedeji B. Badiru, Handbook of Industrial and Systems Engineering, 2013
Hyperboloid of one sheet x2a2+y2b2-z2c2=1
General Characteristics
Published in Victor S. Chernyak, Fundamentals of Multisite Radar Systems, 2018
It is well known, that both monostatic and bistatic radars can determine only Directions Of signal Arrival (DOAs) in a passive mode, i.e. bearings on radiation sources. Unlike those radars, MSRSs can obtain three coordinates and their derivatives. This may be achieved by triangulation, or hyperbolic methods, or their combination [18,67]. The triangulation method determines the position of a radiation source in 3D space from the intersection of DOAs from spatially separated receiving stations. The hyperbolic method determines the position of a source from the intersection of hyperboloids of revolution which have their foci at receiving stations (see Section 11.1). A fixed Time Difference Of signal Arrival (TDOA) at a pair of stations (corresponding to the fixed source range difference relative to those stations) determines a hyperboloid of revolution on which surface the source must lie. (A hyperboloid of revolution is a body obtained by revolving of a hyperbola about the axis passing through its foci.) The TDOA is estimated by the signal delay in one station which is necessary to maximize the mutual correlation of signals received by the two stations. It is important to note, that when range R of a radiation source is several times greater than effective baselength Leff between stations, then angular errors of both methods are independent of the source range, so that linear cross-range errors are proportional to range, while range errors are proportional to squared range. A simple relationship can be written for the approximate comparison of source position fix accuracy attainable by triangulation and hyperbolic method. Under the R/Leff ≫1 condition a range difference measurement with the r.m.s. error σΔR (for the hyperbolic method) is approximately equivalent to a bearing measurement (for the triangulation) with the r.m.s. error () σθeq≈σΔR/Leff where Leff is, as earlier, the effective baselength between two receiving stations.
Determination of particular singular configurations of Stewart platform type of fixator by the stereographic projection method
Published in Inverse Problems in Science and Engineering, 2021
Doğan Dönmez, İbrahim Deniz Akçalı, Ercan Avşar, Ahmet Aydın, Hüseyin Mutlu
It is shown by construction in 3-dimensional (3D) geometry that under general skew conditions, two straight lines intersecting four given straight lines can be found [34]. It is also well known that, in 3D space, the set of lines, intersecting three pairwise skew lines (not parallel to a plane) forms a hyperboloid of one sheet [7,35]. Hyperboloid of one sheet is a doubly-ruled surface, meaning that at each point of the surface there are two lines lying on the surface. In fact all the lines on the hyperboloid can be divided into two groups, so that, (i) any two lines in the same group are skew and (ii) any line in one group intersect all but one line in the other group (it is parallel to the remaining line). The first four lines in the rotated SP considered here lie on such a hyperboloid and since they are pairwise skew, they belong to the same group. Each one of the fifth and sixth lines intersects this hyperboloid at two points (fifth line at and and the sixth line at and ). This permits us to conclude that there are exactly two lines intersecting : one intersecting at and the other at . Similarly, there are exactly two lines intersecting and : one intersecting at and the other at and these lines lie on the hyperboloid. These lines can be found in the following way: conditions for intersection points to lie on line and , to lie on lines are written as follows: where and are constants for lines and , respectively. Following the simultaneous solution of equation sets (42) and (43) for the unknown constants, intersection points corresponding to the cases and , and hence lines in question are determined.