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Isarithmic Mapping
Published in Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard, Thematic Cartography and Geovisualization, 2022
Terry A. Slocum, Robert B. McMaster, Fritz C. Kessler, Hugh H. Howard
The simple (no sectors) strategy requires that control points fall within an ellipse (typically, a circle) of fixed size; normally, only a subset of all control points is used to make an estimate (here, we have assumed that eight control points will be used). Quadrant and octant strategies also require that control points fall within an ellipse, but the ellipse is divided into four and eight sectors, respectively, with a specified number of control points used within each sector. For this example, we have again assumed a total of eight control points, and so two points are located in each sector in the quadrant strategy and one point is located in each sector in the octant strategy. For these hypothetical data, it appears that either the quadrant or octant strategy is preferable to the simple strategy because the latter does not use any control points southwest of the grid point.
Metal Crystals—III Energies and Processes
Published in Alan Cottrell, An Introduction to Metallurgy, 2019
which gives EF. In three dimensions we replace the quadrant by an octant (one eighth of a sphere) of volume πrm3. Eqn. 19.2 then changes to N2=πrm36=π6(8mL2EFh2)3/2
Points
Published in Christopher M. Gold, Spatial Context: An Introduction to Fundamental Computer Algorithms for Spatial Analysis, 2018
Various attempts have been made to ameliorate this problem, for example selecting one point from each quadrant, or octant – see Figure 90. Case a) selects the nearest 6; case b selects all within a specified counting circle; case c) selects all within a square region; case d) selects the nearest in each octant. Many interpolation programs offer some of these choices, but the underlying problem remains: how to guarantee that the selection of neighbouring data points matches the weighting function, so that there are no discontinuities when the counting circle, or octant search, etc., adds or removes a point as the query location is perturbed – the weighting function should reach zero by the time this happens. Similarly, when the query location precisely matches a data point, its weighting function must equal 100% (and the others 0%) or the surface will not pass precisely through the data point: here we assume the data is precise, with no error component.
Monteiro da Rocha and the international debate in the 1760s on astronomical methods to find the longitude at sea: his proposals and criticisms to Lacaille’s lunar-distance method
Published in Annals of Science, 2022
Fernando B. Figueiredo, Guy Boistel
Monteiro da Rocha also devotes particular attention to instrumental practice, namely, problems related to instrumental errors, those coming from observations and those coming from astronomical ephemerides, and how they can affect longitude determination.53 Monteiro da Rocha's motto to avoid gross errors is ‘practice, and you will be master’; because this kind of errors is due to lack of attention and the observer's practice, they can be avoided by training. Wrong observations may also be due to parallax. To minimize that effect, Monteiro da Rocha emphasizes the importance of taking any measure in a perpendicular sight to the octant's scale or the clock’s face. However, he notes that there are other types of errors that are unavoidable even for a qualified observer. They are due to inadequate construction and improper calibration of the instruments. That kind of errors (that we call systematic errors nowadays) only can be reduced «through repeated observations», which is «the only resource of all practical operations, where there is a need for great accuracy.» Taking the mean value of several measurements, or «the average of different measurements made by different observers» would be the only way to reduce this kind of error and thus «eliminate any uncertainty that the longitude may have, based on the observations.» For the second case, it would only be possible on board large warships where there were many qualified officers, as Monteiro da Rocha points out.54 Lacaille was the first astronomer interested in calculating errors, in what is called today the propagation of errors. Following his long voyage across the Indian Ocean with the officers of the East India Company in 1753–54 on his way back to France, Lacaille tried several instruments and came to the conclusion that he could not obtain less than four minutes arc of error on the measurement of a celestial arc with a reflection quarter of 20 in. radius (Hadley quadrant or octant 55 cm radius). In his crucial memory of 1759, Lacaille ends up giving the primary sources of errors on the measurement of a celestial arc: 1’ error on the reading of the instrument (1’ for 1 / 29th of the line, less than a tenth of a millimeter for extremely tenuous graduations); the errors of parallelism and alignment of the mirrors leads to 0.5’ error on the angle; the movements of the ship make it challenging to stabilize the image in the telescope of the sextant; thus the error amplified by the device leads to an error of 3’ in the measurement of the arc. Lacaille estimates as 4.5’ the total error on the measurement of the arc, « précision en-dessous de laquelle nul observateur ne peut prétendre descendre» (Lacaille 1765: 69).