scholarly journals On the Implicit Equation of Conics and Quadrics Offsets

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1784
Author(s):  
Jorge Caravantes ◽  
Gema M. Diaz-Toca ◽  
Mario Fioravanti ◽  
Laureano Gonzalez-Vega
Keyword(s):  
Three Dimensions ◽  
Analytic Geometry ◽  
Implicit Equation ◽  
Determinantal Representation ◽  
Point Positioning ◽  
Geometric Queries ◽  
Expanded Form

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in “A Treatise on the Analytic Geometry of Three Dimensions” by G. Salmon for offsets to non-degenerate conics and central quadrics.

A Treatise on the Analytic Geometry of Three Dimensions.

Biometrika ◽  
1959 ◽  
Vol 46 (1/2) ◽  
pp. 271
Author(s):  
F. N. David ◽  
George Salmon ◽  
C. H. Rowe
Keyword(s):  
Three Dimensions ◽  
Analytic Geometry

Tracking Einhard's ‘Life of Charlemagne’ Einharts Vita Karoli. Studien zur Entstehung, Überlieferung und Rezeption. 2 vols. By Matthias M. Tischler. (Monumenta Germaniae Historica, Schriften, 48.) Pp. lxx+896+8 plates; vii+898–1828. Hanover: Hahnsche Buchhandlung, 2001. €140. 3 7752 5448 X; 0080 6951

2006 ◽  
Vol 57 (2) ◽  
pp. 301-307
Author(s):  
JANET L. NELSON
Keyword(s):  
Original Writing ◽  
Three Dimensions ◽  
Doctoral Thesis ◽  
Political Situation ◽  
Expanded Form

‘Habent sua fata libelli’. Had Einhard been able to contemplate the fate of his own libellus of 41 pages (in the standard modern edition) as revealed in Matthias Tischler's 1,828-page two-volume liber, the gamut of his responses from angst to wonderment would surely have included gratitude. In all the voluminous historiography on the Vita Karoli, no one has paid Einhard the compliment of taking his little book as seriously as has Tischler in bringing this Heidelberg doctoral thesis of 1998, now in expanded form, to a wider audience. The subtitle indicates the three dimensions of Tischler's vast enquiry: the immediate context of the VK's original writing, as Einhard responded creatively to an urgent political situation; the VK's own evolving life (if books have fates, Tischler says they also have lives) through preservation and transmission in an extraordinary number of manuscripts; and the VK's afterlife in its variegated reception across eight centuries by scribes, scholars and patrons. Tischler has taken his cue from the literary historian Paul Aebischer: ‘la Vita Karoli est un immense continent’. In Tischler, the improbable continent – vaster than ever realised before and with larger prospects of cultural riches – has found its hinterland explorer and topographer.

scholarly journals A Treatise on the Analytic Geometry of Three Dimensions. Vol. I

1912 ◽  
Vol 6 (99) ◽  
pp. 351 ◽  
Author(s):  
G. Salmon ◽  
R. A. P. Rogers
Keyword(s):  
Three Dimensions ◽  
Analytic Geometry

scholarly journals Performance Evaluation of Precise Point Positioning for BeiDou-3 B1c/B2a Signals in the Global Range

Sensors ◽  
2021 ◽  
Vol 21 (17) ◽  
pp. 5780
Author(s):  
Ershen Wang ◽  
Tao Yang ◽  
Zhi Wang ◽  
Yize Zhang ◽  
Jing Guo ◽  
...  
Keyword(s):  
Precise Point Positioning ◽  
Satellite System ◽  
Three Dimensions ◽  
Convergence Time ◽  
The North ◽  
Significant Difference ◽  
Global Positioning ◽  
System Bias ◽  
Point Positioning ◽  
Kinematic Ppp

With the construction and development of the BeiDou navigation satellite system (BDS), the precise point positioning (PPP) performance of the BDS is worthy of research. In this study, observational data from 17 stations around the world across 20 days are used to comprehensively evaluate the PPP performance of BDS B1c/B2a signals. For greater understanding, the results are also compared with the Global Positioning System (GPS) and BDS PPP performance of different signals and system combinations. The evaluation found root mean square (RMS) values of the static PPP in the north (N), east (E), and upward (U) components, based on the B1c/B2a frequency of BDS-3, to be 6.9 mm, 4.7 mm, and 26.6 mm, respectively. Similar to the static positioning, the RMS values of kinematic PPP in the three directions of N, E, and U are 2.6 cm, 6.0 cm, and 8.5 cm, respectively. Besides this, the static PPP of BDS-3 (B1cB2a) and BDS-2 + BDS-3 (B1IB3I) have obvious system bias. Compared with static PPP, kinematic PPP is more sensitive to the number of satellites, and the coordinate accuracy in three dimensions can be increased by 27% with the combination of GPS (L1L2) and BDS. Compared with BDS-2+BDS-3 (B1IB3I), the convergence time of BDS-3 (B1CB2a) performs better in both static and kinematic modes. The antenna model does not show a significant difference in terms of the effect of the convergence speed, though the number of satellites observed has a certain influence on the convergence time.

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