3017:(DOP) analysis to inform decisions on the number and location of the stations and the system's service area (two dimensions) or service volume (three dimensions). Fig. 5 shows horizontal DOPs (HDOPs) for a 2-D, two-station true-range multilateration system. HDOP is infinite along the baseline and its extensions, as only one of the two dimensions is actually measured. A user of such a system should be roughly broadside of the baseline and within an application-dependent range band. For example, for DME/DME navigation fixes by aircraft, the maximum HDOP permitted by the U.S. FAA is twice the minimum possible value, or 2.828, which limits the maximum usage range (which occurs along the baseline bisector) to 1.866 times the baseline length. (The plane containing two DME ground stations and an aircraft in not strictly horizontal, but usually is nearly so.) Similarly, surveyors select point
1815:) is a 3-D aircraft surveillance application. 3-D true-range multilateration has been used on an experimental basis with GPS satellites for aircraft navigation. The requirement that an aircraft be equipped with an atomic clock precludes its general use. However, GPS receiver clock aiding is an area of active research, including aiding over a network. Thus, conclusions may change. 3-D true-range multilateration was evaluated by the International Civil Aviation Organization as an aircraft landing system, but another technique was found to be more efficient. Accurately measuring the altitude of aircraft during approach and landing requires many ground stations along the flight path.
204:, it is known that if a point lies on two circles, then the circle centers and the two radii provide sufficient information to narrow the possible locations down to two – one of which is the desired solution and the other is an ambiguous solution. Additional information often narrow the possibilities down to a unique location. In three-dimensional geometry, when it is known that a point lies on the surfaces of three spheres, then the centers of the three spheres along with their radii also provide sufficient information to narrow the possible locations down to no more than two (unless the centers lie on a straight line).
2833:
3006:
294:. Its major limitations are that: (a) the target does not identify itself, and in a multiple target situation, mis-assignment of a return can occur; (b) the return signal is attenuated (relative to the transmitted signal) by the fourth power of the vehicle-station range (thus, for distances of tens of miles or more, stations generally require high-power transmitters and/or large/sensitive antennas); and (c) many systems utilize line-of-sight propagation, which limits their ranges to less than 20 miles when both parties are at similar heights above sea level.
2888:. This can be understood from Fig. 4. The two stations are shown as dots, and BLU denotes baseline units. (The measurement pattern is symmetric about both the baseline and the perpendicular bisector of the baseline, and is truncated in the figure.) As is commonly done, individual range measurement errors are taken to be independent of range, statistically independent and identically distributed. This reasonable assumption separates the effects of user-station geometry and range measurement errors on the error in the calculated
386:
1964:
DME/DME navigation (roughly 0.1 seconds). Generally, implementations for repetitive use: (a) employ a 'tracker' algorithm (in addition to the multilateration solution algorithm), which enables measurements collected at different times to be compared and averaged in some manner; and (b) utilize an iterative solution algorithm, as they (b1) admit varying numbers of measurements (including redundant measurements) and (b2) inherently have an initial guess each time the solution algorithm is invoked.
1918:(NLLS) problems is generally preferred when there are more 'good' measurements than the minimum necessary. An important advantage of the Gauss–Newton method over many closed-form algorithms is that it treats range errors linearly, which is often their nature, thereby reducing the effect of range errors by averaging. The Gauss–Newton method may also be used with the minimum number of measured ranges. Since it is iterative, the Gauss–Newton method requires an initial solution estimate.
2802:
330:
1153:
43:
1161:
2434:
2141:
1789:
1824:
1496:
3121:
Navigation and surveillance systems typically involve vehicles and require that a government entity or other organization deploy multiple stations that employ a form of radio technology (i.e., utilize electromagnetic waves). The advantages and disadvantages of employing true-range multilateration for
1921:
In 3-D Cartesian space, a fourth sphere eliminates the ambiguous solution that occurs with three ranges, provided its center is not co-planar with the first three. In 2-D Cartesian or spherical space, a third circle eliminates the ambiguous solution that occurs with two ranges, provided its center is
947:
used the same principle to guide aircraft based on measured ranges to two ground stations. SHORAN was later used for off-shore oil exploration and for aerial surveying. The
Australian Aerodist aerial survey system utilized 2-D Cartesian true-range multilateration. This 2-D scenario is sufficiently
1963:
is the same vehicle. Example applications (and selected intervals between measurements) are: multiple radar aircraft surveillance (5 and 12 seconds, depending upon radar coverage range), aerial surveying, Loran-C navigation with a high-accuracy user clock (roughly 0.1 seconds), and some aircraft
1867:
While intended as a 'spherical' pseudo range multilateration system, Loran-C has also been used as a 'spherical' true-range multilateration system by well-equipped users (e.g., Canadian
Hydrographic Service). This enabled the coverage area of a Loran-C station triad to be extended significantly
1868:(e.g., doubled or tripled) and the minimum number of available transmitters to be reduced from three to two. In modern aviation, slant ranges rather than spherical ranges are more often measured; however, when aircraft altitude is known, slant ranges are readily converted to spherical ranges.
3175:
True-range multilateration is often contrasted with (pseudo range) multilateration, as both require a form of user ranges to multiple stations. Complexity and cost of user equipage is likely the most important factor in limiting use of true-range multilateration for vehicle navigation and
1855:
As the earth is better modeled as an ellipsoid of revolution than a sphere, iterative techniques may be used in modern implementations. In high-altitude aircraft and missiles, a celestial navigation subsystem is often integrated with an inertial navigation subsystem to perform automated
2939:
Without redundant measurements, a true-range multilateration system can be no more accurate than the range measurements, but can be significantly less accurate if the measurement geometry is not chosen properly. Accordingly, some applications place restrictions on the location of point
2429:{\displaystyle {\begin{aligned}x^{\prime }&={\frac {(r_{1}^{\prime }+r_{2}^{\prime })(r_{1}^{\prime }-r_{2}^{\prime })}{2U}}\\y^{\prime }&=\pm {\frac {{\sqrt {(r_{1}^{\prime }+r_{2}^{\prime })^{2}-U^{2}}}{\sqrt {U^{2}-(r_{1}^{\prime }-r_{2}^{\prime })^{2}}}}{2U}}\end{aligned}}}
1514:
891:
955:
The baseline containing the centers of the circles is a line of symmetry. The correct and ambiguous solutions are perpendicular to and equally distant from (on opposite sides of) the baseline. Usually, the ambiguous solution is easily identified. For example, if
1835:
problem (Fig. 3). It's the spherical geometry equivalent of the trilateration method of surveying (although the distances involved are generally much larger). A solution at sea (not necessarily involving the Sun and Moon) was made possible by the
738:
960:
is a vehicle, any motion toward or away from the baseline will be opposite that of the ambiguous solution; thus, a crude measurement of vehicle heading is sufficient. A second example: surveyors are well aware of which side of the baseline that
2732:). While eliminating one transmitter is a benefit, there is a countervailing 'cost': the synchronization tolerance for the two stations becomes dependent on the propagation speed (typically, the speed of light) rather that the speed of point
1244:
1139:
must be measured within a synchronization tolerance that depends on the vehicle speed and the allowable vehicle position error. Alternatively, vehicle motion between range measurements may be accounted for, often by dead reckoning.
312:(TOF) of electromagnetic energy between multiple stations and the vehicle is measured based on transmission by one party and reception by the other. This is the most recently developed method, and was enabled by the development of
1930:
This article largely describes 'one-time' application of the true-range multilateration technique, which is the most basic use of the technique. With reference to Fig. 1, the characteristic of 'one-time' situations is that point
274:. However, it is often more difficult or costly to measure true-ranges than it is to measure pseudo ranges. For distances up to a few miles and fixed locations, true-range can be measured manually. This has been done in
269:
For similar ranges and measurement errors, a navigation and surveillance system based on true-range multilateration provide service to a significantly larger 2-D area or 3-D volume than systems based on pseudo-range
1172:
There are multiple algorithms that solve the 3-D Cartesian true-range multilateration problem directly (i.e., in closed-form) – e.g., Fang. Moreover, one can adopt closed-form algorithms developed for pseudo range
196:
problem. Moreover, if more than the minimum number of ranges are available, it is good practice to utilize those as well. This article addresses the general issue of position determination using multiple ranges.
2051:
1784:{\displaystyle {\begin{aligned}x&={\frac {r_{1}^{2}-r_{2}^{2}+U^{2}}{2U}}\\y&={\frac {r_{1}^{2}-r_{3}^{2}+V_{x}^{2}+V_{y}^{2}-2V_{x}x}{2V_{y}}}\\z&=\pm {\sqrt {r_{1}^{2}-x^{2}-y^{2}}}\end{aligned}}}
1143:
A trigonometric solution is also possible (side-side-side case). Also, a solution employing graphics is possible. A graphical solution is sometimes employed during real-time navigation, as an overlay on a map.
2129:
1519:
1249:
281:
For longer distances and/or moving vehicles, a radio/radar system is generally needed. This technology was first developed circa 1940 in conjunction with radar. Since then, three methods have been employed:
1943:
change from one application of the true-range multilateration technique to the next. This is appropriate for surveying, celestial navigation using manual sightings, and some aircraft DME/DME navigation.
2146:
757:
617:
3028:
Errors in trilateration surveys are discussed in several documents. Generally, emphasis is placed on the effects of range measurement errors, rather than on the effects of algorithm numerical errors.
752:
3643:
1156:
Fig. 2 3-D True-Range
Multilateration Scenario. C1, C2 and C3 are known centers of spheres in the x,y plane. P is point whose (x,y,z) coordinates are desired based on its ranges to C1, C2 and C3.
3472:
2784:
1972:
Hybrid multilateration systems – those that are neither true-range nor pseudo range systems – are also possible. For example, in Fig. 1, if the circle centers are shifted to the left so that
2960:: The ideal is a right angle, which occurs at distances from the baseline of one-half or less of the baseline length; maximum allowable deviations from the ideal 90 degrees may be specified.
2722:
2672:
939:
is the most fundamental true-range multilateration relationship. Aircraft DME/DME navigation and the trilateration method of surveying are examples of its application. During World War II
182:
from two known locations can be used to locate a third point in a two-dimensional
Cartesian space (plane), which is a frequently applied technique (e.g., in surveying). Similarly, two
2991:
2963:
The horizontal dilution of precision (HDOP), which multiplies the range error in determining the position error: For two dimensions, the ideal (minimum) HDOP is the square root of 2 (
2789:
While not implemented operationally, hybrid multilateration systems have been investigated for aircraft surveillance near airports and as a GPS navigation backup system for aviation.
531:
An analytic solution has likely been known for over 1,000 years, and is given in several texts. Moreover, one can easily adapt algorithms for a three dimensional
Cartesian space.
158:) between the vehicle/point and multiple spatially-separated known locations (often termed "stations"). Energy waves may be involved in determining range, but are not required.
2619:
2587:
2501:
2469:
612:
1811:
Many applications of 3-D true-range multilateration involve short ranges—e.g., precision manufacturing. Integrating range measurement from three or more radars (e.g., FAA's
1852:– were employed. These can accommodate more than two measured 'altitudes'. Owing to the difficulty of making measurements at sea, 3 to 5 'altitudes' are often recommended.
1808:
The plane containing the sphere centers is a plane of symmetry. The correct and ambiguous solutions are perpendicular to it and equally distant from it, on opposite sides.
2555:
2528:
1840:(introduced in 1761) and the discovery of the 'line of position' (LOP) in 1837. The solution method now most taught at universities (e.g., U.S. Naval Academy) employs
1491:{\displaystyle {\begin{aligned}r_{1}^{2}&=x^{2}+y^{2}+z^{2}\\r_{2}^{2}&=(x-U)^{2}+y^{2}+z^{2}\\r_{3}^{2}&=(x-V_{x})^{2}+(y-V_{y})^{2}+z^{2}\end{aligned}}}
1232:
161:
True-range multilateration is both a mathematical topic and an applied technique used in several fields. A practical application involving a fixed location occurs in
305:
radar surveillance and DME/DME navigation. It requires that both parties have both transmitters and receivers, and may require that interference issues be addressed.
2918:
2870:
1137:
1110:
1075:
1048:
604:
525:
498:
451:
1848:
measurements of the 'altitude' of two heavenly bodies. This problem can also be addressed using vector analysis. Historically, graphical techniques – e.g., the
928:
560:
471:
415:
3655:
3001:
is 90 degrees; a maximum allowable HDOP value may be specified. (Here, equal HDOPs are simply the locus of points in Fig. 4 having the same crossing angle.)
1016:
can be used as new baselines, and additional points surveyed. Thus, large areas or distances can be surveyed based on multiple, smaller triangles—termed a
316:; it requires that the vehicle (user) and stations having synchronized clocks. It has been successfully demonstrated (experimentally) with Loran-C and GPS.
2936:
is not on a circle, the error in its position is approximately proportional to the area bounded by the nearest two blue and nearest two magenta circles.
3278:, Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Report Number: DOT/TSC-RA-3-8-(1) (Technical note 1973-43), Oct. 11, 1973
3456:
993:
211:, which employs range differences to locate a (typically, movable) point. Pseudo range multilateration is almost always implemented by measuring
3724:
3573:"STELLA (System To Estimate Latitude and Longitude Astronomically)", George Kaplan, John Bangert, Nancy Oliversen; U.S. Naval Observatory, 1999.
1947:
However, in other situations, the true-range multilateration technique is applied repetitively (essentially continuously). In those situations,
3092:
1979:
107:
60:
2060:
534:
The simplest algorithm employs analytic geometry and a station-based coordinate frame. Thus, consider the circle centers (or stations)
79:
17:
1004:
are known in that second system. Both are often done in surveying when the trilateration method is employed. Once the coordinates of
3669:
886:{\displaystyle {\begin{aligned}x&={\frac {r_{1}^{2}-r_{2}^{2}+U^{2}}{2U}}\\y&=\pm {\sqrt {r_{1}^{2}-x^{2}}}\end{aligned}}}
86:
286:
Two-way range measurement, one party active – This is the method used by traditional radars (sometimes termed
297:
Two-way range measurement, both parties active – This method was reportedly first used for navigation by the
93:
2924:. Here, the measurement geometry is simply the angle at which two circles cross—or equivalently, the angle between lines
1827:
Fig. 3 Example of celestial navigation altitude intercept problem (lines of position are distorted by the map projection)
2739:
3014:
1903:
Ambiguous solutions can be identified automatically (i.e., without human involvement) -- requires an additional station
75:
3095:) -- U.S./NATO system that (among other capabilities) locates participants in a network using inter-participant ranges
2677:
2627:
126:
301:
aircraft guidance system fielded in 1941 by the
Luftwaffe. It is now used globally in air traffic control – e.g.,
3631:, Harry B. Lee, Massachusetts Institute of Technology, Lincoln Laboratory, Technical Note 1973-43, Oct. 11, 1973.
2876:
in Fig. 1 -- depends upon two factors: (1) the range measurement accuracy, and (2) the geometric relationship of
2813:
341:
3176:
surveillance. Some uses are not the original purpose for system deployment – e.g., DME/DME aircraft navigation.
64:
2966:
1180:
The simplest algorithm corresponds to the sphere centers in Fig. 2. The figure 'page' is the plane containing
235:
here . That name is selected because it: (a) is an accurate description and partially familiar terminology (
3197:
252:
208:
3550:
A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples
3405:. U.S. DOT National Transportation Library: U.S. DOT John A. Volpe National Transportation Systems Center.
3740:
3295:
3106:
1876:
When there are more range measurements available than there are problem dimensions, either from the same
1861:
150:) is a method to determine the location of a movable vehicle or stationary point in space using multiple
3141:
Accuracy degrades slowly with distance from station (generally better than pseudo-range multilateration)
733:{\displaystyle {\begin{aligned}r_{1}^{2}&=x^{2}+y^{2}\\r_{2}^{2}&=(U-x)^{2}+y^{2}\end{aligned}}}
377:
Any pseudo-range multilateration algorithm can be specialized for use with true-range multilateration.
100:
3147:
Station synchronization is not demanding (based on speed of point of interest, and may be addressed by
1911:
188:
can be used to locate a point on a sphere, which is a fundamental concept of the ancient discipline of
3347:
Escobal, P. R.; Fliegel, H. F.; Jaffe, R. M.; Muller, P. M.; Ong, K. M.; Vonroos, O. H. (2013-08-07).
2944:. For a 2-D Cartesian (trilateration) situation, these restrictions take one of two equivalent forms:
3185:
2592:
2560:
2474:
2442:
216:
3601:
Joint
Workshop of the National Academy of Engineering and the Chinese Academy of Engineering (2012).
542:
in Fig. 1 which have known coordinates (e.g., have already been surveyed) and thus whose separation
239:
is often used in this context); (b) avoids specifying the number of ranges involved (as does, e.g.,
3646:, Robert W. Lilley and Robert Erikson, Federal Aviation Administration, White Paper, July 23, 2012.
3099:
1915:
1857:
3750:
3231:
2533:
2506:
53:
3379:
3225:
1841:
3561:
3491:
3427:
2832:
1199:
3503:
1177:. Bancroft's algorithm (adapted) employs vectors, which is an advantage in some situations.
952:
is often applied to all applications involving a known baseline and two range measurements.
3461:
Proceedings of the 7th
Workshop on Positioning, Navigation and Communication 2010 (WPNC'10)
3191:
3062:
2891:
2843:
1115:
1088:
1053:
1026:
577:
503:
476:
424:
189:
3671:
Comparison of the
Accuracy of Triangulation, Trilateration and Triangulation-Trilateration
8:
3597:; M.J. Narins, L.V. Eldredge, P. Enge, S.C. Lo, M.J. Harrison, and R. Kenagy; Chapter in
3013:
Planning a true-range multilateration navigation or surveillance system often involves a
3005:
3506:; Fang-Cheng Chan, Mathieu Joerger, Samer Khanafseh, Boris Pervan, and Ondrej Jakubov;
3203:
3116:
3085:
2557:-solution. It could be implemented as a true-range multilateration system by measuring
1837:
913:
545:
456:
400:
298:
201:
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3535:
2624:
However, it could also be implemented as a hybrid multilateration system by measuring
380:
3708:
3415:
3329:
3214:
3081:
2725:
940:
291:
3745:
3595:"Alternative Position, Navigation, and Timing: The Need for Robust Radionavigation"
3321:
3208:
3068:
2840:
The position accuracy of a true-range multilateration system—e.g., accuracy of the
1849:
3439:"Trilateration and extension to global positioning system navigation", B.T. Fang,
3239:—Developed as a military aircraft navigation system, later used for civil purposes
2836:
Fig. 4 2-D true-range multi-lateration (trilateration) system ranging measurements
1818:
3697:
Proceedings of the 2008 National
Technical Meeting of The Institute of Navigation
3219:
3200:(DME) -- System used to measure distance between an aircraft and a ground station
3167:
Non-cooperative surveillance involves path losses to the fourth power of distance
1174:
271:
212:
184:
31:
3009:
Fig. 5 HDOP contours for a 2-D true-range multilateration (trilateration) system
207:
True-range multilateration can be contrasted to the more frequently encountered
3519:
3148:
1152:
985:
385:
309:
3683:
3348:
930:
has two values (i.e., solution is ambiguous); this is usually not a problem.
3734:
3333:
3253:
3044:
1831:
This is a classic celestial (or astronomical) navigation problem, termed the
313:
215:(TOAs) of energy waves. True-range multilateration can also be contrasted to
169:
when on-board persons/equipment are informed of its location, and are termed
3660:; William Navidi, William S Murphy, Jr and Willy Hereman; December 20, 1999.
3594:
3138:
Station locations are flexible; they can be placed centrally or peripherally
290:
radars) to determine the range of a non-cooperative target, and now used by
278:
for several thousand years – e.g., using ropes and chains.
27:
Using distance measures along a shape's edges to determine position in space
3626:
3400:
3325:
3273:
3247:
373:
presence of redundant measurements (more than the problem space dimension).
170:
2439:
This form of the solution explicitly depends on the sum and difference of
1896:) stations, or from additional stations, at least these benefits accrue:
1147:
389:
Fig. 1 2-D Cartesian true-range multilateration (trilateration) scenario.
178:
3504:"How a Chip-Scale Atomic Clock Can Help Mitigate Broadband Interference"
2801:
329:
3628:
Accuracy Limitations of Range-Range (Spherical) Multilateration Systems
3296:
Rho-Rho Loran-C Combined with Satellite Navigation for Offshore Surveys
3275:
Accuracy limitations of range-range (spherical) multilateration systems
3161:
Cooperative system accuracy is sensitive to equipment turn-around error
1823:
992:
can be expressed in a second, better-known coordinate system—e.g., the
166:
3686:, YouTube, U.S. National Oceanic and Atmospheric Administration, 2006.
3457:
Closed-form Algorithms in Mobile Positioning: Myths and Misconceptions
574:(e.g., a vehicle or another point to be surveyed) is at unknown point
3381:
Impact of Rubidium Clock Aiding on GPS Augmented Vehicular Navigation
3242:
3144:
Requires one fewer station than a pseudo range multilateration system
3038:
1925:
1906:
Errors in 'good' measurements can be averaged, reducing their effect.
1168:
lateration limits the potential positions amount to two (here A or B)
275:
162:
3674:; K.L. Provoro; Novosibirsk Institnte of Engineers of Geodesy; 1960.
3117:
Advantages and disadvantages for vehicle navigation and surveillance
231:
There is no accepted or widely-used general term for what is termed
42:
3340:
2728:
with one transmitter and two receivers (rather than two monostatic
2046:{\displaystyle x_{1}^{\prime }=-{\tfrac {1}{2}}U,y_{1}^{\prime }=0}
1160:
977:
are on the ground, the ambiguous solution is usually below ground.
381:
Two Cartesian dimensions, two measured slant ranges (trilateration)
155:
3228:—Systems used to measure distance between two points on the ground
3158:
Often a user is required to have both a transmitter and a receiver
2124:{\displaystyle x_{2}^{\prime }={\tfrac {1}{2}}U,y_{2}^{\prime }=0}
173:
when off-vehicle entities are informed of the vehicle's location.
1845:
264:
151:
3349:"A 3-D Multilateration: A Precision Geodetic Measurement System"
1023:
An implied assumption for the above equation to be true is that
3644:"DME/DME for Alternate Position, Navigation, and Timing (APNT)"
3236:
3077:
1819:
Two spherical dimensions, two or more measured spherical ranges
944:
3492:
LaserTracer – A New Type of Self Tracking Laser Interferometer
3402:
Earth-Referenced Aircraft Navigation and Surveillance Analysis
370:
problem space geometry (generally, Cartesian or spherical) and
3494:, Carl-Thomas Schneider, IWAA2004, CERN, Geneva, October 2004
2729:
220:
1234:, then Pythagoras's theorem yields the slant ranges between
3056:
3025:
roughly form an equilateral triangle (where HDOP = 1.633).
1812:
3346:
3194:—ancient technique of navigation based on heavenly bodies
3384:, Zhaonian Zhang; University of Calgary; December, 1997.
3320:(1). American Society of Civil Engineers (ASCE): 81–92.
3088:—Aircraft guidance systems developed for 'blind' bombing
2724:
using different equipment – e.g., for surveillance by a
3312:
Wirtanen, Theodore H. (1969). "Laser Multilateration".
1148:
Three Cartesian dimensions, three measured slant ranges
3305:
2083:
2005:
3552:, John Casey, Dublin, Hodges, Figgis & Co., 1889.
3477:
IEEE Transactions on Aerospace and Electronic Systems
3428:"The Nature of Geographic Information: Trilateration"
3071:—Graphical solution to the altitude intercept problem
2969:
2894:
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2742:
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1982:
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403:
165:. Applications involving vehicle location are termed
251:) and (d) and avoids confusion with the more common
243:; (c) avoids implying an application (as do, e.g.,
67:. Unsourced material may be challenged and removed.
3583:Tracking and Data Fusion: A Handbook of Algorithms
2985:
2912:
2864:
2779:{\displaystyle r_{1}^{\prime }\pm r_{2}^{\prime }}
2778:
2716:
2666:
2613:
2581:
2549:
2522:
2495:
2463:
2428:
2123:
2045:
1926:One-time application versus repetitive application
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1042:
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885:
732:
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367:problem space dimension (generally, two or three),
3657:Statistical Methods in Surveying by Trilateration
3479:, Volume: AES-21, Issue: 7 (Jan. 1985), pp 56–59.
3211:—Graphical technique used in celestial navigation
1967:
1900:'Bad' measurements can be identified and rejected
308:One-way range measurement – The
3732:
3540:, Isaac Todhunter, MacMillan; 5th edition, 1886.
3122:such a system are shown in the following table.
1844:to solve an oblique spherical triangle based on
2717:{\displaystyle r_{1}^{\prime }-r_{2}^{\prime }}
2667:{\displaystyle r_{1}^{\prime }+r_{2}^{\prime }}
1871:
397:are centers of circles having known separation
3524:IEEE Aerospace and Electronic Systems Magazine
3093:Joint Tactical Information Distribution System
3314:Journal of the Surveying and Mapping Division
965:lies. A third example: in applications where
3585:; Y. Bar-Shalom, P.K. Willett, X. Tian; 2011
3487:
3485:
3473:"An Algebraic Solution of the GPS Equations"
3256:– Surveying method based on measuring angles
1196:is a 'point of interest' (e.g., vehicle) at
933:While there are many enhancements, Equation
3699:, San Diego, CA, January 2008, pp. 443–451.
980:If needed, the interior angles of triangle
3576:
3441:Journal of Guidance, Control, and Dynamics
3109:aircraft—Employs astro-inertial navigation
3102:aircraft—Employs astro-inertial navigation
994:Universal Transverse Mercator (UTM) system
3639:
3637:
3529:
3482:
3451:
3449:
2503:and does not require 'chaining' from the
127:Learn how and when to remove this message
3497:
3466:
3311:
3222:– Addresses pseudo range multilateration
3188:, similar technique applied to molecules
3004:
2986:{\displaystyle {\sqrt {2}}\approx 1.414}
2831:
1822:
1159:
1151:
384:
363:algorithms may be partitioned based on
30:For broader coverage of this topic, see
3663:
3604:
3409:
3394:
3392:
3390:
3375:
3373:
3290:
3288:
3286:
3284:
3250:—First microwave electronic rangefinder
3164:Cannot be used for stealth surveillance
2993:), which occurs when the angle between
14:
3733:
3684:"Trilateration in Maritime Archeology"
3649:
3634:
3588:
3567:
3555:
3446:
3055:Multiple radar integration (e.g., FAA
2736:, in order to accurately measure both
988:. Also, if needed, the coordinates of
3702:
3543:
3513:
3398:
984:can be found using the trigonometric
562:is known. The figure 'page' contains
3689:
3677:
3620:
3433:
3387:
3370:
3281:
2796:
1507:
745:
324:
219:, which involves the measurement of
65:adding citations to reliable sources
36:
3599:Global Navigation Satellite Systems
3065:using the altitude intercept method
606:, then Pythagoras's theorem yields
320:
258:
24:
3695:"DME/DME Accuracy", Michael Tran,
3421:
3267:
2793:Preliminary and final computations
2771:
2753:
2709:
2691:
2659:
2641:
2606:
2574:
2542:
2515:
2488:
2456:
2394:
2376:
2318:
2300:
2267:
2239:
2221:
2200:
2182:
2154:
2110:
2074:
2032:
1993:
1922:not co-linear with the first two.
25:
3762:
3718:
3430:, Pennsylvania State Univ., 2018.
3300:International Hydrographic Review
3074:Calibrating laser interferometers
570:. If a third 'point of interest'
453:coordinates are desired based on
3052:DME/DME RNAV aircraft navigation
2948:The allowable interior angle at
2800:
1856:navigation—e.g., U.S. Air Force
328:
41:
3031:
2614:{\displaystyle r_{2}^{\prime }}
2582:{\displaystyle r_{1}^{\prime }}
2496:{\displaystyle r_{2}^{\prime }}
2464:{\displaystyle r_{1}^{\prime }}
1077:relate to the same position of
52:needs additional citations for
3112:Experimental Loran-C technique
3041:using the trilateration method
2907:
2895:
2859:
2847:
2400:
2363:
2324:
2287:
2244:
2208:
2205:
2169:
1968:Hybrid multilateration systems
1462:
1442:
1430:
1410:
1349:
1336:
1221:
1203:
704:
691:
593:
581:
440:
428:
226:
13:
1:
3727:, PHP / Python Implementation
3260:
3049:Maritime archeology surveying
1085:is a vehicle, then typically
996:—provided the coordinates of
3526:; Vol. 1, Issue 5; May 1986.
3443:, vol. 9 (1986), pp 715–717.
3399:Geyer, Michael (June 2016).
3198:Distance measuring equipment
1872:Redundant range measurements
253:pseudo-range multilateration
209:pseudo-range multilateration
76:"True-range multilateration"
7:
3179:
2550:{\displaystyle y^{\prime }}
2523:{\displaystyle x^{\prime }}
2131:then the point of interest
1798:
935:
900:
144:range-range multilateration
10:
3767:
3711:, Christian Wolff, undated
3520:"Microwave Landing System"
943:and during the Korean War
361:True-range multilateration
262:
233:true-range multilateration
140:True-range multilateration
29:
18:True range multilateration
3508:GPS World -- Innovations
3186:Distance geometry problem
2880:to the system's stations
1501:Thus, the coordinates of
148:spherical multilateration
3418:retrieved Jan. 22, 2019.
1916:non-linear least squares
1238:and the sphere centers:
948:important that the term
202:two-dimensional geometry
3611:"Dilution of Precision"
3232:Resection (orientation)
1227:{\displaystyle (x,y,z)}
1008:are established, lines
3562:"Vector-based geodesy"
3537:Spherical Trigonometry
3416:Adastra Aerial Surveys
3326:10.1061/jsueax.0000322
3010:
2987:
2914:
2866:
2837:
2780:
2718:
2668:
2615:
2583:
2551:
2524:
2497:
2465:
2430:
2125:
2047:
1959:) remain constant and
1912:Gauss–Newton algorithm
1842:spherical trigonometry
1828:
1785:
1492:
1228:
1169:
1157:
1133:
1106:
1071:
1044:
924:
887:
734:
600:
556:
528:
521:
494:
467:
447:
411:
3617:, May 1999, pp 52–59.
3564:, Chris Veness. 2016.
3015:dilution of precision
3008:
2988:
2915:
2913:{\displaystyle (x,y)}
2872:coordinates of point
2867:
2865:{\displaystyle (x,y)}
2835:
2781:
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2126:
2048:
1826:
1786:
1493:
1229:
1163:
1155:
1134:
1132:{\displaystyle r_{2}}
1107:
1105:{\displaystyle r_{1}}
1072:
1070:{\displaystyle r_{2}}
1045:
1043:{\displaystyle r_{1}}
925:
888:
735:
601:
599:{\displaystyle (x,y)}
557:
522:
520:{\displaystyle r_{2}}
495:
493:{\displaystyle r_{1}}
468:
448:
446:{\displaystyle (x,y)}
412:
388:
3613:, Richard Langeley,
3475:, Stephen Bancroft,
3353:JPL Quart. Tech. Rev
3192:Celestial navigation
3063:Celestial navigation
2967:
2892:
2844:
2740:
2678:
2628:
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2061:
1980:
1935:and at least one of
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753:
613:
578:
546:
504:
477:
473:and measured ranges
457:
425:
401:
190:celestial navigation
61:improve this article
3522:; Thomas E. Evans;
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1997:
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969:is an aircraft and
863:
805:
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683:
634:
3741:Euclidean geometry
3204:Euclidean distance
3021:in Fig. 1 so that
3011:
2983:
2910:
2862:
2838:
2812:. You can help by
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2100:
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2022:
2014:
1983:
1838:marine chronometer
1833:altitude intercept
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340:. You can help by
292:laser rangefinders
245:DME/DME navigation
194:altitude intercept
3725:stackexchange.com
3463:, March 11, 2010.
3459:", Niilo Sirola,
3215:Laser rangefinder
3173:
3172:
2975:
2830:
2829:
2726:multistatic radar
2530:-solution to the
2420:
2409:
2346:
2256:
2091:
2013:
1806:
1805:
1775:
1713:
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923:{\displaystyle y}
908:
907:
877:
828:
555:{\displaystyle U}
466:{\displaystyle U}
410:{\displaystyle U}
358:
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111:
16:(Redirected from
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3209:Intercept method
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3045:Aerial surveying
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321:Solution methods
259:Obtaining ranges
213:times-of-arrival
185:spherical ranges
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3306:
3298:". S.T. Grant,
3293:
3282:
3272:
3268:
3263:
3220:Multilateration
3182:
3119:
3100:SR-71 Blackbird
3034:
2970:
2968:
2965:
2964:
2920:coordinates of
2893:
2890:
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2841:
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2810:needs expansion
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1957:Cn, n = 3,4,...
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1175:multilateration
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237:multilateration
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32:Multilateration
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3751:Geopositioning
3748:
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3720:
3719:External links
3717:
3714:
3713:
3709:"Radar Basics"
3701:
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3149:dead reckoning
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3131:Disadvantages
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2197:
2193:
2189:
2184:
2179:
2175:
2171:
2165:
2162:
2160:
2156:
2152:
2148:
2147:
2120:
2117:
2112:
2107:
2103:
2099:
2096:
2090:
2087:
2081:
2076:
2071:
2067:
2042:
2039:
2034:
2029:
2025:
2021:
2018:
2012:
2009:
2003:
2000:
1995:
1990:
1986:
1969:
1966:
1927:
1924:
1910:The iterative
1908:
1907:
1904:
1901:
1873:
1870:
1820:
1817:
1804:
1803:
1794:
1792:
1772:
1768:
1764:
1759:
1755:
1751:
1746:
1741:
1737:
1731:
1728:
1725:
1723:
1721:
1718:
1717:
1709:
1705:
1701:
1696:
1691:
1687:
1683:
1680:
1675:
1670:
1666:
1662:
1657:
1652:
1648:
1644:
1639:
1634:
1630:
1626:
1621:
1616:
1612:
1605:
1602:
1600:
1598:
1595:
1594:
1588:
1585:
1578:
1574:
1570:
1565:
1560:
1556:
1552:
1547:
1542:
1538:
1531:
1528:
1526:
1524:
1521:
1520:
1499:
1498:
1481:
1477:
1473:
1468:
1464:
1458:
1454:
1450:
1447:
1444:
1441:
1436:
1432:
1426:
1422:
1418:
1415:
1412:
1409:
1406:
1404:
1400:
1395:
1391:
1387:
1386:
1381:
1377:
1373:
1368:
1364:
1360:
1355:
1351:
1347:
1344:
1341:
1338:
1335:
1332:
1330:
1326:
1321:
1317:
1313:
1312:
1307:
1303:
1299:
1294:
1290:
1286:
1281:
1277:
1273:
1270:
1268:
1264:
1259:
1255:
1251:
1250:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1149:
1146:
1126:
1122:
1099:
1095:
1064:
1060:
1037:
1033:
986:law of cosines
919:
906:
905:
896:
894:
874:
870:
866:
861:
856:
852:
846:
843:
840:
838:
836:
833:
832:
826:
823:
816:
812:
808:
803:
798:
794:
790:
785:
780:
776:
769:
766:
764:
762:
759:
758:
741:
740:
723:
719:
715:
710:
706:
702:
699:
696:
693:
690:
687:
685:
681:
676:
672:
668:
667:
662:
658:
654:
649:
645:
641:
638:
636:
632:
627:
623:
619:
618:
595:
592:
589:
586:
583:
551:
514:
510:
487:
483:
462:
442:
439:
436:
433:
430:
406:
382:
379:
375:
374:
371:
368:
356:
355:
335:
333:
322:
319:
318:
317:
310:time of flight
306:
295:
263:Main article:
260:
257:
228:
225:
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
3763:
3752:
3749:
3747:
3744:
3742:
3739:
3738:
3736:
3726:
3723:
3722:
3710:
3705:
3698:
3692:
3685:
3680:
3673:
3672:
3666:
3659:
3658:
3652:
3645:
3640:
3638:
3630:
3629:
3623:
3616:
3612:
3607:
3600:
3596:
3591:
3584:
3579:
3570:
3563:
3558:
3551:
3546:
3539:
3538:
3532:
3525:
3521:
3516:
3509:
3505:
3500:
3493:
3488:
3486:
3478:
3474:
3469:
3462:
3458:
3452:
3450:
3442:
3436:
3429:
3424:
3417:
3412:
3404:
3403:
3395:
3393:
3391:
3383:
3382:
3376:
3374:
3358:
3354:
3350:
3343:
3335:
3331:
3327:
3323:
3319:
3315:
3308:
3301:
3297:
3291:
3289:
3287:
3285:
3277:
3276:
3270:
3266:
3255:
3254:Triangulation
3252:
3249:
3246:
3244:
3241:
3238:
3235:
3233:
3230:
3227:
3224:
3221:
3218:
3216:
3213:
3210:
3207:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3183:
3177:
3166:
3163:
3160:
3157:
3156:
3155:
3150:
3146:
3143:
3140:
3137:
3136:
3135:
3134:
3130:
3127:
3126:
3123:
3111:
3108:
3104:
3101:
3097:
3094:
3090:
3087:
3083:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3054:
3051:
3048:
3046:
3043:
3040:
3036:
3035:
3029:
3026:
3024:
3020:
3016:
3007:
3000:
2996:
2980:
2977:
2972:
2962:
2959:
2955:
2951:
2947:
2946:
2945:
2943:
2937:
2935:
2932:. When point
2931:
2927:
2923:
2904:
2901:
2898:
2887:
2883:
2879:
2875:
2856:
2853:
2850:
2834:
2824:
2815:
2811:
2808:This section
2806:
2803:
2799:
2798:
2790:
2787:
2766:
2762:
2758:
2748:
2744:
2735:
2731:
2727:
2704:
2700:
2696:
2686:
2682:
2654:
2650:
2646:
2636:
2632:
2622:
2601:
2597:
2569:
2565:
2538:
2511:
2483:
2479:
2451:
2447:
2416:
2413:
2404:
2389:
2385:
2381:
2371:
2367:
2360:
2355:
2351:
2341:
2337:
2333:
2328:
2313:
2309:
2305:
2295:
2291:
2279:
2276:
2274:
2263:
2252:
2249:
2234:
2230:
2226:
2216:
2212:
2195:
2191:
2187:
2177:
2173:
2163:
2161:
2150:
2138:
2137:
2136:
2134:
2118:
2115:
2105:
2101:
2097:
2094:
2088:
2085:
2079:
2069:
2065:
2056:
2040:
2037:
2027:
2023:
2019:
2016:
2010:
2007:
2001:
1998:
1988:
1984:
1975:
1965:
1962:
1958:
1955:(and perhaps
1954:
1950:
1945:
1942:
1938:
1934:
1923:
1919:
1917:
1913:
1905:
1902:
1899:
1898:
1897:
1895:
1891:
1887:
1883:
1879:
1869:
1865:
1863:
1859:
1853:
1851:
1847:
1843:
1839:
1834:
1825:
1816:
1814:
1809:
1802:
1795:
1793:
1791:
1770:
1766:
1762:
1757:
1753:
1749:
1744:
1739:
1735:
1729:
1726:
1724:
1719:
1707:
1703:
1699:
1694:
1689:
1685:
1681:
1678:
1673:
1668:
1664:
1660:
1655:
1650:
1646:
1642:
1637:
1632:
1628:
1624:
1619:
1614:
1610:
1603:
1601:
1596:
1586:
1583:
1576:
1572:
1568:
1563:
1558:
1554:
1550:
1545:
1540:
1536:
1529:
1527:
1522:
1510:
1509:
1506:
1504:
1479:
1475:
1471:
1466:
1456:
1452:
1448:
1445:
1439:
1434:
1424:
1420:
1416:
1413:
1407:
1405:
1398:
1393:
1389:
1379:
1375:
1371:
1366:
1362:
1358:
1353:
1345:
1342:
1339:
1333:
1331:
1324:
1319:
1315:
1305:
1301:
1297:
1292:
1288:
1284:
1279:
1275:
1271:
1269:
1262:
1257:
1253:
1241:
1240:
1239:
1237:
1218:
1215:
1212:
1209:
1206:
1195:
1191:
1187:
1183:
1178:
1176:
1167:
1162:
1154:
1145:
1141:
1124:
1120:
1097:
1093:
1084:
1080:
1062:
1058:
1035:
1031:
1021:
1019:
1015:
1011:
1007:
1003:
999:
995:
991:
987:
983:
978:
976:
972:
968:
964:
959:
953:
951:
950:trilateration
946:
942:
938:
937:
931:
917:
904:
897:
895:
893:
872:
868:
864:
859:
854:
850:
844:
841:
839:
834:
824:
821:
814:
810:
806:
801:
796:
792:
788:
783:
778:
774:
767:
765:
760:
748:
747:
744:
721:
717:
713:
708:
700:
697:
694:
688:
686:
679:
674:
670:
660:
656:
652:
647:
643:
639:
637:
630:
625:
621:
609:
608:
607:
590:
587:
584:
573:
569:
565:
549:
541:
537:
532:
512:
508:
485:
481:
460:
437:
434:
431:
420:
404:
396:
392:
387:
378:
372:
369:
366:
365:
364:
362:
352:
343:
339:
336:This section
334:
331:
327:
326:
315:
314:atomic clocks
311:
307:
304:
300:
296:
293:
289:
285:
284:
283:
279:
277:
273:
266:
256:
254:
250:
249:trilateration
246:
242:
238:
234:
224:
222:
218:
217:triangulation
214:
210:
205:
203:
198:
195:
192:— termed the
191:
187:
186:
181:
180:
174:
172:
168:
164:
159:
157:
153:
149:
145:
142:(also termed
141:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
3704:
3696:
3691:
3679:
3670:
3665:
3656:
3651:
3627:
3622:
3614:
3606:
3598:
3590:
3582:
3578:
3569:
3557:
3549:
3545:
3536:
3531:
3523:
3515:
3507:
3499:
3476:
3468:
3460:
3440:
3435:
3423:
3411:
3401:
3380:
3361:. Retrieved
3356:
3352:
3342:
3317:
3313:
3307:
3299:
3274:
3269:
3248:Tellurometer
3174:
3120:
3032:Applications
3027:
3022:
3018:
3012:
2998:
2994:
2957:
2953:
2949:
2941:
2938:
2933:
2929:
2925:
2921:
2885:
2881:
2877:
2873:
2839:
2818:
2814:adding to it
2809:
2788:
2733:
2623:
2438:
2132:
2054:
1973:
1971:
1960:
1956:
1952:
1948:
1946:
1940:
1936:
1932:
1929:
1920:
1914:for solving
1909:
1893:
1889:
1885:
1881:
1877:
1875:
1866:
1854:
1832:
1830:
1810:
1807:
1796:
1511:
1502:
1500:
1235:
1193:
1189:
1185:
1181:
1179:
1171:
1165:
1142:
1082:
1078:
1022:
1017:
1013:
1009:
1005:
1001:
997:
989:
981:
979:
974:
970:
966:
962:
957:
954:
949:
934:
932:
909:
898:
749:
742:
571:
567:
563:
539:
535:
533:
530:
418:
394:
390:
376:
360:
359:
346:
342:adding to it
337:
302:
287:
280:
268:
248:
244:
240:
236:
232:
230:
206:
199:
193:
183:
179:slant ranges
177:
175:
171:surveillance
160:
147:
143:
139:
138:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
3510:; May 2014.
3226:Rangefinder
3128:Advantages
241:range-range
227:Terminology
3735:Categories
3363:2022-11-06
3261:References
3107:B-2 Spirit
1862:B-2 Spirit
910:Note that
167:navigation
87:newspapers
3615:GPS World
3334:0569-8073
3302:, undated
3243:Surveying
3039:surveying
2978:≈
2821:June 2018
2772:′
2759:±
2754:′
2710:′
2697:−
2692:′
2660:′
2642:′
2607:′
2575:′
2543:′
2516:′
2489:′
2457:′
2395:′
2382:−
2377:′
2361:−
2334:−
2319:′
2301:′
2280:±
2268:′
2240:′
2227:−
2222:′
2201:′
2183:′
2155:′
2111:′
2075:′
2033:′
2002:−
1994:′
1763:−
1750:−
1730:±
1679:−
1625:−
1551:−
1449:−
1417:−
1343:−
865:−
845:±
789:−
698:−
349:June 2017
303:secondary
276:surveying
163:surveying
156:distances
117:June 2017
3180:See also
1018:traverse
743:Thus,
3746:Geodesy
3091:JTIDS (
3023:C1-C2-P
1846:sextant
1081:. When
982:C1-C2-P
299:Y-Gerät
288:primary
265:Ranging
101:scholar
3332:
3237:SHORAN
3078:SHORAN
2730:radars
2135:is at
2057:is at
1976:is at
945:SHORAN
221:angles
152:ranges
103:
96:
89:
82:
74:
3105:USAF
3098:USAF
3086:Gee-H
3037:Land
2981:1.414
2956:and
2928:and
1505:are:
1192:. If
108:JSTOR
94:books
3330:ISSN
3082:Oboe
3057:ERAM
2999:P-C2
2997:and
2995:P-C1
2958:P-C2
2954:P-C1
2930:P-C2
2926:P-C1
2884:and
2674:and
2589:and
2471:and
2053:and
1951:and
1939:and
1892:and
1884:(or
1880:and
1860:and
1813:ERAM
1188:and
1164:3-D
1112:and
1050:and
1014:C2-P
1012:and
1010:C1-P
1000:and
973:and
941:Oboe
566:and
538:and
500:and
393:and
176:Two
146:and
80:news
3359:(3)
3322:doi
2816:.
1166:Tri
344:.
247:or
200:In
63:by
3737::
3636:^
3484:^
3448:^
3389:^
3372:^
3355:.
3351:.
3328:.
3318:95
3316:.
3283:^
3084:,
3080:,
2934:P-
2886:C2
2882:C1
2786:.
2621:.
2055:C2
1974:C1
1953:C2
1949:C1
1941:C2
1937:C1
1894:C3
1890:C2
1888:,
1886:C1
1882:C2
1878:C1
1864:.
1190:C3
1186:C2
1184:,
1182:C1
1020:.
1002:C2
998:C1
975:C2
971:C1
568:C2
564:C1
540:C2
536:C1
417:.
395:C2
391:C1
255:.
223:.
3455:"
3366:.
3357:2
3336:.
3324::
3294:"
3151:)
3059:)
3019:P
2973:2
2950:P
2942:P
2922:P
2908:)
2905:y
2902:,
2899:x
2896:(
2878:P
2874:P
2860:)
2857:y
2854:,
2851:x
2848:(
2823:)
2819:(
2767:2
2763:r
2749:1
2745:r
2734:P
2705:2
2701:r
2687:1
2683:r
2655:2
2651:r
2647:+
2637:1
2633:r
2602:2
2598:r
2570:1
2566:r
2539:y
2512:x
2484:2
2480:r
2452:1
2448:r
2417:U
2414:2
2405:2
2401:)
2390:2
2386:r
2372:1
2368:r
2364:(
2356:2
2352:U
2342:2
2338:U
2329:2
2325:)
2314:2
2310:r
2306:+
2296:1
2292:r
2288:(
2277:=
2264:y
2253:U
2250:2
2245:)
2235:2
2231:r
2217:1
2213:r
2209:(
2206:)
2196:2
2192:r
2188:+
2178:1
2174:r
2170:(
2164:=
2151:x
2133:P
2119:0
2116:=
2106:2
2102:y
2098:,
2095:U
2089:2
2086:1
2080:=
2070:2
2066:x
2041:0
2038:=
2028:1
2024:y
2020:,
2017:U
2011:2
2008:1
1999:=
1989:1
1985:x
1961:P
1933:P
1801:)
1799:2
1797:(
1771:2
1767:y
1758:2
1754:x
1745:2
1740:1
1736:r
1727:=
1720:z
1708:y
1704:V
1700:2
1695:x
1690:x
1686:V
1682:2
1674:2
1669:y
1665:V
1661:+
1656:2
1651:x
1647:V
1643:+
1638:2
1633:3
1629:r
1620:2
1615:1
1611:r
1604:=
1597:y
1587:U
1584:2
1577:2
1573:U
1569:+
1564:2
1559:2
1555:r
1546:2
1541:1
1537:r
1530:=
1523:x
1503:P
1480:2
1476:z
1472:+
1467:2
1463:)
1457:y
1453:V
1446:y
1443:(
1440:+
1435:2
1431:)
1425:x
1421:V
1414:x
1411:(
1408:=
1399:2
1394:3
1390:r
1380:2
1376:z
1372:+
1367:2
1363:y
1359:+
1354:2
1350:)
1346:U
1340:x
1337:(
1334:=
1325:2
1320:2
1316:r
1306:2
1302:z
1298:+
1293:2
1289:y
1285:+
1280:2
1276:x
1272:=
1263:2
1258:1
1254:r
1236:P
1222:)
1219:z
1216:,
1213:y
1210:,
1207:x
1204:(
1194:P
1125:2
1121:r
1098:1
1094:r
1083:P
1079:P
1063:2
1059:r
1036:1
1032:r
1006:P
990:P
967:P
963:P
958:P
936:1
918:y
903:)
901:1
899:(
873:2
869:x
860:2
855:1
851:r
842:=
835:y
825:U
822:2
815:2
811:U
807:+
802:2
797:2
793:r
784:2
779:1
775:r
768:=
761:x
722:2
718:y
714:+
709:2
705:)
701:x
695:U
692:(
689:=
680:2
675:2
671:r
661:2
657:y
653:+
648:2
644:x
640:=
631:2
626:1
622:r
594:)
591:y
588:,
585:x
582:(
572:P
550:U
527:.
513:2
509:r
486:1
482:r
461:U
441:)
438:y
435:,
432:x
429:(
419:P
405:U
351:)
347:(
154:(
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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