Editing True-range multilateration (section)

=== Two Cartesian dimensions, two measured slant ranges (trilateration) ===

[[File:2D Trilat Scenario 2019-0116.jpg|thumb|Fig. 1 2-D Cartesian true-range multilateration (trilateration) scenario. '''C1''' and '''C2''' are centers of circles having known separation <math>U</math>. '''P''' is point whose <math>(x,y)</math> coordinates are desired based on <math>U</math> and measured ranges <math>r_1</math> and <math>r_2</math>.]]

An analytic solution has likely been known for over 1,000 years, and is given in several texts.<ref name="Geyer" /> Moreover, one can easily adapt algorithms for a three dimensional Cartesian space.

The simplest algorithm employs analytic geometry and a station-based coordinate frame. Thus, consider the circle centers (or stations) '''C1''' and '''C2''' in Fig. 1 which have known coordinates (e.g., have already been surveyed) and thus whose separation <math>U</math> is known. The figure 'page' contains '''C1''' and '''C2'''. If a third 'point of interest' '''P''' (e.g., a vehicle or another point to be surveyed) is at unknown point <math>(x,y)</math>, then Pythagoras's theorem yields

: <math> \begin{align}
r_1^2 & = x^2 + y^2  \\[4pt]
r_2^2 & = (U-x)^2 + y^2
\end{align} </math>

Thus,  
{{NumBlk|::|
<math> \begin{align}
x & = \frac { r_1^2 - r_2^2 + U^2 } {2 U}   \\[4pt]
y & = \pm \sqrt{r_1^2 - x^2}
\end{align} </math>
|{{EquationRef|1}}}}

Note that <math>y</math> has two values (i.e., solution is ambiguous); this is usually not a problem. 

While there are many enhancements, Equation {{EquationNote|1}} 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 [[Oboe (navigation)|Oboe]] and during the Korean War [[SHORAN]] 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.<ref name="Aerodist">[http://www.adastra.adastron.com/equip/aerodist.htm Adastra Aerial Surveys] retrieved Jan. 22, 2019.</ref>  This 2-D scenario is sufficiently important that the term ''trilateration'' is often applied to all applications involving a known baseline and two range measurements.

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 '''P''' 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 '''P''' lies. A third example: in applications where '''P''' is an aircraft and '''C1''' and '''C2''' are on the ground, the ambiguous solution is usually below ground.

If needed, the interior angles of triangle '''C1-C2-P''' can be found using the trigonometric [[law of cosines]]. Also, if needed, the coordinates of '''P''' can be expressed in a second, better-known coordinate system—e.g., the [[Universal Transverse Mercator coordinate system|Universal Transverse Mercator (UTM) system]]—provided the coordinates of '''C1''' and '''C2''' are known in that second system. Both are often done in surveying when the trilateration method is employed.<ref name="PSU">[https://www.e-education.psu.edu/natureofgeoinfo/c5_p12.html "The Nature of Geographic Information: Trilateration"], Pennsylvania State Univ., 2018.</ref> Once the coordinates of '''P''' are established, lines '''C1-P''' and '''C2-P''' 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 ''traverse''.

An implied assumption for the above equation to be true is that <math>r_1</math> and <math>r_2</math> relate to the same position of '''P'''. When '''P''' is a vehicle, then typically <math>r_1</math> and <math>r_2</math> 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.

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.