Re: Triangulation Problem

*Subject*: [mg3058] Re: [mg2968] Triangulation Problem*From*: nichols at godel.math.cmu.edu (Preston Nichols)*Date*: 30 Jan 1996 05:19:42 -0600*Approved*: usenet@wri.com*Distribution*: local*Newsgroups*: wri.mathgroup*Organization*: Wolfram Research, Inc.*Sender*: daemon at wri.com

David Hoare wrote: <<<< I am looking for the trigonometric / algebreic formulae one would use to triangulate an unknown position, knowing 3 fixed points and the respective distances to the unknown point. (make sense?) Oh - in 3 dimensional space. [....] It's for a program I'm writing, and it's been a while since I've done this sort of thing - - my math has left me! Any Help would be GREATLY appreciated!! >>>> I have seen essentially two responses to this, and I have a few comments on each. 1. Anthony D. Gleckler's approach [mg3007], slightly paraphrased (and the approach I used): If a and b are points (given as lists of coordinates), then the distance between them is, in Mathematica, Sqrt[(a-b).(a-b)] Let the three given points be a = {a1,a2,a3}; b = {b1,b2,b3}; c = {c1,c2,c3}; let the associated distances be A,B,C, and let the unknown point be X = {x,y,z}; Then the desired point is the simultaneous solution of these three equations: spherea := ( (X-a).(X-a) == A^2 ); sphereb := ( (X-b).(X-b) == B^2 ); spherec := ( (X-c).(X-c) == C^2 ); each of which just states the desired equality of distances, but with both sides squared. This is three quadratic equations in three unknowns; sols = Solve[{eqa,eqb,eqc}, X] will return the solution as replacement rules, and soughtpoints = X/.sols will give the unknown point as a list of coordinates -- but not really! First of all, the Mathematica commands given above work fine for numerical values for a, b, c, A, B, and C. If you try to get a general symbolic solution, however, the expressions get unmanageably big. But there's another issue where writing the necessary code is not hard, but deciding just what you want the code to do requires some reflection. Unless the measurements of the given data (a, b, c, A, B, and C) are absolutely perfect, the system of equations will have TWO DISTINCT SOLUTIONS, possibly complex (having equal and opposite imaginary parts), and the soughtpoints as computed above will actually be a list of TWO DISTINCT POINTS, possibly with complex coordinates. Each of the equations shperea, sphereb, and spherec define a sphere; two of the spheres generically intersect in a circle, which will generically intersect the third sphere in two points. These two points (if real) determine a line perpendicular to the plane containing the given points a,b,c, and the two points are equidistant from that plane. If the measurements of a, b, c, A, B, and C are only slightly off, these two points will be very close together, and simply averaging them will give a reasonable, usable result: thepoint = Apply[Plus, soughtpoints]/2 (Conveniently, this will give a real result even if complex solutions are involved!) You might, however, wish to use the distance between the two solutions as an indicator of the quality of your data (i.e. the given points and distances). At the simplest level, a relatively "large" distance between the two solution points indicates that your values for A, B, and C *can't* be accepted as distances of a, b, and c from a common fourth point; you could use that "largeness" simply as a tipoff that something's wrong with the data. With some more effort, you could also convert the size of that distance into a numerical measurement of how well all your pieces are fitting together. --------------------------------------------------- 2. Dave Rusin's approach [mg3015] I really enjoyed Dave's method, because it so closely follows what we might do if we could directly manipulate abstract geometrical objects in 3-space with our hands. He suggests locating and explicitly parametrizing the circle where the first two spheres intersect. (I also think parametrizing curves is underappreciated in general.) This can be done in Mathematica by using some tricks with NullSpace and LinearAlgebra`Orthogonalization`GramSchmidt. Dave's method will lead to the same two solutions for less than perfect data, but this time they appear in the (ususally) multiple solutions of the trigonometric equation > D3^2 = (r cos(t)-a)^2 + (r sin(t)-b)^2 + (x-c)^2 the solving of which involves taking multi-valued arcsines. (In 1, we might have been alerted to multiple solutions by the fact that we were solving quadratic equations.) --------------------------------------------------- Preston Nichols Visiting Assistant Professor Department of Mathematics Carnegie Mellon University (seeking employment for Fall 1996)