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Trilaterate your position

Code Golf Asked by Denker on November 17, 2021

Introduction

Imagine you are on a two dimensional cartesian plane and want to determine your position on it. You know 3 points on that plane and your distance to each of them. While it is always possible to calculate your position from that, doing that in your head is pretty hard. So you decide to write a program for that.

The Challenge

Given 3 points and your distance to them, output the cordinates of your position.

  • Input and output may be in any convenient format, including using complex instead of real numbers. Please clarify in your answer which format you use.
  • You will always get exactly 3 distinct points with their distance to you.
  • The coordinates and distances will be floats with arbitrary precision. Your output has to be correct to 3 decimal places. Rounding is up to you. Please clarify in your answer.
  • You may assume that the three points are not collinear, so there will be always an unique solution.
  • You are not allowed to bruteforce the solution.
  • You may not use any builtins that trivialize this particular problem. Builtins for vector norms, etc. are allowed though.

Hint to get started:

Rules

Test cases

Input format for one point here is [[x,y],d] with x and y being the coordinates and d being the distance to this point. The 3 of those points are arranged in a list. Output will be x and then y in a list.

[[[1, 2], 1.414], [[1, 1], 2.236], [[2, 2], 1.0]] -> [2, 3]
[[[24.234, -13.902], 31.46], [[12.3242, 234.12], 229.953], [[23.983, 0.321], 25.572]] -> [-1.234, 4.567]
[[[973.23, -123.221], 1398.016], [[-12.123, -98.001], 990.537], [[-176.92, 0], 912.087]] -> [12.345, 892.234]

You can generate additional test cases with this Pyth program. Location goes on the first line of the input and the 3 points are on the following 3 lines.

Happy Coding!

4 Answers

APL (Dyalog Unicode) 18.0, 23 bytes

-∘(+/)⍥(×⍨)⌹⍥(2-⌿⊢)¯2×⊢

Try it online!

A tacit infix function that can be called like dists f pts, where pts contains three centers' coordinates as a 3-row 2-column matrix, and dists contains three distances to the respective centers.

Uses 18.0 feature . TIO's Dyalog Unicode is not updated yet from 17.1, so TIO link uses Dyalog Extended for demonstration purposes.

How it works

Uses the same method as R.T.'s Python answer, i.e. constructing a linear system of equations and solving it.

The distance equations are necessarily quadratic. Then how did we get linear ones?

Let's consider just two circles, and denote the point we want to find as $P$, two circles' centers as $C, D$, and their radii as $r_C, r_D$. Also, let's use $A_x, A_y$ for the $x$- and $y$-coordinates of a point $A$. Then the following equations hold:

$$ (P_x-C_x)^2+(P_y-C_y)^2 =r_C^2 \ (P_x-D_x)^2+(P_y-D_y)^2 =r_D^2 \ $$

Then we expand them and subtract the second from the first:

$$ P_x^2-2P_xC_x+C_x^2+P_y^2-2P_yC_y+C_y^2=r_C^2\ P_x^2-2P_xD_x+D_x^2+P_y^2-2P_yD_y+D_y^2=r_D^2\ -2P_x(C_x-D_x)+(C_x^2-D_x^2)-2P_y(C_y-D_y)+(C_y^2-D_y^2)=r_C^2-r_D^2\ -2P_x(C_x-D_x)-2P_y(C_y-D_y)=(r_C^2-C_x^2-C_y^2)-(r_D^2-D_x^2-D_y^2) $$

We've just got one linear equation on $P_x$ and $P_y$. Since we have three circles given, we can take any two pairs and construct two equations, which then solving the system of equations gives the coordinates we want.

-∘(+/)⍥(×⍨)⌹⍥(2-⌿⊢)¯2×⊢  ⍝ Left: dists, Right: pts
-∘(+/)⍥(×⍨) ⍥(2-⌿⊢)      ⍝ Constants part
      ⍥(×⍨)              ⍝ Square each number in both dists and pts
 ∘(+/)                   ⍝ Row sums of squared pts
-                        ⍝ Subtract above from squared dists
            ⍥(2-⌿⊢)      ⍝ Take pairwise difference

            ⍥(2-⌿⊢)¯2×⊢  ⍝ Coefficients part
                   ¯2×⊢  ⍝ -2 times pts
            ⍥(2-⌿⊢)      ⍝ Take pairwise difference between rows
           ⌹             ⍝ Solve linear equations

Answered by Bubbler on November 17, 2021

Python - 172

Takes input as a list of tuples of form (x,y,d). Let me know if you see a way to golf this further, I feel like there must be but I can't figure it out!

import numpy as N
def L(P):
    Z=[p[2]**2-p[0]**2-p[1]**2 for p in P];return N.linalg.solve([[P[i][0]-P[0][0],P[i][1]-P[0][1]]for i in[1,2]],[(Z[0]-Z[i])*0.5 for i in [1,2]])

Answered by R.T. on November 17, 2021

C, 362 348 345 bytes

Input is given as a sequence of space-separated floats on stdin: x1 y1 d1 x2 y2 d2 x3 y3 d3. Output is similar on stdout: x y.

#define F"%f "
#define G float
#define T(x)(b.x*b.x-a.x*a.x)
typedef struct{G a;G b;G c;}C;G f(C a,C b,G*c){G x=b.b-a.b;*c=(T(a)+T(b)-T(c))/x/2;return(a.a-b.a)/x;}main(){C a,b,c;G x,y,z,t,m;scanf(F F F F F F F F F,&a.a,&a.b,&a.c,&b.a,&b.b,&b.c,&c.a,&c.b,&c.c);x=f(a,a.b==b.b?c:b,&y);z=f(b.b==c.b?a:b,c,&t);m=t-y;m/=x-z;printf(F F"n",m,x*m+y);}

C is a structure type whose members are an x-coordinate a, a y-coordinate b, and a distance (radius) c. The function f takes two C structures and a pointer to a float, and determines the line at which the C (circles) intersect. The y-intercept of this line is placed into the pointed-to float, and the slope is returned.

The program calls f on two pairs of circles, then determines the intersection of the produced lines.

Answered by Fox on November 17, 2021

Desmos, 122 bytes

Online use. Copy+paste each equation into an equation box, click "add all" for each box, then click on the point of intersection, then enter in each value as appropriate. each of A, B, and C are the distances for the points (a,b), (c,d), and (E,f), respectively. To get a square root in the value, type sqrt then the value in the box.

left(x-aright)^2+left(y-bright)^2=AA
left(x-cright)^2+left(y-dright)^2=BB
left(x-Eright)^2+left(y-fright)^2=CC

Verify the first test case.

Or you can take a look here:

circles!

Answered by Conor O'Brien on November 17, 2021

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