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Hello, First off, I apologize for posting this math question because I know this is not the right forum for it. But I appreciate it if anybody could help me with it. Or if you know of another forum more appropriate for this question, please let me know. Let's first limit ourselves to only the first quadrant of a Cartesian coordinate system. I have a point G that's located on the x-axis, and it's position (x_G, 0) is known (x_G>0). Now, I have four points A, B, C, and D in this quadrant, and I can measure the four distances GA, GB, GC, and GD. The four distance measurements each contain some errors. But if error-free (as shown in the attached figure below; I accidentally posted it twice), they're supposed to be all on a straight line, i.e., a linear trajectory (i.e., the line can be described as y=ax+b). Moreover, these four distances are periodically sampled, meaning that (in the error-free case) AB=BC=CD. And, just for the sake of the argument, let's say GA>GB>GC>GD. However, I do not know the coordinates of these four points in the Cartesian system (denoted as (x_A, y_A), (x_B, y_B), (x_C, y_C), (x_D, y_D), respectively). I want to find the slope of the linear trajectory in the Cartesian system. The way I'm thinking about doing is to cast this as a minimization problem to first find the points' coordinates on the linear trajectory (denote them as (x'_A, y'_A), (x'_B, y'_B), (x'_C, y'_C), (x'_D, y'_D)). That way, the slope can be estimated to be = (y'_A - y'_B) / (x'_A - x'_B). However, I'm not familiar with formulating the minimization (what should be the objective function, and what should be the constraint function). Can anyone help me with this? Thanks in advance.[diagram=361_0][diagram=361_0]

DonBrown, the distance from G to the line AD (let's call it GE, which is the shortest distance and perpendicular to AD), and the values for AB, BC, and CD, ...if there are indeed infinite solutions, then the fact that you were able to compute these GE, AB, BC, and CD from (GA, GB, GC, and GD) must imply that somehow you have implicitly assumed a trajectory slope already before the program was run...?