Calculate distance to a smooth line

Question

I am trying to find the distance to a point (in 4 dimensions, only 2 are shown here) (any colored crosses in the figure) to the supposed Pareto border (black line). This line represents the best representation of the Pareto boundaries in the optimization process.

Pareto = [[0.3875575798354123, -2.4122340425531914], [0.37707675586149786, -2.398936170212766], [0.38176077842761763, -2.4069148936170213], [0.4080534133844003, -2.4914285714285715], [0.35963459448268725, -2.3631532329495126], [0.34395217638838566, -2.3579931972789114], [0.32203302106516224, -2.344858156028369], [0.36742404637441123, -2.3886054421768708], [0.40461156254852226, -2.4141156462585034], [0.36387868122767975, -2.375], [0.3393199109776927, -2.348404255319149]] 

Right now, I am calculating the distance from any point to the Pareto border as follows:

 def dominates(row, rowCandidate): return all(r >= rc for r, rc in zip(row, rowCandidate)) def dist2Pareto(pareto,candidate): listDist = [] dominateN = 0 dominatePoss = 0 if len(pareto) >= 2: for i in pareto: if i != candidate: dominatePoss += 1 dominate = dominates(candidate,i) if dominate == True: dominateN += 1 listDist.append(np.linalg.norm(np.array(i)-np.array(candidate))) listDist.sort() if dominateN == len(pareto): print "beyond" return listDist[0] else: return listDist[0] 

Where I calculate the distance to each point of the black line and get the shortest distance (distance to the nearest point of a known border).

However, I believe that I should calculate the distance to the nearest line segment. How can I achieve this?

Answer 1

The formula for the coordinates of the nearest point on the line is given here . In particular, you are interested in one that is called a "line defined by two points". For posterity, the formula:

Since the border is relatively simple, you can scroll through each two-point line segment on the border and calculate the closest distance for each, keeping the smallest value. You can enter other restrictions / preliminary calculations to limit the number of calculations needed.