# How to get the coordinates of the closest point along a line?

I have a plot as below (the plot is just for illustration purpose. in my real problem, I just have pts1):-

pts1 = BlockRandom[SeedRandom; RandomReal[2.9, {30, 2}]];
pts1L = MapThread[Labeled[#1, #2] &, {pts1, Range[Length@pts1]}];
plot1 = ListPlot[pts1L, PlotRange -> {{0, 3.5}, {0, 3}}];
plot2 = ParametricPlot[{3, t}, {t, 0, 3}, PlotStyle -> Darker[Red]];
Show[plot1, plot2] When I have a vector representing the red line, let say redLine = {0, 0.5, 1, 1.5, 2, 2.5, 3}, I want to have the coordinates of the closet points as output, i.e. {pts1[], pts1[], pts1[], pts1[], pts1[], pts1[], pts1[]} as output.

My current solution is as below, but that's too clumsy and will be very slow when the scale of the problem increase. What are the more efficient ways to get it done? (For example, is it possible to calculate the DistanceMatrix of just "frontline points v.s. red line" instead of "all points v.s. red line"?)

redLine = {0, 0.5, 1, 1.5, 2, 2.5, 3}
redLineCood = Transpose@ArrayFlatten[{ConstantArray[3, Length@redLine], redLine}]
dM = DistanceMatrix[redLineCood, pts1];
min = Min /@ dM;


Many thanks!

• pts1=BlockRandom[SeedRandom;RandomReal[2.9,{30,2}]]; redLine={0,0.5,1,1.5,2,2.5,3}; redLineCood=Map[{3,#}&,redLine]; f[p_]:=MinimalBy[pts1,Norm[p-#]&][]; f/@redLineCood – Bill Jun 17 '18 at 0:18
• Thanks. This is so elegant. – H42 Jun 17 '18 at 0:59
• If the shortest possible distance from pi to the red line is greater than the greatest possible distance from pj to the red line then pi could be discarded. But that is still roughly an n^2 process of comparing all pi against all pj. I don't see any way of overcoming that. Ah, and redLineCood={3,#}&/@redLine; – Bill Jun 17 '18 at 1:07
• ‘Nearest[pts, #, 1]& /@ Thread[{3, redLine}]’? – kglr Jun 17 '18 at 1:35
• Thanks, Nearest[pts1] /@ Thread[{3, redLine}] works so well! – H42 Jun 17 '18 at 1:41

 Nearest[pts1] /@ Thread[{3, redLine}]