select from a list of disks

Question

I have the following list of centers of disks.

r=0.03;
pts = {{0.10420089319018544`, -0.024872674177014872`}, \
{0.9743669105930046`, 0.9169054125547074`}, {0.028760526736240563`, 
    0.45959879163736717`}, {-0.0059035632830851115`, 
    0.2922099255180086`}, {0.41615337459441437`, 
    0.9928402345083696`}, {0.23798002911834915`, 
    0.028922198016334083`}, {0.9439796979256947`, 
    0.014706790894535735`}, {1.0038820168274474`, 
    0.677501854561434`}, {0.5927003061040934`, 
    1.0077273741847754`}, {1.0067437297432018`, 
    0.9752013154551493`}, {0.6938266053169369`, 
    1.0266240439258578`}, {0.98338794538608`, 
    0.26086568952319356`}, {0.9782652227588948`, 
    0.5171196676396621`}, {0.2545283903372031`, 
    1.0169130378365447`}, {-0.004475792541620699`, 
    0.08252251782338371`}, {0.0792067543327557`, 
    1.0190929886159708`}, {0.01012007453604613`, 
    0.9742793220312869`}, {0.37795635641517067`, 
    0.0006807388703829187`}, {0.16475658437291774`, 
    1.0073414058455288`}, {0.7109091765268767`, 
    0.010031721670787197`}, {0.751636715216994`, 
    0.9718377389063747`}, {-0.02868599270782108`, 
    0.8001977126969162`}, {0.9951076171959166`, 
    0.06361811727917832`}, {0.48856448143911413`, 
    0.9822360639397044`}, {0.8498119226349552`, 
    0.9954192191276157`}, {0.9343505459643862`, 
    1.026090808866221`}, {-0.02352512434168942`, 
    0.3638162649853829`}, {0.015464116868890446`, 
    0.8880954853506882`}, {0.5397449644933747`, 
    0.0075650003442971625`}, {1.0124037284145468`, 
    0.3445056232122783`}, {0.012611151682823951`, 
    0.22876062170703393`}, {0.01349201390129795`, 
    0.012484449700555664`}, {1.0287174186840569`, 
    0.4444472847751937`}, {1.0148117241333423`, 
    0.8423869907442485`}, {0.028166977268371904`, 
    0.749683634621167`}, {0.17535597671574954`, \
-0.02307375065511537`}, {0.6373054881521303`, -0.02231729523620847`}, \
{0.9791564983854446`, 0.7765347566257843`}, {-0.028389156500012468`, 
    0.7133375307394074`}, {0.47391260398278434`, \
-0.0029577636868491908`}, {0.009099484184960496`, 
    0.5770732674239099`}, {1.013402544772932`, 
    0.17610957824632423`}, {0.9714646177489623`, 
    0.6166237906203449`}, {0.30769518373972393`, \
-0.008573400690361568`}, {0.8261937637371302`, 
    0.001386073976981178`}, {0.8914586354858451`, \
-0.026330731282082395`}, {0.02725861951844763`, 
    0.6605283794945094`}, {-0.026575383043093995`, 
    0.5203231079582309`}, {0.004468178226517766`, 
    0.16296092728164324`}, {0.3189756417708023`, 0.9723496550714829`}};

Graphics[{FaceForm@Lighter[Blue, 0.8], 
  EdgeForm[{Thickness[0.004], Black}], Disk[#, r] & /@ pts}, 
 Background -> Lighter[Gray], Frame -> True, 
 PlotRange -> {{0, 1}, {0, 1}}]

I want to delete from this list the centers that are above the line in the figure below

I use

ptsxy = DeleteCases[
   pts, {x_, 
     y_} /; (0.05 <= x <= 1 && y >= 0.9) || (0 <= y <= 1 && x >= 0.9)];
Graphics[{FaceForm@Lighter[Blue, 0.8], 
  EdgeForm[{Thickness[0.004], Black}], Disk[#, r] & /@ ptsxy}, 
 Background -> Lighter[Gray], Frame -> True, 
 PlotRange -> {{0, 1}, {0, 1}}]

But I am sure that there are better ways.

Answer 1

You can use geometry property. Let's say your equation of line is x+y-1=0. You can distinguish whether a point lies on same side as that of the origin by substituting the points into the equation of line. If the value is same that when origin 0,0 is substituted, then the point lies on the same side.

Here is a simple code that you can use:

eqline = (x + y - 1);
snOrigin = Sign[(eqline /. {x -> 0, y -> 0})];
Select[pts, (eqline /. {x -> First@#,y -> Last@#})*snOrigin > 0 &];

This should give you all the points which lie on the side of the line containing the origin.

Answer 2

Given the definition of the circle centres in pts, and a line parameterised by y=m*x+b.

Manipulate[
   Module[{r = 0.03, x = pts[[All, 1]], y = pts[[All, 2]], p},
      p = Pick[pts, UnitStep[y - m*x - b], 0];
      Graphics[{
         EdgeForm[{Thickness[0.004], Black}], FaceForm[Lighter[Blue, 0.8]],
         Map[Disk[#, r] &, p],
         Thin, Map[Circle[#, r] &, pts],
         Red, Line[Select[Transpose[{x, m*x + b}], -0.4 < #[[2]] < 1.4 &]]
         }, PlotRange -> {{-0.1, 1.1}, {-0.1, 1.1}}, Frame -> True]],
  {{m, -1.0}, -10., 10., Appearance -> "Labeled"},
  {{b, 1.0}, -5., 5., Appearance -> "Labeled"}
]

Answer 3

This is a good place to use RegionMember and HalfPlane. First, I reproduced your diagram, after explicitly drawing the square and tweaking the formatting a bit:

square = {
   FaceForm[{Lighter@Gray, Opacity[0.2]}],
   EdgeForm[{Thickness[0.002], Black}],
   Rectangle[{0.0, 0.0}, {1.0, 1.0}]};

disks[pts_] := {
   EdgeForm[{Thickness[0.001], Black}],
   Disk[#, r] & /@ its};

Graphics[{
  square,
  {FaceForm[{Lighter@Blue, Opacity[0.2]}],
   disks[pts]}}]

Then I used the drawing tools to draw a line similar to the one you provided, and assigned the edited Graphics object to a variable using copy and paste:

This allowed me to use cases to pick out the points defining that line, and then construct a RegionMember function to see which centers were in the HalfPlane above it:

With[{hp = FirstCase[pic, Line[ps_] :> HalfPlane[ps, {0, 1}], 
    Missing["NotFound"], Infinity]}, With[{fn = RegionMember[hp]}, 
  With[{split = GroupBy[pts, fn]}, 
   Graphics[
    {square, 
     {EdgeForm[{Red, Thick}], Opacity[0.2], 
      Darker[Gray], hp}, 
     {FaceForm[{Lighter[Green], Opacity[0.3]}], 
      disks[split[False]]}, 
     {FaceForm[{Lighter[Red], Opacity[0.3]}], 
      disks[split[True]]}}]]]]

Here, I used GroupBy to allow me to separate members from non-members in one swell foop. I also drew the HalfPlane, and drew the rejected disks in red instead of just omitting them.