# How to connect boundary points of graph

I have the following data;

pnts = {{-1.8, 0.2}, {-1.8, 0.4}, {-1.8, 0.6}, {-1.8, 0.8}, {-1.8,
1.}, {-1.8, 1.2}, {-1.8, 1.4}, {-1.8, 1.6}, {-1.8, 1.8}, {-1.8,
2.}, {-1.6, 0.2}, {-1.6, 0.4}, {-1.6, 0.6}, {-1.6, 0.8}, {-1.6,
1.}, {-1.6, 1.2}, {-1.6, 1.4}, {-1.6, 1.6}, {-1.6, 1.8}, {-1.6,
2.}, {-1.4, 0.2}, {-1.4, 0.4}, {-1.4, 0.6}, {-1.4, 0.8}, {-1.4,
1.}, {-1.4, 1.2}, {-1.4, 1.4}, {-1.4, 1.6}, {-1.4, 1.8}, {-1.4,
2.}, {-1.2, 0.2}, {-1.2, 0.4}, {-1.2, 0.6}, {-1.2, 0.8}, {-1.2,
1.}, {-1.2, 1.2}, {-1.2, 1.4}, {-1.2, 1.6}, {-1.2, 1.8}, {-1.2,
2.}, {-1., 0.2}, {-1., 0.4}, {-1., 0.6}, {-1., 0.8}, {-1.,
1.}, {-1., 1.2}, {-1., 1.4}, {-1., 1.6}, {-1., 1.8}, {-1.,
2.}, {-0.8, 0.2}, {-0.8, 0.4}, {-0.8, 0.6}, {-0.8, 0.8}, {-0.8,
1.}, {-0.8, 1.2}, {-0.8, 1.4}, {-0.8, 1.6}, {-0.8, 1.8}, {-0.8,
2.}, {-0.6, 0.2}, {-0.6, 0.4}, {-0.6, 0.6}, {-0.6, 0.8}, {-0.6,
1.}, {-0.6, 1.2}, {-0.6, 1.4}, {-0.6, 1.6}, {-0.6, 1.8}, {-0.6,
2.}, {-0.4, 0.2}, {-0.4, 0.4}, {-0.4, 0.6}, {-0.4, 0.8}, {-0.4,
1.}, {-0.4, 1.2}, {-0.4, 1.4}, {-0.4, 1.6}, {-0.4, 1.8}, {-0.4,
2.}, {-0.2, 0.2}, {-0.2, 0.4}, {-0.2, 0.6}, {-0.2, 0.8}, {-0.2,
1.}, {-0.2, 1.2}, {-0.2, 1.4}, {-0.2, 1.6}, {-0.2, 1.8}, {-0.2,
2.}, {0., 0.2}, {0., 0.4}, {0., 0.6}, {0., 0.8}, {0., 1.}, {0.,
1.2}, {0., 1.4}, {0., 1.6}, {0., 1.8}, {0., 2.}, {0.2, 0.2}, {0.2,
0.4}, {0.2, 0.6}, {0.2, 0.8}, {0.2, 1.}, {0.2, 1.2}, {0.2,
1.4}, {0.2, 1.6}, {0.2, 1.8}, {0.2, 2.}, {0.4, 0.2}, {0.4,
0.4}, {0.4, 0.6}, {0.4, 0.8}, {0.4, 1.}, {0.4, 1.2}, {0.4,
1.4}, {0.4, 1.6}, {0.4, 1.8}, {0.4, 2.}, {0.6, 0.2}, {0.6,
0.4}, {0.6, 0.6}, {0.6, 0.8}, {0.6, 1.}, {0.6, 1.2}, {0.6,
1.4}, {0.6, 1.6}, {0.6, 1.8}, {0.6, 2.}, {0.8, 0.2}, {0.8,
0.4}, {0.8, 0.6}, {0.8, 0.8}, {0.8, 1.}, {0.8, 1.2}, {0.8,
1.4}, {0.8, 1.6}, {0.8, 1.8}, {0.8, 2.}, {1., 0.2}, {1.,
0.4}, {1., 0.6}, {1., 0.8}, {1., 1.}, {1., 1.2}, {1., 1.4}, {1.,
1.6}, {1., 1.8}, {1., 2.}, {1.2, 0.2}, {1.2, 0.4}, {1.2,
0.6}, {1.2, 0.8}, {1.2, 1.}, {1.2, 1.2}, {1.2, 1.4}, {1.2,
1.6}, {1.2, 1.8}, {1.2, 2.}, {1.4, 0.2}, {1.4, 0.4}, {1.4,
0.6}, {1.4, 0.8}, {1.4, 1.}, {1.4, 1.2}, {1.4, 1.4}, {1.4,
1.6}, {1.4, 1.8}, {1.4, 2.}, {1.6, 0.2}, {1.6, 0.4}, {1.6,
0.6}, {1.6, 0.8}, {1.6, 1.}, {1.6, 1.2}, {1.6, 1.4}, {1.6,
1.6}, {1.6, 1.8}, {1.6, 2.}, {1.8, 0.2}, {1.8, 0.4}, {1.8,
0.6}, {1.8, 0.8}, {1.8, 1.}, {1.8, 1.2}, {1.8, 1.4}, {1.8,
1.6}, {1.8, 1.8}, {1.8, 2.}, {2., 0.2}, {2., 0.4}, {2., 0.6}, {2.,
0.8}, {2., 1.}, {2., 1.2}, {2., 1.4}, {2., 1.6}, {2., 1.8}, {2.,
2.}};
newpnts = Select[pnts, ! #[[1]]^2 + #[[2]]^2 < 1 &];
ListPlot[newpnts];


The graph of newpnts is essentially a uniform spread of points with increment .2, that has had points excluded where x^2+y^2<1. I wanted to know if there was an easy way to get mathematica to connect the boundary points for me. Essentially it would be creating 4 line segments and a semi circle. Is the only way to do this to create functions for each segment? If so how do I lay all 5 functions on the same plot?

{xmin, xmax} = MinMax[newpnts[[All, 1]]];

{ymin, ymax} = MinMax[newpnts[[All, 2]]];

RegionPlot[
xmin <= x <= xmax && ymin <= y <= ymax && x^2 + y^2 >= 1,
{x, -2, 2}, {y, 0, 2},
PlotStyle -> White,
BoundaryStyle -> Red,
Epilog -> {Blue, Point[newpnts]},
AspectRatio -> 1/2]


If I understand correctly you are interested in simply drawing lines with know equations for this specific set of points:

Show[
ListPlot[newpnts],
Graphics[{Red,Thick,Line[{{-1,.2},{-1.8,.2},{-1.8,2},{2,2},{2,.2},{1,.2}}]}],
ContourPlot[x^2+y^2==1,{x,-1,1},{y,.2,1},ContourStyle->Directive[Red,Thick]]
]