How to plot a function with different conditions (if or which or piecewise)

Question

The given function is

This function is plotted to be

V[x_, r_] := 
 Piecewise[{{-Sqrt[2] r + Sqrt[r^2 - x^2], 
    Abs[x] <= Sqrt[2]/2 r}, {-1 + Sqrt[2] x - Sqrt[
     r^2 - (Abs[x] - 1)^2], Abs[x] >= 1 - Sqrt[2]/2 r}, {-Abs[x],}}]

My problem is how to write the otherwise condition as given in the paper for the Mathematica and plot the function for different values of r.

Answer 1

• -1 + Sqrt[2] x - Sqrt[ r^2 - (Abs[x] - 1) should be -1 + Sqrt[2] r - Sqrt[r^2 - (Abs[x] - 1)^2]
• The otherwise condition -Abs[x] is at the end of all List.

expr = Piecewise[{{-Sqrt[2] r + Sqrt[r^2 - x^2], 
    Abs[x] <= Sqrt[2]/2 r}, {-1 + Sqrt[2] r - 
     Sqrt[r^2 - (Abs[x] - 1)^2], Abs[x] >= 1 - Sqrt[2]/2 r}}, -Abs[
    x]]
Plot[Table[expr, {r, {.01, .1, .2}}] // Evaluate, {x, -1, 1}, 
 PlotRange -> {-1, 0}, PlotStyle -> {Black, Red, Blue}, Frame -> True,
  Axes -> False, 
 PlotLegends -> 
  Placed[LineLegend[{Black, Red, Blue}, {"r=0.2", "r=0.1", "r=0.01"}, 
    LegendMarkers -> "OpenMarkers"], {.2, .8}]]