Since in the original parametric equation,{x[0],y[0]} is not equal to {x[8],y[8]}, it means that such curve is not a closed curve. We need to replace some Less to LessEqual.

Now we can directly use BoundaryDiscretizeGraphics by add Exclusions -> None to contain the boundary point of segment to make the curves be a closed curve.

• Since in the original parametric equation,{x[0],y[0]} is not equal to {x[8],y[8]}, it means that such curve is not a closed curve. We need to replace some Less to LessEqual.

{x[0], y[0]} == {x[8], y[8]}

True.

• Now we can directly use BoundaryDiscretizeGraphics by add Exclusions -> None to contain the boundary point of segment to make the curves be a closed curve.

Since in the original parametric equation,{x[0],y[0]} is not equal to {x[8],y[8]}, it means that such curve is not a close curve. We need to replace some Less to LessEqual.

Clear[x,y];
x[t_] = -(Piecewise[{{-1, 
         Inequality[0, LessEqual, t, LessEqual, 2]}, {-3 + t, 
         Inequality[2, Less, t, LessEqual, 4]}, {1, 
         Inequality[4, Less, t, LessEqual, 6]}, {7 - t, 
         Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2]) + 
   Piecewise[{{-1 + t, Inequality[0, LessEqual, t, LessEqual, 2]}, {1,
        Inequality[2, Less, t, LessEqual, 4]}, {5 - t, 
       Inequality[4, Less, t, LessEqual, 6]}, {-1, 
       Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2];

y[t_] = Piecewise[{{-1, 
       Inequality[0, LessEqual, t, LessEqual, 2]}, {-3 + t, 
       Inequality[2, Less, t, LessEqual, 4]}, {1, 
       Inequality[4, Less, t, LessEqual, 6]}, {7 - t, 
       Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2] + 
   Piecewise[{{-1 + t, Inequality[0, LessEqual, t, LessEqual, 2]}, {1,
        Inequality[2, Less, t, LessEqual, 4]}, {5 - t, 
       Inequality[4, Less, t, LessEqual, 6]}, {-1, 
       Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2];

Now we can directly use BoundaryDiscretizeGraphics by add Exclusions -> None to contain the boundary point to make the curves be a close curve.

ParametricPlot[{x[t], y[t]}, {t, 0, 8}, 
  Exclusions -> None] // BoundaryDiscretizeGraphics

Since in the original parametric equation,{x[0],y[0]} is not equal to {x[8],y[8]}, it means that such curve is not a closed curve. We need to replace some Less to LessEqual.

Clear[x,y];
x[t_] = -(Piecewise[{{-1, 
         Inequality[0, LessEqual, t, LessEqual, 2]}, {-3 + t, 
         Inequality[2, Less, t, LessEqual, 4]}, {1, 
         Inequality[4, Less, t, LessEqual, 6]}, {7 - t, 
         Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2]) + 
   Piecewise[{{-1 + t, Inequality[0, LessEqual, t, LessEqual, 2]}, {1,
        Inequality[2, Less, t, LessEqual, 4]}, {5 - t, 
       Inequality[4, Less, t, LessEqual, 6]}, {-1, 
       Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2];

y[t_] = Piecewise[{{-1, 
       Inequality[0, LessEqual, t, LessEqual, 2]}, {-3 + t, 
       Inequality[2, Less, t, LessEqual, 4]}, {1, 
       Inequality[4, Less, t, LessEqual, 6]}, {7 - t, 
       Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2] + 
   Piecewise[{{-1 + t, Inequality[0, LessEqual, t, LessEqual, 2]}, {1,
        Inequality[2, Less, t, LessEqual, 4]}, {5 - t, 
       Inequality[4, Less, t, LessEqual, 6]}, {-1, 
       Inequality[6, Less, t, LessEqual, 8]}}, 0]/Sqrt[2];

Now we can directly use BoundaryDiscretizeGraphics by add Exclusions -> None to contain the boundary point of segment to make the curves be a closed curve.

ParametricPlot[{x[t], y[t]}, {t, 0, 8}, 
  Exclusions -> None] // BoundaryDiscretizeGraphics