Producing a closed 2D parametric region

Question

I would like to create a 2D contour plot inside a closed region where the latter is produced by generating a parametric plot of a piecewise parametric function and then using BoundaryDiscretizeGraphics. If I plot each parametric component over its domain and then combine the plots with Show, the closed region is produced. However, if I use Piecewise to define the function and then produce the parametric plot, the closed region is not produced. I would much rather work with a single piecewise defined function, and thus would like to know what I am doing incorrectly. Any suggestion would be greatly appreciated. Below is a simple example of a triangular region that illustrates the problem.

With Piecewise

x[s_] = Piecewise[{{s, 0 <= s < 1}, {s, 1 <= s < 2}, {2 (3 - s), 
    2 <= s < 3}}]

y[s_] = Piecewise[{{s, 0 <= s < 1}, {2 - s, 1 <= s < 2}, {0, 
    2 <= s < 3}}]
plot01 = ParametricPlot[{x[s], y[s]}, {s, 0, 3}]

BoundaryDiscretizeGraphics[plot01]

Without Piecewise

plot01 = ParametricPlot[{s, s}, {s, 0, 1}];
plot02 = ParametricPlot[{s, 2 - s}, {s, 1, 2}];
plot03 = ParametricPlot[{2 (3 - s), 0}, {s, 2, 3}];
plot04 = Show[{plot01, plot02, plot03}, PlotRange -> All]

BoundaryDiscretizeGraphics[plot04]

Answer 1

We need to use Exclusions -> None to contain the boundary point to make a cycle curve to construct a region.

x[s_] = Piecewise[{{s, 0 <= s < 1}, {s, 1 <= s < 2}, {2 (3 - s), 
     2 <= s < 3}}];
y[s_] = Piecewise[{{s, 0 <= s < 1}, {2 - s, 1 <= s < 2}, {0, 
     2 <= s < 3}}];
{plot01 = ParametricPlot[{x[s], y[s]}, {s, 0, 3}, Exclusions -> None],
  BoundaryDiscretizeGraphics[plot01]} // GraphicsRow

Edit

We can compare the two cases.

x[s_] = Piecewise[{{s, 0 <= s < 1}, {s, 1 <= s < 2}, {2 (3 - s), 
     2 <= s < 3}}];
y[s_] = Piecewise[{{s, 0 <= s < 1}, {2 - s, 1 <= s < 2}, {0, 
     2 <= s < 3}}];
plot01 = ParametricPlot[{x[s], y[s]}, {s, 0, 3}];
plot02 = ParametricPlot[{x[s], y[s]}, {s, 0, 3}, Exclusions -> None];
Cases[plot01, Line[a_] :> a, Infinity] 
Cases[plot, Line[a_] :> a, Infinity] 

( * plot01 *)
{{{0.0009566326530612245`, 
   0.0009566326530612245`}, ..., {0.9990433673469388`, 
   0.9990433673469388`}},
 {{1.0009566326530612`, 
   0.9990433673469388`}, ..., {1.9990433673469388`, 
   0.000956632653061229`}},
 {{1.998086734693878`, 0.`}, ..., {0.`, 0.`}}}

(* plot *)
{{{0., 0.},...,{0., 0.}}}

(* plot04 *)
{{{0.`, 0.`}, ..., {1.`, 1.`}},
 {{1.`, 1.`}, ..., {2.`, 0.`}},
 {{2.`, 0.`}, ..., {0.`, 0.`}}}

It means that plot is one cyclic curve and plot04 is end to end but plot01 just three independent curves and not end to end.