Pxy[u_, ax_, bx_, cx_, dx_, ay_, by_, cy_, 
        dy_] :=  {ax*u^3 + bx*u^2 + cx*u  + dx,  
        ay*u^3 + by*u^2 + cy*u + dy}

P0 = {2, 2};
P1  =  {3, 4};
P2  =  {4, 1};
P3  =  {5, 2};

v  =  
        Solve[
{Pxy[0, ax, bx,  cx, dx, ay, by, cy, 
                dy]  ==  
              P0, 
Pxy[1/3, ax, bx,  cx, dx, ay, by, cy, 
                dy] == P1, 
 
            Pxy[2/3, ax, bx,  cx, dx, ay, by, cy, dy] == 
              P2, 
Pxy[1, ax, bx,  cx, dx, ay, by, cy, 
                dy] == P3}, 
{ax, bx,  cx, dx, ay, 
            by, cy, dy}];

ParametricPlot[{Pxy[u, ax, bx,  cx, dx, ay, by, cy, dy]  /.  
          v[[1]]}, 
{u, 0, 1}]

t =  
      Table[Pxy[u, ax, bx,  cx, dx, ay, by, cy, dy]  /.  
          v[[1]], 
{u, 0, 1, 0.1}]

ListPlot[t]

Options[%]

s = Table[Point[t[[i]]], {i, 1, 11}]

Graphics[{PointSize[0.025], s}]

Show[%, Axes -> Automatic]