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                              From Three Dimensions to Two and Back

The cell below shows how Mathematica may be used to find the location of the shadow produced by an object located at the point

                                                                        P = (2, 1, 1)

illuminated by a light source located at the point

                                                                        S = (1, -2, 8)

Evaluate it now.
;[s]
10:0,1;67,0;94,2;105,0;269,1;282,0;408,1;422,0;424,3;439,0;441,-1;
4:5,13,9,Times,0,12,0,0,0;3,13,9,Times,1,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect]
Clear[P, S, t]

P = {2, 1, 1};
S = {1, -2, 8};

Solve[S + t (P - S) == {x, y, 0}, {t, x, y}]

{x, y} /. %
:[font = special1; inactive; preserveAspect]
The next  cell packages the computations above as a function, so that they can easily be applied to a sequence of points.  Evaluate it now.
;[s]
3:0,0;123,1;139,0;140,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect]
Clear[P, S]

Shadow[P_, S_] :=
  ({x, y} /. 
     Solve[S + t (P - S) == {x, y, 0}, {t, x, y}])[[1]]
   
Shadow[{2, 1, 1}, {1, -2, 8}]
:[font = special1; inactive; preserveAspect]
The next cell defines the cube used in the first set of exercises in the textbook and then solves the first exercise using the function defined above.  Notice that it gives numeric results and it draws the shadow of the cube.  Evaluate it now.
;[s]
2:0,0;227,1;244,-1;
2:1,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect]
Cube = {{0, 0, 0},
        {0, 1, 0},
        {1, 1, 0},
        {1, 0, 0},
        {0, 0, 1},
        {0, 1, 1},
        {1, 1, 1},
        {1, 0, 1}};
        
P = {0, 0, 2};
        
Ground   := 
  Table[Shadow[Cube[[i]], P], {i, 1, 8}]
  
Vertices := 
  Table[{PointSize[0.03], Point[Ground[[i]]]}, {i, 1, 8}]
  
Base := {
  {RGBColor[1, 0, 0], Line[{Ground[[1]], Ground[[2]], Ground[[3]],
                            Ground[[4]], Ground[[1]]}]}
        }
        
Topp := {
  {RGBColor[0, 0, 1], Line[{Ground[[5]], Ground[[6]], Ground[[7]],
                            Ground[[8]], Ground[[5]]}]}
        }
                            
Sides := Table[{RGBColor[0, 0, 0], Line[{Ground[[i]], Ground[[i + 4]]}]},
               {i, 1, 4}]
  
Show[Graphics[Join[Vertices, Base, Topp, Sides]],
        PlotRange -> {{-1, 3}, {-1, 3}},
        AspectRatio -> Automatic,
        Axes -> False]
^*)