/*
    The following scene is a quick-and-dirty brute force approximation
    for Minkowski sums. Two arbitrary shapes are minkowski-added by
    chosing many random points in the first given object, then adding
    them to the second object, i.e. making a translated copy of the
    second object. The number of copies should be high (1000 to 10000),
    the objects should be simple, esp. the second object. Inner points
    of the first object usually don't contribute anything visible to
    the sum because they translate the second object to a place where
    it will be surrounded by copies belonging to points in the vicinity
    of this inner point. So it's a good idea to remove inner points of
    the first object; this will increase parse time (because it's
    difficult to find points of a thin object), but will greatly
    increase the quality of the surface of the minkowski sum (because
    no copies are wasted). Modify the #switch value to see the
    examples.

       Sputnik

*/


// Minkowski sum

// +SP8 +EP8 +D -F +W320 +H240

#macro Minkowski_sum (Copies, O1, O2)

  // preparations for sampling the bounding box of O1
  #local Min = min_extent (O1);
  #local Size = max_extent (O1)-Min;
  #local Rand = seed(123);

  union {

    #local I = 0; // counts inside-points
    #local N = 0; // counts tested points
    #while (I<Copies)
      // get a random point in the bounding box of O1
      #local Point = Min+Size*<rand(Rand),rand(Rand),rand(Rand)>;
      #local N = N+1;
      // if this point is inside O1 then ...
      #if (inside(O1, Point))
        // ... minkowski-add it to O2 and count it
        object { O2 translate Point }
        #local I = I+1;
        #end//if
      #end//while I

    // no "}" for union to allow application of a texture

    #debug concat ( str(I*100/N,10,2), "% successful inside tests\n" )

  #end//macro Minkowski_sum


#switch (2) // <-- choose an example here (1 .. 3)


  #case (1) // rounded box
    Minkowski_sum ( 10000,
      box { -4, 4 }
      sphere { 0, 1 }
      )
      texture { pigment { color rgb 1 } }
      }//the missing "}" for the union
    #break//1


  #case (2) // rounded box, improved
    Minkowski_sum ( 2000,
      difference { box { -4, 4 } box { -3.9, 3.9 } }
      sphere { 0, 1 }
      )
      texture { pigment { color rgb 1 } }
      }
    #break//2


  #case (3) // "coned" torus
    Minkowski_sum ( 1000,
      difference {
        torus { 5, 1 }
        torus { 5, 0.9 }
        }
        cone { 0, 2, 3*y, 0.5 }
      )
      texture { pigment { color rgb 1 } }
      }
    #break//3


  #case (4) // "torused" cone = 1.0*cone + 1.0*torus
    Minkowski_sum ( 1000,
      difference {
        cone { 0  *y, 2  , 3  *y, 0.5 }
        cone { 0.1*y, 1.9, 2.9*y, 0.4 }
        }
      torus { 5, 1 }
      )
      texture { pigment { color rgb 1 } }
      }
    #break//4


  #case (5) // 1.6*cone + 0.4*torus
    Minkowski_sum ( 1000,
      difference {
        cone { 0  *y, 2  , 3  *y, 0.5 }
        cone { 0.1*y, 1.9, 2.9*y, 0.4 }
        scale 1.6
        }
      torus { 5, 1 scale 0.4 }
      )
      texture { pigment { color rgb 1 } }
      }
    #break//5


  #case (6) // 0.4*cone + 1.6*torus
    Minkowski_sum ( 1000,
      difference {
        cone { 0  *y, 2  , 3  *y, 0.5 }
        cone { 0.1*y, 1.9, 2.9*y, 0.4 }
        scale 0.4
        }
      torus { 5, 1 scale 1.6 }
      )
      texture { pigment { color rgb 1 } }
      }
    #break//6


  #end//switch


light_source { <-100, 150, -50>, rgb 1 }

camera { location <-5, 10, -15> look_at -1.5*y }

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