Well, I played a bit with that, and I tried to see what's going on with
clipped_by.

Clipped_By didn't gave the warning that Bounded_By gave: uneccary
bouding object
removed. If I clipped the box with the paraboloid, I got the cap of the
paraboloid,
and if I clipped the paraboloid with the box I got the cup.

I can merge them and get the full paraboloid. check this file:

--------------------------

// Persistence of Vision Ray Tracer Scene Description File
// File: ?.pov
// Vers: 3
// Desc: Checkered Floor Example
// Date: mm/dd/yy
// Auth: ?
//

#version 3

#include "colors.inc"
#include "shapes.inc"

global_settings
{
  assumed_gamma 1.0
}

// ----------------------------------------
camera
{
  location  <0.0, 25.5, -20.0>
  direction 1.5*z
  right     4/3*x
  look_at   <0.0, 0.0,  0.0>
}

sky_sphere
{
  pigment
  {
    gradient y
    color_map { [0.0 color blue 0.6] [1.0 color rgb 1] }
  }
}

light_source
{
  0*x // light's position (translated below)
  color red 1.0  green 1.0  blue 1.0  // light's color
  translate <-30, 30, -30>
}

// ----------------------------------------
plane
{
  y, -1
  texture
  {
    pigment {checker color rgb 1 color blue 1 scale 4}
    finish {reflection 0.2}
  }
}

#declare Paraboloid_Cup =
object {Paraboloid_Y
pigment {color rgb <0,1,0>}
clipped_by { box { <10,10,10>, <-10,-10,-10> }}
}

#declare Paraboloid_Cap =
box {<10,10,10>, <-10,-10,-10>
pigment {color rgb <1,0,0>}
clipped_by {object {Paraboloid_Y}}
}

#declare Cutten_Paraboloid =
union {
  object {Paraboloid_Cup}
  object {Paraboloid_Cap}
}

object {Cutten_Paraboloid translate <-10,0,0>}
object {Paraboloid_Cup translate <0,0,0>}
object {Paraboloid_Cap translate <10,0,0>}

-------------

was that bound bug because the paraboloid is infinite or doesn't have
any region in it? (hollow?)
if it has no region, why would clipped_by work?

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