/* Hi, friends and creators of POV-Ray


as "t" alone, and the letter following the "u" looks like a
greek beta, spoken as a sharp "s". If your computer doesn't know
it, use "ss" instead [ or "$" ;-) ].
Usually I'm called "Sputnik".

Please look at this and check it:

To create a drop, I used a quartic as described in the scene
below, and to see the effect of varying the threshold constant
T, I "differenced" this with a drop having T=0, which should
give an hollow object similar to a drop-shaped soap bubble.
But I saw strange effects I couldn't understand.

To show these effects, I prepared an environment with light,
a camera (one of 3), the coordinate axes (x/y/z, colored R/G/B),
a shadow-casting grey box, a flat box, and two scenes: the first
contains a difference of a drop and two planes, below it a full
drop for comparison. The second scene contains a difference of
two drops. A third (empty) scene is provided for your own
explorations.

Perhaps you gurus can render these scenes (simply copy this
*whole* text to POV-Ray - all non-SDL is a comment;
"320x240, No AA" is sufficient; 16 sec. at 533 MHz) and tell me
if you see what I see (WISIWYG: what I see is what you get?!):
_the_planes_use_a_larger_drop_to_cut_from_! (It seems to be a
drop with T increased by 0.0001: compare with the lower drop
Drop(T+0.0001).) T=-0.00003 even gives a visible cutting surface
*completely outside* the drop; the drop itself is invisible,
because the true intersection with the planes is empty.
The second scene (use #switch(2)) shows an coincident-surface-
effect on the surface of the outer drop.

Is this an error in my visual perception, a misunderstanding
of quartics/difference, a bug in all three computers mentioned
below, a problen of numerical accuracy or an error in POV-Ray
3.1g? Does this also happen with POV-Ray 3.5 (I haven't
installed it yet)?

Awaiting your comments

   Sputnik

fr### [at] computermuseumfh-kielde

P.S. Change the color of the yellow box to "rgbt<1,1,0,.7>",
use a "camera { location <0.002, 0.004, 0> look_at 0.002*y }",
select the empty scene by "#switch(3)" and look at the shadows
of the green cylinder: in my pictures they don't start at the
cylinder, but look as if the cylinder ends above the box. What
is going on there?
*/


/* ========================
=========================
===========

POV-Ray scene showing a bug (?) in a difference
with a drop-shaped quartic
using POV-Ray 3.1g.watcom.win32 on
- a Pentium-Pro-PC, 233 MHz, 32 MB, with Windows 95
- a Pentium-Laptop, 133 MHz, 40 MB, with Windows 95
- a Pentium-PC, 533 MHz, 1 GB, with Windows 2000 Professional



quartic for a lying drop along the x-axis ---------------------
(use a non-proportional font to see this ASCII-picture)

R^2 ^
    !  __                  ___
    ! /  \                /   \
  --+=---+---> x   =>   -=     )
  / 0    1\               \___/

-x^3*(x-1)+T = x^3-x^4+T = R^2 = (Radius at x)^2 >= y^2+z^2
x^4-x^3+y^2+z^2-T <= 0
T=0 => tip at <0, 0, 0>, bottom at <1, 0, 0>
T>0 => larger drop, T<0 => smaller drop
*/
#macro Drop ( T )
  quartic {
    <1, 0,0, -1, // x^4, x^3
    0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
    1, 0,0,0,0,0,0, 1, 0, -T > // y^2, z^2, 1
    sturm
    }
  #end//macro Drop


// transparent texture for drops ------------------------------

#declare Tex = 

  texture  { 

    pigment { color rgbt <.6, .7, 1, .5 > }
    finish { ambient .3 specular .2 }
    }


// lights and camera ------------------------------------------

light_source { <1, 1.5, -2>, rgb 1.5 }
light_source { <-1, 1, 1>, rgb 1.5 }

// look at the whole scene
//camera { location <1.2, 0.5, -0.5> look_at 0.7*x }

// closely look at the tip
camera { location <0.04, 0.05, -0.04> look_at 0.01*y }

// closely look at the tip from the other side
//camera { location <-0.065, 0.055, 0.055> look_at -0.01*x }


// auxiliary objects ------------------------------------------

box { -0.002, 0.002 texture { pigment { color rgbt <1,1,0,0> } } } 

cylinder { 0, 1.1*x, 0.0004 texture { pigment { color red 1 } } } 

cylinder { -y, y, 0.0004 texture { pigment { color green 1 } } } 

cylinder { 0, z, 0.0004 texture { pigment { color blue 1 } } }

box { <-0.049, -0.025, -0.005>, <-0.0491, 0.025, 0.005>
  texture { Tex }
  } 

box { <0.01, 0.034, 0>, <0.011, 0.035, 0.033>
  texture { pigment { color rgb .7 } }
  }

// the drops --------------------------------------------------

#switch ( 1 ) // 1: difference of drop and planes;
              // 2: difference of drops
              // 3: empty (for your experiments)


// SCENE 1: difference of drop and planes -----------
#case ( 1 )
  #declare T = 0.00003; // try  0.00003, -0.00003, 0.0001, -0.0001
  intersection {
    Drop ( T )
    plane { x, 0.03 }
    plane { -z, 0 }
    texture { Tex }
    translate 0.015*y
    }
  // drop with T+0.0001 for comparison
  object {
    Drop ( T+0.0001 )
    texture { Tex }
    translate -0.015*y
    }
  #break


// SCENE 2: difference of drops ---------------------
#case ( 2 )
  difference {
    Drop ( 0.0001 )
    Drop ( 0 )
    //plane { z, 0 } // that's what I wanted to do
    texture { Tex }
    }
  #break


// scene 3: this is the place for your experiments --
#case ( 3 )
  // ...
  #end//switch


// E N D ======================
=========================
=======

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