Warp wrote:
>...
>   And I can believe that there could be a shortcut to see if a line
> intersects a triangle. It's just too complicated for me to think being
> this tired (and lazy) :)

I think that my RayIntersectsTriangle() macro in the code below
provides such a shortcut (*), but I'm not sure yet if it is optimal.

(*) Not really for lines, but for rays (which has a defined direction)


Tor Olav


// ===== 1 ======= 2 ======= 3 ======= 4 ======= 5 ======= 6 ======= 7
// Copyright 2002 by Tor Olav Kristensen
// Email: tor### [at] hotmailcom
// http://www.crosswinds.net/~tok/povray
// ===== 1 ======= 2 ======= 3 ======= 4 ======= 5 ======= 6 ======= 7

#version 3.5;
#include "colors.inc"

// ===== 1 ======= 2 ======= 3 ======= 4 ======= 5 ======= 6 ======= 7
// Intersection macros

#macro RayIntersectsTriangle(pRay, vRay, pA, pB, pC)

  #local vRA = pA - pRay;
  #local vRB = pB - pRay;
  #local vRC = pC - pRay;
  #local vABN = vcross(vRA, vRB);
  #local vBCN = vcross(vRB, vRC);
  #local vCAN = vcross(vRC, vRA);
  #local STP = vdot(vABN, vRC);
  #if (STP = 0)
    #debug "Macro RayIntersectsTriangle: "
    #debug "Degenerate triangle or "
    #debug "origin of ray lies in same plane as triangle\n"
    #local Intersect = false;
  #else
    #local AB = vdot(vRay, vABN);
    #local BC = vdot(vRay, vBCN);
    #local CA = vdot(vRay, vCAN);
    #if (AB = 0 | BC = 0 | CA = 0)
      #local Intersect = true; // Intersection at one or two edges
    #else
      #if (STP > 0)
        #local Intersect = (AB > 0 & BC > 0 & CA > 0);
      #else
        #local Intersect = (AB < 0 & BC < 0 & CA < 0);
      #end // if
    #end // if
  #end // if

  Intersect
      
#end // macro RayIntersectsTriangle


#macro LinePlaneIntersection(pLine, vLine, pPlane, vPlane)

  (pLine + vLine*vdot(pPlane - pLine, vPlane)/vdot(vLine, vPlane))

#end // macro LinePlaneIntersection

// ===== 1 ======= 2 ======= 3 ======= 4 ======= 5 ======= 6 ======= 7
// Set up and show triangle

#declare p0 = <1, 3, -2>;
#declare p1 = <2, -3, 3>;
#declare p2 = <-2, 1, -3>;

triangle {
  p0, p1, p2
  pigment { color rgbf <1, 1, 0, 0.2> }
}

union {
  sphere { p0, 0.03 }
  sphere { p1, 0.03 }
  sphere { p2, 0.03 }
  pigment { color Magenta*2 }
}

union {
  cylinder { p0, p1, 0.02 }
  cylinder { p1, p2, 0.02 }
  cylinder { p2, p0, 0.02 }
  pigment { color Cyan*2 }
}

// ===== 1 ======= 2 ======= 3 ======= 4 ======= 5 ======= 6 ======= 7
// Shoot some random rays and highlight those that hits the triangle

#declare R = 4;
#declare S = seed(5);

#declare Cnt = 0;
#while (Cnt < 150)
  #declare pR = (<1, 1, 1> - 2*<rand(S), rand(S), rand(S)>)*R;
  #declare vR = (<1, 1, 1> - 2*<rand(S), rand(S), rand(S)>);
  #if (vlength(vR) > 0)
    sphere {
      pR, 0.04
      pigment { color Green*2 }
    }
    cylinder {
      pR, pR + 100*vnormalize(vR), 0.02
      pigment { color White }
    }
    #if (RayIntersectsTriangle(pR, vR, p0, p1, p2))
      #local vPlaneNormal = vcross(p1 - p0, p2 - p0);
      #local pIntersect = 
        LinePlaneIntersection(pR, vR, p0, vPlaneNormal);
      cylinder {
        pR, pIntersect, 0.021
        pigment { color White*4 }
      }
      sphere {
        pIntersect, 0.06
        pigment { color Red*2 }
      }
    #end // if
  #end // if  
  #declare Cnt = Cnt + 1;
#end // while

// ===== 1 ======= 2 ======= 3 ======= 4 ======= 5 ======= 6 ======= 7

background { color Blue/2 }

light_source {
  <1, 1, -3>*100 color White
  shadowless
}

camera {
  location -10*z
  look_at <0, 0, 0>
}

// ===== 1 ======= 2 ======= 3 ======= 4 ======= 5 ======= 6 ======= 7

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