Mike Horvath <mik### [at] gmailcom> wrote:
> On 7/21/2018 6:15 AM, Mike Horvath wrote:
> > Is there already a function/macro for determining the intersection of
> > two coplanar lines or vectors? (So I don't have to program it again.)
> >
> >
> > Thanks.
>
>
> I forgot to say that the lines are not in y = mx + b form. Rather, I
> just have two pairs of points.
>
>
> Mike

Oh. God.
Some people just want EVERYTHING.   :P

I think I coded this from Graphics Gems.




#version 3.7;
global_settings {assumed_gamma 1.0}
#include "Math.inc"
#include "Determinant.mcr"

#macro IntersectionOfLines (pa, pb, PA, PB)
 /*
 INTERSECTION OF
 TWO LINES IN
 THREE-SPACE
 Ronald Goldman
 University of Waterloo
 Waterloo, Ontario, Canada
 [From Graphics Gems (1)]
 Let each line be defined by a point Pk and a unit direction vector Vk,
 k = 1,2.
 */
 #declare P1 = pa;
 #declare V1 = vnormalize (pb-pa);
 //#debug concat ( " V1 ", vstr (3, V1, ",", 3, 1),  "\n")
 #declare P2 = PA;
 #declare V2 = vnormalize (PB-PA);
 //#debug concat ( " V2 ", vstr (3, V2, ",", 3, 1),  "\n")

 // Then we can express each line parametrically by writing
 // L1(_t) = P1 + V1 * _t
 // and
 // L2(_s) = P2 + V2 * _s.

 // The intersection occurs when L1(t) = L2(s) or equivalently when P1 + V1t =
P2 + V2s
 // Subtracting Pl from both sides and crossing with V2 yields
 // vcross(V1, V2) * _t = vcross ((P2 � P1), V2)
 // Now dotting with vcross(V1, V2) and dividing by pow(Det(vcross(V1, V2)), 2)
gives us
 // _t = vdot ( vcross((P2 � P1), V2), vcross(V1, V2) ) / pow (vdot (V1,
V2), 2);
 // Symmetrically, solving for s, we obtain
 // _s = vdot ( vcross((P2 � P1), V1), vcross(V1, V2) ) / pow (vdot (V1,
V2), 2);

 /*
 Two important observations follow:
 � If the lines are parallel, the denominator |Vl x V2|2 = 0.
 � If the lines are skew, s and t represent the parameters of the points
of
 closest approach.
 */

 //#if (pow (vdot (V1, V2), 2) = 0) // check for parallel

 #local Denom = pow (Det (V1, V2), 2);
 #if (Denom = 0) // check for parallel
  #debug concat ( " Det (V1, V2) ", str (vdot (V1, V2), 3, 1),  "\n")
  #debug concat ( " pow (Det (V1, V2), 2) ", str (Denom, 3, 1),  "\n")
  #debug "No intersection - parallel lines. \n"
  #local Intersection = (x+y+z)*666;
 #else
  //#local _t = Det ( vcross((P2 - P1), V2), vcross(V1, V2) ) / pow (Det (V1,
V2), 2);
  //#local _s = Det ( vcross((P2 - P1), V1), vcross(V1, V2) ) / pow (Det (V1,
V2), 2);
  #local _t = Det3 ( (P2 - P1), V2, vcross(V1, V2) ) / Denom;
  #local _s = Det3 ( (P2 - P1), V1, vcross(V1, V2) ) / Denom;
  //#debug concat ( " _t =", str (_t, 3, 1),  "\n")
  //#debug concat ( " _s =", str (_s, 3, 1),  "\n")

  #local I1 = P1 + V1 * _t;
  #local I2 = P2 + V2 * _s;
  //#debug concat ( "I1 = ", vstr(3, I1, ", ", 3, 2), " \n")
  //#debug concat ( "I2 = ", vstr(3, I2, ", ", 3, 2), " \n")
  #if (VEq (I1, I2)) // check for intersection, else skew
   #local Intersection = P1 + V1 * _t;
  #else
   #debug "No intersection - skew lines. \n"
   #local Intersection = (x+y+z)*666;
  #end
 #end // end if parallel

 #debug concat ( " Intersection of Line 1 [from <", vstr (3, pa, ", ", 3, 1), ">
to <", vstr (3, pb, ", ", 3, 1), ">] and Line 2 [from <", vstr (3, PA, ", ", 3,
1), "> to <", vstr (3, PB, ", ", 3, 1), ">] is the point <", vstr (3,
Intersection, ", ", 3, 1),  "> \n")

 Intersection

#end // end macro IntersectionOfLines

#include "colors.inc"

camera {
 location  <0, 0, -20>
 right    x*image_width/image_height
 look_at   <0, 0, 0>
}

light_source { <0, 10, -20>  color rgb <1, 1, 1>}

background {color rgb <1, 1, 1>*0.1}


#declare Point1 = <-5,  1,  0>;
#declare Point2 = < 5,  1,  0>;
#declare Point3 = < 0, -5,  0>;
#declare Point4 = < 0,  5,  0>;

#declare I = IntersectionOfLines (Point1, Point2, Point3, Point4);

cylinder {Point1, Point2, 0.1 pigment {Green}}
cylinder {Point3, Point4, 0.1 pigment {Blue}}
sphere {I, 0.2 pigment {Red}}

Post a reply to this message