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Mike Horvath <mik### [at] gmail com> 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}}
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