The routine finds the intersection between two lines, two planes, a line and a plane, a line and a sphere, or three planes.

In general, the output is assigned to the first argument obj. If the routine is unable to determine the intersection(s) of given objects, it will return FAIL.

If l1 and l2 are two lines, the output is either NULL, the point of intersection, or a line in case l1 and l2 are the same.

If p1 and p2 are two planes, the output is either NULL, the line of intersection, or a plane in case p1 and p2 are the same.

If l1 is a line and p1 a plane, the output is either NULL, the point of intersection, or a line in case l1 lies in p1.

If l1 is a line and s a sphere, the output is either NULL, one point of intersections, a list of two points of intersection.

If p1, p2 and p3 are three planes, the output is NULL, or the point of intersection.

For more details on the output, use detail.

The command with(geom3d,intersection) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{geom3d}\right)\:$

$\mathrm{plane}\left(\mathrm{p1}\,4x+4y-5z=12\,\left[x\,y\,z\right]\right)\:$

$\mathrm{plane}\left(\mathrm{p2}\,8x+12y-13z=32\,\left[x\,y\,z\right]\right)\:$

$\mathrm{intersection}\left(l\,\mathrm{p1}\,\mathrm{p2}\right)\:$

$\mathrm{detail}\left(l\right)$

$\begin{array}{ll}\text{name of the object}& l\\ \text{form of the object}& \mathrm{line3d}\\ \text{equation of the line}& \left[x=1+8\mathrm{\_t}\,y=2+12\mathrm{\_t}\,z=16\mathrm{\_t}\right]\end{array}$

$\mathrm{point}\left(A\,0\,0\,0\right),\mathrm{point}\left(B\,1\,0\,0\right),\mathrm{point}\left(C\,0\,1\,0\right),\mathrm{point}\left(E\,0\,0\,1\right)\:$

$\mathrm{plane}\left(\mathrm{oxy}\,\left[A\,B\,C\right]\right),\mathrm{plane}\left(\mathrm{oyz}\,\left[A\,C\,E\right]\right),\mathrm{plane}\left(\mathrm{oxz}\,\left[A\,B\,E\right]\right)\:$

$\mathrm{intersection}\left(P\,\mathrm{oxy}\,\mathrm{oyz}\,\mathrm{oxz}\right)$

$P$

$\mathrm{coordinates}\left(P\right)$

$\left[0\,0\,0\right]$

$\mathrm{plane}\left(\mathrm{p1}\,2x-y+3z+3=0\,\left[x\,y\,z\right]\right)\:$$\mathrm{plane}\left(\mathrm{p2}\,x+10y-21=0\,\left[x\,y\,z\right]\right)\:$

$\mathrm{line}\left(\mathrm{l1}\,\left[\mathrm{p1}\,\mathrm{p2}\right]\right)\:$

$\mathrm{plane}\left(\mathrm{p3}\,2x-y=0\,\left[x\,y\,z\right]\right)\:$$\mathrm{plane}\left(\mathrm{p4}\,7x+z-6=0\,\left[x\,y\,z\right]\right)\:$

$\mathrm{line}\left(\mathrm{l2}\,\left[\mathrm{p3}\,\mathrm{p4}\right]\right)\:$

$\mathrm{intersection}\left(P\,\mathrm{l1}\,\mathrm{l2}\right)$

$P$

$\mathrm{coordinates}\left(P\right)$

$\left[1\,2\,\mathrm{-1}\right]$

$\mathrm{plane}\left(p\,\left[\mathrm{l1}\,\mathrm{l2}\right]\right)\:$

$\mathrm{Equation}\left(p\right)$

$-378+63x+189y+63z=0$

$\mathrm{assume}\left(\mathrm{ap}\ne 0\,a\ne 0\right)\:$

$\mathrm{line}\left(\mathrm{l1}\,\left[\mathrm{point}\left(\mathrm{o1}\,a\,b\,c\right)\,\left[\mathrm{ap}\,\mathrm{bp}\,\mathrm{cp}\right]\right]\right)\:$

$\mathrm{line}\left(\mathrm{l2}\,\left[\mathrm{point}\left(\mathrm{o2}\,\mathrm{ap}\,\mathrm{bp}\,\mathrm{cp}\right)\,\left[a\,b\,c\right]\right]\right)\:$

$\mathrm{intersection}\left(P\,\mathrm{l1}\,\mathrm{l2}\right)$

$P$

$\mathrm{coordinates}\left(P\right)$

$\left[\mathrm{a~}+\mathrm{ap~}\,b+\mathrm{bp}\,c+\mathrm{cp}\right]$