intersection - Maple Help

geom3d

 intersection
 find the intersections between two or three given objects.

 Calling Sequence intersection(obj, l1, l2) intersection(obj, p1, p2) intersection(obj, l1, p1) intersection(obj, l1, s) intersection(obj, p1, p2, p3)

Parameters

 obj - name l1, l2 - lines p1, p2, p3 - planes s - sphere

Description

 • 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.

Examples

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

intersection of two planes

 > $\mathrm{plane}\left(\mathrm{p1},4x+4y-5z=12,\left[x,y,z\right]\right):$
 > $\mathrm{plane}\left(\mathrm{p2},8x+12y-13z=32,\left[x,y,z\right]\right):$
 > $\mathrm{intersection}\left(l,\mathrm{p1},\mathrm{p2}\right):$
 > $\mathrm{detail}\left(l\right)$
 $\begin{array}{ll}{\text{name of the object}}& {l}\\ {\text{form of the object}}& {\mathrm{line3d}}\\ {\text{equation of the line}}& \left[{x}{=}{1}{+}{8}{}{\mathrm{_t}}{,}{y}{=}{2}{+}{12}{}{\mathrm{_t}}{,}{z}{=}{16}{}{\mathrm{_t}}\right]\end{array}$ (1)

intersection of three planes

 > $\mathrm{point}\left(A,0,0,0\right),\mathrm{point}\left(B,1,0,0\right),\mathrm{point}\left(C,0,1,0\right),\mathrm{point}\left(E,0,0,1\right):$
 > $\mathrm{plane}\left(\mathrm{oxy},\left[A,B,C\right]\right),\mathrm{plane}\left(\mathrm{oyz},\left[A,C,E\right]\right),\mathrm{plane}\left(\mathrm{oxz},\left[A,B,E\right]\right):$
 > $\mathrm{intersection}\left(P,\mathrm{oxy},\mathrm{oyz},\mathrm{oxz}\right)$
 ${P}$ (2)
 > $\mathrm{coordinates}\left(P\right)$
 $\left[{0}{,}{0}{,}{0}\right]$ (3)

Prove that the lines $2x-y+3z+3$=0=$x+10y-21$ and $2x-y$=0=$7x+z-6$ intersect. Find the coordinates of their common point, and the equation of the plane containing them.

Define the line l1: 2*x-y+3*z+3=0=x+10*y-21

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

Define the line l2: $2x-y$=0=$7x+z-6$

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

Find the intersection of l1 and l2

 > $\mathrm{intersection}\left(P,\mathrm{l1},\mathrm{l2}\right)$
 ${P}$ (4)
 > $\mathrm{coordinates}\left(P\right)$
 $\left[{1}{,}{2}{,}{-1}\right]$ (5)

Find the equation of the plane containing l1 and l2

 > $\mathrm{plane}\left(p,\left[\mathrm{l1},\mathrm{l2}\right]\right):$
 > $\mathrm{Equation}\left(p\right)$
 ${-}{378}{+}{63}{}{x}{+}{189}{}{y}{+}{63}{}{z}{=}{0}$ (6)

Prove that the lines $\frac{x-a}{\mathrm{ap}}$=$\frac{y-b}{\mathrm{bp}}$=$\frac{z-c}{\mathrm{cp}}$ and $\frac{x-\mathrm{ap}}{a}$=$\frac{y-\mathrm{bp}}{b}$=$\frac{z-\mathrm{cp}}{c}$ intersect, and find the coordinates of the point of intersection.

 > $\mathrm{assume}\left(\mathrm{ap}\ne 0,a\ne 0\right):$
 > $\mathrm{line}\left(\mathrm{l1},\left[\mathrm{point}\left(\mathrm{o1},a,b,c\right),\left[\mathrm{ap},\mathrm{bp},\mathrm{cp}\right]\right]\right):$
 > $\mathrm{line}\left(\mathrm{l2},\left[\mathrm{point}\left(\mathrm{o2},\mathrm{ap},\mathrm{bp},\mathrm{cp}\right),\left[a,b,c\right]\right]\right):$
 > $\mathrm{intersection}\left(P,\mathrm{l1},\mathrm{l2}\right)$
 ${P}$ (7)
 > $\mathrm{coordinates}\left(P\right)$
 $\left[{\mathrm{a~}}{+}{\mathrm{ap~}}{,}{b}{+}{\mathrm{bp}}{,}{c}{+}{\mathrm{cp}}\right]$ (8)