When two lines l1, l2 do not intersect, we define the angle determined by them as the angle between two lines through the origin parallel to the given lines. It is the convention that the angle returned is in the interval [0,Pi/2].

The angle between two planes p1, p2 is equal to the angle between their normals.

The angle of intersection of two spheres s1 and s2 at a common point is the angle between the tangent-planes to the spheres at that points. Note that at all common points, the angle of intersection is the same.

The angle between a straight line l1 and a plane p1 is equal to the complement of the angle between the straight line and the normal of the plane.

If T is a triangle, and A a vertex of T, FindAngle(A,T) returns the internal angle of T at A.

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

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

$\mathrm{assume}\left(l\ne 0\,\mathrm{lp}\ne 0\right)\:$

$\mathrm{plane}\left(\mathrm{p1}\,lx+my+nz+p=0\,\left[x\,y\,z\right]\right)\:$

$\mathrm{line}\left(\mathrm{l1}\,\left[\mathrm{point}\left(A\,\mathrm{x1}\,\mathrm{y1}\,\mathrm{z1}\right)\,\left[\mathrm{lp}\,\mathrm{mp}\,\mathrm{np}\right]\right]\right)\:$

$\mathrm{FindAngle}\left(\mathrm{l1}\,\mathrm{p1}\right)$

$\arcsin \left(\frac{\mathrm{lp~}\mathrm{l~}+m\mathrm{mp}+n\mathrm{np}}{\sqrt{\left({\mathrm{l~}}^2+m^2+n^2\right)\left({\mathrm{lp~}}^2+{\mathrm{mp}}^2+{\mathrm{np}}^2\right)}}\right)$

$l≔'l'\:$$\mathrm{lp}≔'\mathrm{lp}'\:$

$\mathrm{assume}\left(a\ne 0\,b\ne 0\,c\ne 0\right)$

$\mathrm{point}\left(P\,\left[a\,b\,c\right]\right)\:$

$\mathrm{point}\left(o\,0\,0\,0\right),\mathrm{point}\left(X\,1\,0\,0\right),\mathrm{point}\left(Y\,0\,1\,0\right),\mathrm{point}\left(Z\,0\,0\,1\right)\:$

$\mathrm{plane}\left(\mathrm{oxz}\,\left[o\,X\,Z\right]\right),\mathrm{plane}\left(\mathrm{oxy}\,\left[o\,X\,Y\right]\right)\:$

$\mathrm{projection}\left(M\,P\,\mathrm{oxz}\right)\:$$\mathrm{projection}\left(N\,P\,\mathrm{oxy}\right)\:$

$\mathrm{plane}\left(p\,\left[o\,M\,N\right]\right)$

$p$

$\mathrm{Equation}\left(p\,\left[x\,y\,z\right]\right)$

$\mathrm{a~}\mathrm{b~}z+\mathrm{a~}\mathrm{c~}y-\mathrm{b~}\mathrm{c~}x=0$

$\mathrm{line}\left(\mathrm{OP}\,\left[o\,P\right]\right)\:$

$\mathrm{FindAngle}\left(\mathrm{OP}\,p\right)$

$\arcsin \left(\frac{\mathrm{b~}\mathrm{c~}\mathrm{a~}}{\sqrt{\left({\mathrm{a~}}^2{\mathrm{b~}}^2+{\mathrm{a~}}^2{\mathrm{c~}}^2+{\mathrm{b~}}^2{\mathrm{c~}}^2\right)\left({\mathrm{a~}}^2+{\mathrm{b~}}^2+{\mathrm{c~}}^2\right)}}\right)$