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geom3d

 FindAngle
 find the angle between two given objects

 Calling Sequence FindAngle(l1, l2) FindAngle(p1, p2) FindAngle(s1, s2) FindAngle(l1, p1) FindAngle(A, T)

Parameters

 l1, l2 - lines p1, p2 - planes s1, s2 - sphere A - a point T - a triangle

Description

 • 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.

Examples

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

Find the angle between a line and a plane

make necessary assumptions

 > $\mathrm{assume}\left(l\ne 0,\mathrm{lp}\ne 0\right):$
 > $\mathrm{plane}\left(\mathrm{p1},lx+my+nz+p=0,\left[x,y,z\right]\right):$
 > $\mathrm{line}\left(\mathrm{l1},\left[\mathrm{point}\left(A,\mathrm{x1},\mathrm{y1},\mathrm{z1}\right),\left[\mathrm{lp},\mathrm{mp},\mathrm{np}\right]\right]\right):$
 > $\mathrm{FindAngle}\left(\mathrm{l1},\mathrm{p1}\right)$
 ${\mathrm{arcsin}}{}\left(\frac{{\mathrm{lp~}}{}{\mathrm{l~}}{+}{m}{}{\mathrm{mp}}{+}{n}{}{\mathrm{np}}}{\sqrt{\left({{\mathrm{l~}}}^{{2}}{+}{{m}}^{{2}}{+}{{n}}^{{2}}\right){}\left({{\mathrm{lp~}}}^{{2}}{+}{{\mathrm{mp}}}^{{2}}{+}{{\mathrm{np}}}^{{2}}\right)}}\right)$ (1)
 > $l≔'l':$$\mathrm{lp}≔'\mathrm{lp}':$

If the point P(a,b,c) perpendiculars PM, PN are drawn to the planes of zx, xy, find the equation of the plane OMN and the angle which OP makes with it.

 > $\mathrm{assume}\left(a\ne 0,b\ne 0,c\ne 0\right)$
 > $\mathrm{point}\left(P,\left[a,b,c\right]\right):$
 > $\mathrm{point}\left(o,0,0,0\right),\mathrm{point}\left(X,1,0,0\right),\mathrm{point}\left(Y,0,1,0\right),\mathrm{point}\left(Z,0,0,1\right):$
 > $\mathrm{plane}\left(\mathrm{oxz},\left[o,X,Z\right]\right),\mathrm{plane}\left(\mathrm{oxy},\left[o,X,Y\right]\right):$
 > $\mathrm{projection}\left(M,P,\mathrm{oxz}\right):$$\mathrm{projection}\left(N,P,\mathrm{oxy}\right):$
 > $\mathrm{plane}\left(p,\left[o,M,N\right]\right)$
 ${p}$ (2)
 > $\mathrm{Equation}\left(p,\left[x,y,z\right]\right)$
 ${\mathrm{a~}}{}{\mathrm{b~}}{}{z}{+}{\mathrm{a~}}{}{\mathrm{c~}}{}{y}{-}{\mathrm{b~}}{}{\mathrm{c~}}{}{x}{=}{0}$ (3)
 > $\mathrm{line}\left(\mathrm{OP},\left[o,P\right]\right):$
 > $\mathrm{FindAngle}\left(\mathrm{OP},p\right)$
 ${\mathrm{arcsin}}{}\left(\frac{{\mathrm{b~}}{}{\mathrm{c~}}{}{\mathrm{a~}}}{\sqrt{\left({{\mathrm{a~}}}^{{2}}{}{{\mathrm{b~}}}^{{2}}{+}{{\mathrm{a~}}}^{{2}}{}{{\mathrm{c~}}}^{{2}}{+}{{\mathrm{b~}}}^{{2}}{}{{\mathrm{c~}}}^{{2}}\right){}\left({{\mathrm{a~}}}^{{2}}{+}{{\mathrm{b~}}}^{{2}}{+}{{\mathrm{c~}}}^{{2}}\right)}}\right)$ (4)