The routine returns true if the given objects are perpendicular to each other; false if they are not; and FAIL if it is unable to reach a conclusion.

In case of FAIL, if the third optional argument is given, the condition that makes the given objects perpendicular to each other is assigned to this argument.

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

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

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

$\mathrm{sphere}\left(\mathrm{s1}\,x^2+y^2+z^2=a^2\,\left[x\,y\,z\right]\right)\:$

$\mathrm{sphere}\left(\mathrm{s2}\,x^2+y^2+z^2-2ay-2az+a^2=0\,\left[x\,y\,z\right]\right)\:$

$\mathrm{sphere}\left(\mathrm{s3}\,x^2+y^2+z^2-2ax-2az+a^2=0\,\left[x\,y\,z\right]\right)\:$

$\mathrm{sphere}\left(\mathrm{s4}\,x^2+y^2+z^2-2ax-2ay+a^2=0\,\left[x\,y\,z\right]\right)\:$

$$\begin{gathered} \mathbf{for}i\mathbf{to}3\mathbf{do} \\ \mathbf{for}j\mathbf{from}i+1\mathbf{to}4\mathbf{do} \\ \mathbf{if}\mathrm{ArePerpendicular}\left(s\Vert i\,s\Vert j\right)\mathbf{then} \\ \mathrm{print}\left(i\,j\,\mathrm{true}\right) \\ \mathbf{else} \\ \mathrm{print}\left(i\,j\,\mathrm{false}\right) \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do} \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$1,2,\mathrm{true}$

$1,3,\mathrm{true}$

$1,4,\mathrm{true}$

$2,3,\mathrm{true}$

$2,4,\mathrm{true}$

$3,4,\mathrm{true}$