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The polar of a point A with respect to a sphere is defined to be the locus of all points conjugate to A.

For a detailed description of the polar of p, use the routine detail.

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

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

$\mathrm{point}\left(A\,1\,2\,-1\right),\mathrm{point}\left(B\,3\,5\,-2\right),\mathrm{point}\left(C\,0\,\frac{1}{2}\,-\frac{1}{2}\right)$

$A,B,C$

$\mathrm{sphere}\left(s\,x^2+y^2+z^2-3x+y+z-5=0\,\left[x\,y\,z\right]\right)\:$

$\mathrm{polar}\left(\mathrm{p1}\,A\,s\right)$

$\mathrm{p1}$

$\mathrm{polar}\left(\mathrm{p2}\,B\,s\right)$

$\mathrm{p2}$

$\mathrm{polar}\left(\mathrm{p3}\,C\,s\right)$

$\mathrm{p3}$

$\mathrm{detail}\left(\mathrm{p3}\right)$

$\begin{array}{cc}\text{name of the object}& \mathrm{p3}\\ \text{form of the object}& \mathrm{plane3d}\\ \text{equation of the plane}& -5-\frac{3x}{2}+y=0\end{array}$

$\mathrm{intersection}\left(\mathrm{l1}\,\mathrm{p1}\,\mathrm{p2}\right)$

$\mathrm{l1}$

$\mathrm{intersection}\left(\mathrm{l2}\,\mathrm{p1}\,\mathrm{p3}\right)$

$\mathrm{l2}$

$\mathrm{intersection}\left(\mathrm{l3}\,\mathrm{p2}\,\mathrm{p3}\right)$

$\mathrm{l3}$

$\mathrm{AreDistinct}\left(\mathrm{l1}\,\mathrm{l2}\,\mathrm{l3}\right)$

$\mathrm{false}$