If P is a point that is not the same as the center O of sphere $s\left(r\right)$, the inverse of P in, or with respect to, sphere $s\left(r\right)$ is the point Q lying on the line OP such that $\mathrm{OP}\mathrm{OQ}=r^2$.

Sphere $s\left(r\right)$ is called the sphere of inversion, point O the center of inversion, r the radius of inversion, and $r^2$ the power of inversion.

For a detailed description of Q the object created, use the routine detail (i.e., detail(Q))

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

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

$\mathrm{sphere}\left(s\,\left[\mathrm{point}\left(o\,0\,0\,0\right)\,1\right]\right)$

$s$

$\mathrm{plane}\left(p\,\left[\mathrm{point}\left(A\,1\,0\,-1\right)\,\mathrm{point}\left(B\,0\,0\,-1\right)\,\mathrm{point}\left(C\,0\,1\,-1\right)\right]\right)$

$p$

$\mathrm{inversion}\left(\mathrm{s1}\,p\,s\right)$

$\mathrm{s1}$

$\mathrm{detail}\left(\mathrm{s1}\right)$

Warning, assume that the name of the axes are _x, _y and _z

$\begin{array}{ll}\text{name of the object}& \mathrm{s1}\\ \text{form of the object}& \mathrm{sphere3d}\\ \text{name of the center}& \mathrm{center\_s1\_1}\\ \text{coordinates of the center}& \left[0\,0\,-\frac{1}{2}\right]\\ \text{radius of the sphere}& \frac{1}{2}\\ \text{surface area of the sphere}& \mathrm{\pi }\\ \text{volume of the sphere}& \frac{\mathrm{\pi }}{6}\\ \text{equation of the sphere}& {\mathrm{\_x}}^2+{\mathrm{\_y}}^2+{\mathrm{\_z}}^2+\mathrm{\_z}=0\end{array}$

$\mathrm{IsOnObject}\left(o\,\mathrm{s1}\right)$

$\mathrm{true}$

$\mathrm{draw}\left(\left[s\,\mathrm{s1}\,p\left(\mathrm{style}=\mathrm{patchnogrid}\,\mathrm{color}=\mathrm{maroon}\right)\right]\,\mathrm{style}=\mathrm{wireframe}\,\mathrm{view}=\left[-1..1\,-1..1\,-2..1\right]\,\mathrm{title}=\mathrm{`inversion\; of\; a\; plane\; with\; respect\; to\; a\; sphere`}\right)$