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geom3d

 inversion
 find the inversion of a point, plane, or sphere with respect to a given sphere.

 Calling Sequence inversion(Q, P, s)

Parameters

 Q - the name of the object to be created P - point, line, or sphere s - sphere

Description

 • 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.

Examples

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

Define the sphere s with center (0,0,0), radius 1

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

Define a plane passing through A, B, C

 > $\mathrm{plane}\left(p,\left[\mathrm{point}\left(A,1,0,-1\right),\mathrm{point}\left(B,0,0,-1\right),\mathrm{point}\left(C,0,1,-1\right)\right]\right)$
 ${p}$ (2)

Find the inversion of the plane with respect to the sphere s

 > $\mathrm{inversion}\left(\mathrm{s1},p,s\right)$
 ${\mathrm{s1}}$ (3)

Sine the plane p does not pass through the center of inversion, its inversion is a sphere through the center of inversion.

Checking:

 > $\mathrm{detail}\left(\mathrm{s1}\right)$
 Warning, assume that the name of the axes are _x, _y and _z
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{s1}}\\ {\text{form of the object}}& {\mathrm{sphere3d}}\\ {\text{name of the center}}& {\mathrm{center_s1_1}}\\ {\text{coordinates of the center}}& \left[{0}{,}{0}{,}{-}\frac{{1}}{{2}}\right]\\ {\text{radius of the sphere}}& \frac{{1}}{{2}}\\ {\text{surface area of the sphere}}& {\mathrm{\pi }}\\ {\text{volume of the sphere}}& \frac{{\mathrm{\pi }}}{{6}}\\ {\text{equation of the sphere}}& {{\mathrm{_x}}}^{{2}}{+}{{\mathrm{_y}}}^{{2}}{+}{{\mathrm{_z}}}^{{2}}{+}{\mathrm{_z}}{=}{0}\end{array}$ (4)
 > $\mathrm{IsOnObject}\left(o,\mathrm{s1}\right)$
 ${\mathrm{true}}$ (5)

Plotting:

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