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geom3d

 find the radical plane of two given spheres
 find the radical line of three given spheres
 find the radical center of four given spheres

Parameters

 p - name s1, s2, s3, s4 - spheres

Description

 • The locus of points $P\left(x,y,z\right)$ which have the same power with respect to the two given spheres s1, s2 is a plane called radical plane.
 • Let us introduce a third sphere s3. Now we have three radical planes that form a pencil whose axis is the straight line. This line is called the radical line of the three sphere.
 • Now add a fourth sphere s4, and we have four radical lines. These four lines are clearly concurrent at the radical center.

Examples

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

Define two spheres s1, s2

 > $\mathrm{sphere}\left(\mathrm{s1},{x}^{2}+{y}^{2}+{z}^{2}=1,\left[x,y,z\right]\right):$
 > $\mathrm{sphere}\left(\mathrm{s2},\left[\mathrm{point}\left(B,\left[5,5,5\right]\right),2\right]\right):$

Find the radical plane of s1 and s2

 > $\mathrm{RadicalPlane}\left(p,\mathrm{s1},\mathrm{s2}\right)$
 ${p}$ (1)
 > $\mathrm{Equation}\left(p\right)$
 ${-}{72}{+}{10}{}{x}{+}{10}{}{y}{+}{10}{}{z}{=}{0}$ (2)
 > $\mathrm{NormalVector}\left(p\right)$
 $\left[{10}{,}{10}{,}{10}\right]$ (3)

Simple check:

Generate a randpoint point on the radical plane:

 > $\mathrm{randpoint}\left(P,p\right)$
 ${P}$ (4)

The power of point P with respect to two spheres s1 and s2 must be the same:

 > $\mathrm{powerps}\left(P,\mathrm{s1}\right)-\mathrm{powerps}\left(P,\mathrm{s2}\right)$
 ${0}$ (5)

Plotting:

 > $\mathrm{draw}\left(\left[p,\mathrm{s1},\mathrm{s2}\right],\mathrm{style}=\mathrm{patchnogrid},\mathrm{orientation}=\left[-26,96\right],\mathrm{lightmodel}=\mathrm{light1},\mathrm{title}=\mathrm{Radical plane of two given spheres}\right)$

Find the radical line of three spheres:

 > $\mathrm{sphere}\left(\mathrm{s3},\left[\mathrm{point}\left(A,1,2,3\right),3\right]\right)$
 ${\mathrm{s3}}$ (6)
 > $\mathrm{RadicalLine}\left(l,\mathrm{s1},\mathrm{s2},\mathrm{s3}\right)$
 ${l}$ (7)
 > $\mathrm{detail}\left(l\right)$
 Warning, assume that the parameter in the parametric equations is _t
 $\begin{array}{ll}{\text{name of the object}}& {l}\\ {\text{form of the object}}& {\mathrm{line3d}}\\ {\text{equation of the line}}& \left[{x}{=}\frac{{57}}{{5}}{+}{20}{}{\mathrm{_t}}{,}{y}{=}{-}\frac{{21}}{{5}}{-}{40}{}{\mathrm{_t}}{,}{z}{=}{20}{}{\mathrm{_t}}\right]\end{array}$ (8)

Find the radical center of four given spheres:

 > $\mathrm{sphere}\left(\mathrm{s4},\left[\mathrm{point}\left(A,-3,7,1\right),3\right]\right)$
 ${\mathrm{s4}}$ (9)
 > $\mathrm{RadicalCenter}\left(P,\mathrm{s1},\mathrm{s2},\mathrm{s3},\mathrm{s4}\right)$
 ${P}$ (10)
 > $\mathrm{form}\left(P\right)$
 ${\mathrm{point3d}}$ (11)
 > $\mathrm{coordinates}\left(P\right)$
 $\left[\frac{{249}}{{100}}{,}\frac{{471}}{{100}}{,}{0}\right]$ (12)