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geom3d

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

 Calling Sequence RadicalPlane(p1, s1, s2) RadicalLine(p1, s1, s2, s3) RadicalCenter(p1, s1, s2, s3, s4)

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)

 See Also

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