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geom3d

 polar
 find the polar of a given point with respect to a given sphere

 Calling Sequence polar(p, A, s)

Parameters

 p - the name of the polar A - point s - sphere

Description

 • 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.

Examples

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

Write down the equations of the polars of the points $1,2,-1$, $3,5,-2$ and $0,\frac{1}{2},-\frac{1}{2}$ with respect to the sphere ${x}^{2}+{y}^{2}+{z}^{2}-3x+y+z-5=0$ and show that they form a pencil of planes.

 > $\mathrm{point}\left(A,1,2,-1\right),\mathrm{point}\left(B,3,5,-2\right),\mathrm{point}\left(C,0,\frac{1}{2},-\frac{1}{2}\right)$
 ${A}{,}{B}{,}{C}$ (1)
 > $\mathrm{sphere}\left(s,{x}^{2}+{y}^{2}+{z}^{2}-3x+y+z-5=0,\left[x,y,z\right]\right):$
 > $\mathrm{polar}\left(\mathrm{p1},A,s\right)$
 ${\mathrm{p1}}$ (2)
 > $\mathrm{polar}\left(\mathrm{p2},B,s\right)$
 ${\mathrm{p2}}$ (3)
 > $\mathrm{polar}\left(\mathrm{p3},C,s\right)$
 ${\mathrm{p3}}$ (4)
 > $\mathrm{detail}\left(\mathrm{p3}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{p3}}\\ {\text{form of the object}}& {\mathrm{plane3d}}\\ {\text{equation of the plane}}& {-}{5}{-}\frac{{3}{}{x}}{{2}}{+}{y}{=}{0}\end{array}$ (5)
 > $\mathrm{intersection}\left(\mathrm{l1},\mathrm{p1},\mathrm{p2}\right)$
 ${\mathrm{l1}}$ (6)
 > $\mathrm{intersection}\left(\mathrm{l2},\mathrm{p1},\mathrm{p3}\right)$
 ${\mathrm{l2}}$ (7)
 > $\mathrm{intersection}\left(\mathrm{l3},\mathrm{p2},\mathrm{p3}\right)$
 ${\mathrm{l3}}$ (8)
 > $\mathrm{AreDistinct}\left(\mathrm{l1},\mathrm{l2},\mathrm{l3}\right)$
 ${\mathrm{false}}$ (9)

The answers show that three planes p1, p2, and p3 form a pencil of planes.