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geometry

 inversion
 find the inversion of a point, line, or circle with respect to a given circle

 Calling Sequence inversion(Q, P, c)

Parameters

 Q - the name of the object to be created P - point, line, or circle c - circle

Description

 • If P is a point that is not the same as the center O of circle $c\left(r\right)$, the inverse of P in, or with respect to, circle $c\left(r\right)$ is the point Q lying on the line OP such that $\mathrm{SensedMagnitude}\left(\mathrm{OP}\right)\mathrm{SensedMagnitude}\left(\mathrm{OQ}\right)={r}^{2}$.
 • If P is a line passing through center O of circle $c\left(r\right)$, the inverse of P is P itself. In case P is a line not passing through center O of circle $c\left(r\right)$, the inverse of P is a circle Q passing though O perpendicular to P
 • If P is a circle passing through the center O of circle $c\left(r\right)$, the inverse of P is a straight line Q not passing through O and perpendicular to the diameter of $c\left(r\right)$ through O. In case P is a line not passing through the center O of circle $c\left(r\right)$, the inverse of P is a circle Q not passing through O and homothetic to circle $c\left(r\right)$ with O as center of homothety.
 • For a detailed description of Q the object created, use the routine detail (i.e., detail(Q);)
 • The command with(geometry,inversion) allows the use of the abbreviated form of this command.

Examples

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

Inversion of a point with respect to a circle

 > $\mathrm{point}\left(A,2,0\right):$$\mathrm{circle}\left(\mathrm{c1},{x}^{2}+{y}^{2}=16,\left[x,y\right]\right):$
 > $\mathrm{inversion}\left(B,A,\mathrm{c1}\right):$$\mathrm{inversion}\left(C,B,\mathrm{c1}\right):$
 > $\mathrm{coordinates}\left(A\right)=\mathrm{coordinates}\left(C\right)$
 $\left[{2}{,}{0}\right]{=}\left[{2}{,}{0}\right]$ (1)

Inversion of a line with respect to a circle

 > $\mathrm{line}\left(\mathrm{l1},y=x,\left[x,y\right]\right):$
 > $\mathrm{IsOnLine}\left(\mathrm{center}\left(\mathrm{c1}\right),\mathrm{l1}\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{inversion}\left(\mathrm{l2},\mathrm{l1},\mathrm{c1}\right):$
 > $\mathrm{Equation}\left(\mathrm{l1}\right)=\mathrm{Equation}\left(\mathrm{l2}\right)$
 $\left({y}{-}{x}{=}{0}\right){=}\left({y}{-}{x}{=}{0}\right)$ (3)
 > $\mathrm{line}\left(k,x=2,\left[x,y\right]\right):$
 > $\mathrm{inversion}\left(\mathrm{k1},k,\mathrm{c1}\right):$$\mathrm{inversion}\left(\mathrm{kk1},\mathrm{k1},\mathrm{c1}\right):$
 > $\mathrm{form}\left(\mathrm{k1}\right)$
 ${\mathrm{circle2d}}$ (4)
 > $\mathrm{Equation}\left(k\right),\mathrm{Equation}\left(\mathrm{kk1}\right)$
 ${x}{-}{2}{=}{0}{,}{-}{16}{+}{8}{}{x}{=}{0}$ (5)

inversion of a circle with respect to a circle

 > $\mathrm{circle}\left(\mathrm{c2},\left[\mathrm{point}\left(A,4,0\right),1\right],\left[x,y\right]\right):$
 > $\mathrm{IsOnCircle}\left(\mathrm{center}\left(\mathrm{c2}\right),\mathrm{c1}\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{inversion}\left(\mathrm{c3},\mathrm{c2},\mathrm{c1}\right):$
 > $\mathrm{form}\left(\mathrm{c3}\right)$
 ${\mathrm{circle2d}}$ (7)
 > $\mathrm{circle}\left(\mathrm{c2},{\left(x-3\right)}^{2}+{y}^{2}=36,\left[x,y\right]\right):$
 > $\mathrm{inversion}\left(\mathrm{c3},\mathrm{c1},\mathrm{c2}\right):$$\mathrm{inversion}\left(\mathrm{c4},\mathrm{c3},\mathrm{c2}\right):$
 > $\mathrm{Equation}\left(\mathrm{c1}\right)=\mathrm{Equation}\left(\mathrm{c4}\right)$
 $\left({{x}}^{{2}}{+}{{y}}^{{2}}{-}{16}{=}{0}\right){=}\left({{x}}^{{2}}{+}{{y}}^{{2}}{-}{16}{=}{0}\right)$ (8)
 > $\mathrm{detail}\left(\left[\mathrm{c1},\mathrm{c2},\mathrm{c3}\right]\right)$
 $\left[\begin{array}{ll}{\text{name of the object}}& {\mathrm{c1}}\\ {\text{form of the object}}& {\mathrm{circle2d}}\\ {\text{name of the center}}& {\mathrm{center_c1}}\\ {\text{coordinates of the center}}& \left[{0}{,}{0}\right]\\ {\text{radius of the circle}}& \sqrt{{16}}\\ {\text{equation of the circle}}& {{x}}^{{2}}{+}{{y}}^{{2}}{-}{16}{=}{0}\end{array}{,}\begin{array}{ll}{\text{name of the object}}& {\mathrm{c2}}\\ {\text{form of the object}}& {\mathrm{circle2d}}\\ {\text{name of the center}}& {\mathrm{center_c2}}\\ {\text{coordinates of the center}}& \left[{3}{,}{0}\right]\\ {\text{radius of the circle}}& \sqrt{{36}}\\ {\text{equation of the circle}}& {{x}}^{{2}}{+}{{y}}^{{2}}{-}{6}{}{x}{-}{27}{=}{0}\end{array}{,}\begin{array}{ll}{\text{name of the object}}& {\mathrm{c3}}\\ {\text{form of the object}}& {\mathrm{circle2d}}\\ {\text{name of the center}}& {\mathrm{center_c3}}\\ {\text{coordinates of the center}}& \left[\frac{{129}}{{7}}{,}{0}\right]\\ {\text{radius of the circle}}& {-}\frac{{36}{}\sqrt{{16}}}{{7}}\\ {\text{equation of the circle}}& \frac{{49}}{{1296}}{}{{x}}^{{2}}{-}\frac{{301}}{{216}}{}{x}{-}\frac{{455}}{{144}}{+}\frac{{49}}{{1296}}{}{{y}}^{{2}}{=}{0}\end{array}\right]$ (9)

 See Also