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geometry

 StretchReflection
 find the stretch-reflection of a geometric object

 Calling Sequence StretchReflection(Q, P, l, O, k )

Parameters

 Q - the name of the object to be created P - geometric object l - line O - point on l k - number which is the ratio of the stretch-reflection

Description

 • Let l be a fixed line of the plane and O a fixed point on l, and let k be a given nonzero real number. By the stretch-reflection $S\left(\mathrm{O},k,l\right)$ we mean the product $R\left(l\right)H\left(\mathrm{O},k\right)$ where $R\left(l\right)$ is the reflection with respect to line l, and $H\left(\mathrm{O},k\right)$ is the dilatation with center O and ratio k. The line l is called the axis of the stretch-reflection, the point O is called the center of the stretch-reflection, and k is called the ratio of the stretch-reflection
 • For a detailed description of Q (the object created), use the routine detail (i.e., detail(Q))
 • The command with(geometry,StretchReflection) allows the use of the abbreviated form of this command.

Examples

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

Assign the name of the horizontal and vertical axis:

 > $\mathrm{_EnvHorizontalName}≔a:$$\mathrm{_EnvVerticalName}≔b:$
 > $\mathrm{parabola}\left(p,b={a}^{2}\right):$
 > $\mathrm{point}\left(A,0,0\right),\mathrm{line}\left(l,b=0\right):$

dilate p with center A and ratio 1/2, then reflect this object with respect to the line l

 > $\mathrm{StretchReflection}\left(\mathrm{p1},p,l,A,\frac{1}{2}\right):$
 > $\mathrm{detail}\left(\left\{p,\mathrm{p1}\right\}\right)$
 $\left\{\begin{array}{ll}{\text{name of the object}}& {p}\\ {\text{form of the object}}& {\mathrm{parabola2d}}\\ {\text{vertex}}& \left[{0}{,}{0}\right]\\ {\text{focus}}& \left[{0}{,}\frac{{1}}{{4}}\right]\\ {\text{directrix}}& {b}{+}\frac{{1}}{{4}}{=}{0}\\ {\text{equation of the parabola}}& {-}{{a}}^{{2}}{+}{b}{=}{0}\end{array}{,}\begin{array}{ll}{\text{name of the object}}& {\mathrm{p1}}\\ {\text{form of the object}}& {\mathrm{parabola2d}}\\ {\text{vertex}}& \left[{0}{,}{0}\right]\\ {\text{focus}}& \left[{0}{,}{-}\frac{{1}}{{8}}\right]\\ {\text{directrix}}& {-}{2}{}{b}{+}\frac{{1}}{{4}}{=}{0}\\ {\text{equation of the parabola}}& {-}{4}{}{{a}}^{{2}}{-}{2}{}{b}{=}{0}\end{array}\right\}$ (1)
 > $\mathrm{draw}\left(\left\{p\left(\mathrm{style}=\mathrm{LINE}\right),\mathrm{p1}\left(\mathrm{style}=\mathrm{POINT}\right)\right\}\right)$