Let O be a fixed point in the plane, k a given nonzero real number, theta and dir denote a given sensed angle. By the homology (or stretch-rotation, or spiral-rotation) $H\left(\mathrm{O}\,k\,\mathrm{\theta }\right)$ we mean the product $R\left(\mathrm{O},\mathrm{theta}\right)H\left(\mathrm{O},k\right)$ where $R\left(\mathrm{O}\,\mathrm{\theta }\,\mathrm{dir}\right)$ is the rotation with respect to O an angle theta in direction dir and $H\left(\mathrm{O}\,k\right)$ is the dilatation with respect to the center O and ratio k.

Point O is called the center of the homology, k the ratio of the homology, theta and dir the angle of the homology.

For a detailed description of Q (the object created), use the routine detail (i.e., detail(Q))

The command with(geometry,StretchRotation) allows the use of the abbreviated form of this command.

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

$\mathrm{point}\left(\mathrm{OO}\,0\,0\right)\:$

$\mathrm{parabola}\left(\mathrm{p1}\,\left['\mathrm{vertex}'=\mathrm{point}\left('\mathrm{ver}'\,0\,0\right)\,'\mathrm{focus}'=\mathrm{point}\left('\mathrm{fo}'\,0\,\frac{1}{2}\right)\right]\right)\:$

$\mathrm{Equation}\left(\mathrm{p1}\,\left[x\,y\right]\right)$

$\frac{x^2}{4}-\frac{y}{2}=0$

$\mathrm{homology}\left(\mathrm{p2}\,\mathrm{p1}\,\mathrm{OO}\,\frac{\mathrm{\pi }}{2}\,'\mathrm{counterclockwise}'\,2\right)\:$

$\mathrm{Equation}\left(\mathrm{p2}\right)$

$\frac{y^2}{16}+\frac{x}{4}=0$

$\mathrm{homology}\left(\mathrm{p3}\,\mathrm{p1}\,\mathrm{OO}\,\mathrm{\pi }\,'\mathrm{counterclockwise}'\,2\right)\:$

$\mathrm{Equation}\left(\mathrm{p3}\right)$

$\frac{x^2}{16}+\frac{y}{4}=0$

$\mathrm{homology}\left(\mathrm{p4}\,\mathrm{p1}\,\mathrm{OO}\,\frac{\mathrm{\pi }}{2}\,'\mathrm{clockwise}'\,2\right)\:$

$\mathrm{Equation}\left(\mathrm{p4}\right)$

$\frac{y^2}{16}-\frac{x}{4}=0$

$\mathrm{homology}\left(\mathrm{p5}\,\mathrm{p1}\,\mathrm{OO}\,0\,'\mathrm{clockwise}'\,2\right)\:$

$\mathrm{Equation}\left(\mathrm{p5}\right)$

$\frac{x^2}{16}-\frac{y}{4}=0$

$\mathrm{draw}\left(\left\{\mathrm{p2}\,\mathrm{p3}\,\mathrm{p4}\,\mathrm{p5}\,\mathrm{p1}\left(\mathrm{color}=\mathrm{green}\,\mathrm{style}=\mathrm{LINE}\,\mathrm{thickness}=2\,\mathrm{numpoints}=50\right)\right\}\,\mathrm{style}=\mathrm{POINT}\,\mathrm{numpoints}=200\,\mathrm{color}=\mathrm{brown}\,\mathrm{title}=\mathrm{`homology\; of\; a\; parabola`}\right)$