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.

with(geometry):

point(OO,0,0):

parabola(p1,['vertex'=point('ver',0,0),'focus'=point('fo',0,1/2)]):

Equation(p1,[x,y]);

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

homology(p2,p1,OO,Pi/2,'counterclockwise',2):

Equation(p2);

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

homology(p3,p1,OO,Pi,'counterclockwise',2):

Equation(p3);

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

homology(p4,p1,OO,Pi/2,'clockwise',2):

Equation(p4);

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

homology(p5,p1,OO,0,'clockwise',2):

Equation(p5);

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

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