Let O be a fixed point of the plane and k a given nonzero real number. By the dilatation (or expansion, or homothety, or stretch) $H\left(\mathrm{O}\,k\right)$ we mean the transformation of S onto itself which carries each point P of the plane into the point Q of the plane such that $\mathrm{SensedMagnitude}\left(\mathrm{OQ}\right)=k\mathrm{SensedMagnitude}\left(\mathrm{OP}\right)$. The point O is called the center of the dilatation, and k is called the ratio of the dilatation.

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

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

with(geometry):

point(A,1,1): dilatation(B,A,3,point(OO,3,3)):

detail(B);

$\begin{array}{ll}\text{name of the object}& B\\ \text{form of the object}& \mathrm{point2d}\\ \text{coordinates of the point}& \left[\mathrm{-3}\,\mathrm{-3}\right]\end{array}$

circle(c,[point(OO,0,0),1]):

homothety(c1,c,3,OO):

draw({c(color=red,style=POINT,symbol=DIAMOND),
      c1(color=blue,style=POINT,symbol=CROSS,numpoints=100)},
      title= `dilatation of a circle`);

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

expansion(p2,p1,2,OO):

expansion(p3,p1,1/2,OO):

expansion(p4,p1,1/4,OO):

draw([p1(color=green,style=LINE,thickness=2),p2,p3,p4],
      style=POINT,color=brown,view=[-1/2..1/2,.0..2/5],
      numpoints = 400, title = `dilatation of a hyperbola`);