Editing Homothety (section)

==== using a pantograph ====
Before computers became ubiquitous, scalings of drawings were done by using a [[pantograph]], a tool similar to a [[Compass (drawing tool)|compass]].

''Construction and geometrical background:''

#Take 4 rods and assemble a mobile ''parallelogram'' with vertices <math>P_0,Q_0,H,P</math> such that the two rods meeting at <math>Q_0</math> are prolonged at the other end as shown in the diagram. Choose the ''ratio'' <math>k</math>.
#On the prolonged rods mark the two points <math>S,Q</math> such that <math>|SQ_0|=k|SP_0|</math> and <math>|QQ_0|=k|HQ_0|</math>. This is the case if <math>|SQ_0|=\tfrac{k}{k-1}|P_0Q_0|.</math> (Instead of <math>k</math> the location of the center <math>S</math> can be prescribed. In this case the ratio is <math>k=|SQ_0|/|SP_0|</math>.) 
#Attach the mobile rods rotatable at point <math>S</math>.
#Vary the location of point <math>P</math> and mark at each time point <math>Q</math>.

Because of <math>|SQ_0|/|SP_0|=|Q_0Q|/|PP_0|</math> (see diagram) one gets from the ''intercept theorem'' that the points <math>S,P,Q</math> are collinear (lie on a line) and equation <math>|SQ|=k|SP|</math> holds. That shows: the mapping  <math>P\to Q</math> is a homothety with center <math>S</math> and ratio <math>k</math>.