Lab Work F6. 

Question: 
Question 1. Find the composition of two dilatations,
 $f = \delta_{P,s}\delta_{Q,t}$?

Hint: Apply this transformation to a test point 
$X$ i.e. $Y = Q + t(X-Q)$, $Z= P+s(Y-P)$, substitute!

Answer: $Z =(P+sQ-stQ-sP )+(st)X$, 
 

If $st \ne 1$ then

$r:=st$, $M = \frac{(P+sQ-rQ-sP )}{1-st} $

$Z= (1-r)M + rX$  therefore $f =\delta_{M,r}$


Question 2:
What is this composition when $st=1$?

$f(X) = (P+sQ-Q-sP)+X = X + D$ where $D=(1-s)(P-Q)$


Question 3: 
What if $P=Q$?
Answer: $f(X) = (1-r)Q + rX $ therefore

$\delta_{Q,s}\delta_{Q,t}=\delta_{Q,st} $
Question 4:
What if $s=t=-1$?