Demostración de la ecuación normalizada de Smith

1) r = 1− u2¿−v2

(1−u)2+v2¿

r= 1−u2−v2

1−2u+u2+v2

r−2ur+ru2+r v2=1−u2−v2

u2 (1+r )+v2 (1+r )−2ur=1−r

u2+v2−2ur1+r

=1−r1+r

Completando cuadrados:

u2+v2−2ur1+r

+( r1+r )2

=1−r1+r

+( r1+r

)2

(u− r1+r

)2

+v2= 1(1+r )2

2) x=2v

(1−u)2+v2

(1−u)2+v2−2vx

=0

1−u¿¿2+v2−2vx

+ 1x2

= 1

x2

(1−u)2+(v−1x)2

= 1x2

(u−1)2+(v−1x)2

= 1x2