Jesús de la Peña Hernández

OP

ON

1

=− (2)

In same Fig. 2 we see:

(^) OP=xcosβ+ycosα (3)
As also it is:
ON
u
cosβ= ;
ON
v
cosα= (4)
substituting (2) and (4) in (3), we have:
ON
v
y
ON
u
OP=x + ; ux+vy+ 1 = 0 (5)
(5) is the Plückerian equation of straight line t and (u,v) its covariant co-ordinates. If we
compare now (1) and (5) we have the relation between Cartesian and Plückerian equations of t:
a
u
1
=− ;
b
v
1
=−
If (5) takes the form of ux 1 +vy 1 + 1 = 0 , it represents a pencil of lines in the plane with
variables (u,v) passing through the particular point (x 1 ,y 1 )
Let ́s find the tangential equation (i.e. in Plückerian co-ordinates) of the circumference
with center O (0,0) to which t is the tangent on P. For that it ́s enough to equalise OP and the
radius r of the circumference:
OP=r ; r
ON
− =
1
; r
u v

• −
  2 2
  1
  ;^222
  1
  u v
  r


• 
  

  = ;^222
  1
  r
  u +v =
  Y
  P
  (x,y)
  t
  X
  O u (a,0)
  v
  N
  (0,b)
  2
  O
  (0,b)
  Y
  (x,y)
  (a,0)
  t
  1
  X
  (0,0)
  O
  B
  v
  X
  A
  t
  3
  Y
  u
  P