Polar_On_Two

1. Polar of point with respect to two lines

Given two lines {L₁, L₂} and a point P not lying on them, consider all secants L trhough P intersecting the lines respectively at points {Q₁, Q₂}. The geometric locus of the harmonic conjugate P' of P with respect to {Q₁, Q₂} is another line L_P called the polar of P with respect to {L₁, L₂}.

The easiest way to prove this is to express the cross ratio (Q₁,Q₂,P,P') through the angles of the lines at Q. It is then (see CrossRatioLines.html )

Hence if this cross ratio is -1 for a line through P it is also -1 for all lines through P. Line QP' is called also the harmonic conjugate of line QP with respect to {L₁, L₂}.

2. The equation of the polar

Assume that the two lines {L₁, L₂} are given by equations f₁(X)=0 and f₂(X)=0. Then the polar of P with respect to {L₁,L₂} is given by
                                                                 f₂(P)f₁(X) + f₁(P)f₂(X) = 0.
In fact, this is a line through the intersection point Q of {L₁, L₂} and consequently belongs to the pencil of lines
                                                                  sf₁(X) + tf₂(x) = 0.
To find it notice that line PQ belongs also to that pencil, hence satisfies
                                                                  sf₁(P) + tf₂(P) = 0,
and consequently
                                                                  (s,t) = k(-f₂(P), f₁(P)).
Thus, line PQ has the equation
                                                                 -f₂(P)f₁(X) + f₁(P)f₂(X) = 0.
The result follows by applying the previous relation of the sines and the fact that the ratio of the distances of P from the two lines is proportional to the ratio of sines:

3. The conjugate reflection

With the polar of a point P with respect to two lines {L₁, L₂} we have the possibility to define a naturally to this configuration related affine transformation. In fact, if L_P denotes the polar of point P, then define the affine reflection R_P by
R_P = (PQ, L_P).
The symbol refers to the above figure and defines the affine reflection (see Affine_Reflexion_Basics.html ) with axes of fixed points the line PQ and conjugate direction that of line L_P. By definition this means that for every point X its image X' under the transformation has XX' parallel to L_P and the middle of XX' is on line PQ.

4. Expression in coordinates

Assume that the line-equation is given in the form
                                                                           f(X) =  (X,a) + c = 0,
where (X,Y)=x₁y₁+x₂y₂ is the usual inner product and c a constant. Assume further that lines {PQ, L_P} are respectively given by two equations  {f₁(X)=(X,a₁)+c₁=0  ,  f₂(X)= (X,a₂)+c₂ = 0}. Then it is easily seen that the reflection is represented by the equation

Here JX denotes the π/2-rotated of vector X. Thus if X=(x₁,x₂), then JX = (-x₂, x₁). To express the transformation in terms of the original lines assume that {L₁, L₂} are given by equations { g₁(X) = (b₁,X)+d₁=0 ,   g₂(X)=2,X>+d₂=0 }. Introducing the constants {s = g₁(P), t = g₂(P)} we saw that the axis PQ is given by
                                     (a₁,X)+c₁ = -t((b₁,X)+d₁) + s((b₂,X)+d₂) = 0,
and the conjugate direction
                                     (a₂,X)+c₂ = t((b₁,X)+d₁) + s((b₂,X)+d₂) = 0.
Hence
                                      a₁ = -tb₁ + sb₂, c₁ = -td₁+st₂,
                                      a₂ =  tb₁ + sb₂, c₁ =   td₁+st₂.
From these follows that
                                                   (a₁, Ja₂) = 2st(b₂,Jb₁)
and finally

5. The conjugate polygon

Given a polygon p=P₀...P_n-1 with n sides a₀=P₀P₁, a₁=P₁P₂, ... etc. (indices exceeding n are taken modulo n) and a point Q, not on its side-lines, we can construct another polygon q=Q₀...Q_n₁ with sides b₀=Q₀Q₁, b₁=Q₁Q₂, ... etc. by taking the polars of Q with respect to the side-pairs (a_i, a_i+1) of p. The resulting polygon circumscribes p and is called the conjugate of p with respect to Q.

1. Polar of point with respect to two lines

2. The equation of the polar

3. The conjugate reflection

4. Expression in coordinates

5. The conjugate polygon

See Also