LinePencils

1. Pencils of lines

A point P of the plane defines the pencil P* of all lines passing through it. Thus, if P=(s, t) this is the set of all lines:
                                              ax+by+c=0   whoese coefficients (a,b,c)    satisfy   as + bt + c = 0.    This is a homogeneous equation in (a,b,c) and setting arbitrary (a,b) we get  c = -as-bt and the solutions:
                                             (a,b,c) = a(1,0,-s) + b(0,1,-t).
This means that all lines of the pencil are expressible as a linear combination of the two lines:
                                              x-s = 0    and   y-t = 0.
Inversely given two lines   f(x,y) = ax+by+c =0 ,   g(x,y) = a'x+b'y+c' = 0, every line of the form:
                                           h(x,y) = sf(x,y) + tg(x,y) = (sa+ta')x+(sb+tb')x+(sc+tc') = 0,
defines a line of the pencil P*, where P is the intersection point of the two lines  f(x,y)=0 and g(x,y)=0.

A particular member h(x,y) = sf(x,y)+tg(x,y) of the pencil P* is determined by the single condition to pass through a point X₀ different from P. In fact (s,t) are (modulo a multiplicative constant) determined by the condition:
h(x₀,y₀) = sf(x₀,y₀) + tg(x₀,y₀) = 0.
This defines (s, t) up to a multiplicative constant (s,t) = k(-g(x₀,y₀), f(x₀,y₀)). The constant k is not essential since h(x,y)=0 and kh(x,y)=0 define the same line.

2. The meaning of (s,t)

The parameters (s,t) defining the members of the pencil P* have a geometrical meaning, since the functions f(x,y)=ax+by+c, g(x,y)=a'x+b'y+c' give appropriate multiples of the signed distances of the point X=(x,y) from the corresponding lines:

Thus the ratio which is essential is equal to:
(s/t) = -g(x₀,y₀)/f(x₀,y₀) = -(k_g/k_f)(d_g(X₀)/d_f(X₀).
Normalizing the equations of the lines (i.e. taking k_g=k_f=1) we get for the corresponding ratio:

The formula with the sines is correct if the normals of the lines point both inside or outside, otherwise it has to be multiplied by -1.
Corollary If lines f(x,y)=g(x,y)=0 are in normal form, then the two lines f(x,y)+g(x,y)=0, f(x,y)-g(x,y)=0 represent the bisectors of the angle formed by the first two lines. In general line h(x,y) = f(x,y)-kg(x,y)=0 coincides with the locus of points for which the ratio of signed distances from the first two lines is k:

3. Ratio on a third line

Line f(x,y)-kg(x,y)=0 , together with f(x,y)=0 and g(x,y)=0 define a ratio on every transversal line j(x,y)=0.

The ratio |CA|/|CB| = area(ACP)/area(CPB) = |PA|sin(CPA)/(|PB|sin(CPB)) and since by the previous section the ratio of sines is equal to k:

This has the consequence to give directly the cross ratio of four lines in terms of the factor k. If the lines are f(x,y) = 0, g(x,y) = 0, f(x,y)-kg(x,y)=0, f(x,y)-k'g(x,y)=0, defining on j(x,y)=0 points {A,B,C,C'}, then the cross ratio (A,B,C,C') is:

Corollary The cross ratio of the traces of four lines on any transversal is the same. It depends only on the relative position of the four lines and not on the transversal.
Last formula for the cross ratio remains true even in the case in which lines f(x,y)=0, g(x,y)=0 are not in normal form. Because of the division the corresponding factors k_f, k_g disapear and the result is the same.
The discussion here is a sequel to the one started in Lines.html . It continues with the consideration of the quadratic equation defined by two lines contained in ConicsDegenerate.html .

1. Pencils of lines

2. The meaning of (s,t)

3. Ratio on a third line

See Also