Can a line be characterized by a single point? Generally, I think of a line in $\mathbb R^2$ as characterized by any two points.  Yet, this question proves that any line not through the origin is characterized by a single point.  And a line through the origin can likewise be characterized through a single point.  This implies that any line can be characterized by a single point and a single bit (indicating whether the line is through the origin or not).
This seems mystifying: To define a line, can we choose any one point (plus a bit) or any two points?
More precisely: Is there a continuous transformation $(\mathbb R^2 \times \mathbb R^2) \longleftrightarrow (\mathbb R^2 \times \{0,1\})$ that is injective? How does this fit with the concept of dimension and degrees of freedom?
I may indeed be struggling to ask the right question here.  Indeed, turning this into a well formed question may almost provide the answer.  So help turning this baffling (at least to me) situation into a rigorous question is a very good way to start.
 A: There is more than one way to characterize a line in
two dimensions. One of those ways is to use projective
geometry. Consider Euclidean three dimensional space
with an origin. Fix a plane which does not contain the
origin. Any line in that plane uniquely determines a new plane that contains that line and also the origin. Now
suppose that new plane is not perpendicular to the
fixed plane. Then the unique line from the origin which
is perpendicular to the new plane intersects the fixed
plane in a unique point. This is known as the "polar"
of the original line in the fixed plane. Thus every
line in the fixed plane (with exceptions) is associated
with a unique point in the fixed plane.
The question you linked to is closely related. In that
case, the closest point of the line to the origin of
the plane is reciprocally associated with the polar
point in the previous paragraph. They are equivalent ways
to associate a unique point to every line (with exceptions)
in a plane. They also share the exceptional lines which are
without any associated points. The main difference is that
in this case only points in the plane are needed, while in the "polar" case, the plane must be embedded in a three
dimensional Euclidean space.
A: As Greg Martin has already pointed out in the comments, two points of a plane is actually too much information for determining a line in the plane, since many pairs of points determine the same line.
Two points on a plane is 2+2=4 dimensions. How much redundancy is there? Well, any two pairs of points on a fixed line determine that same line. This gives 1+1=2 redundant dimensions, leaving 4-2=2 dimensions, which is the number of dimensions for a single point. This agrees with the observation you made.
This also agrees with other standard representations of a line. For example, a line can be expressed in terms of two real numbers $m$ and $b$ as $y = mx + b$, so this is another way to see that the set of lines on a plane is 2 dimensional. In this latter representation, the vertical lines need to be treated differently, in the same way that in your example, lines through the origin are a special case that needs to be treated differently.  These exceptions go away if we go into the projective plane rather than the affine plane.
