2. More on the relational form

Let's write $f(x,y) = ax + b y + c$, and let's always assume that at least one of $a$ and $b$ is not zero. Let $L$ be the line with equation $f(x,y) = 0$.

Observation 1. For any such $f$ with not both $a$ and $b$ equal to $0$, the plane    R$^ 2$ is divided into three subsets: $L$ itself, with equation $f(x,y) = 0$; the half-plane $f(x,y) > 0$, and the half-plane $f(x,y) < 0$.

Observation 2. For any constant $k \neq 0$, the equation $f(x,y) = k$ describes a line parallel to $L$. (In terms of concepts from calculus, $f(x,y) = k$ describes a level curve of $f$. Thus, for such an $f$ the level curves happen to be parallel lines.)

Observation 3. The vector    N$= [a,b]$ is a normal (i.e., a perpendicular) to $L$.

(This is the two-dimensional version of what you are used to with normals for planes in    R$^ 3$.)

Observation 4. The normal    N$= [a,b]$ points in the direction of the positive half-plane $f > 0$. (This makes it easy to tell which side is the one with $f > 0$.)

Observation 5. If $P _ 0$ is any point on $L$, the equation of $L$ can be written as    N$\cdot ($x$- P _ {0})$ $=$ $0$ (the point-normal form).

Observation 6. The absolute value of $f$ at each point equals the perpendicular distance from $L$ times $\vert$   N$\vert$ (the length of the normal).

Remark. A function of the form $f(x,y) = ax + b y + c$ is really an affine transformation $f:$   R$^ 2 \rightarrow$   R$$. Let's call such a function an affine function.

Because of the possibility of scaling $a,b,c$, there are many choices of $f$ that give the same line $L$. Which choice is best? Actually, there are several appropriate choices of $f$, depending on circumstances. One is the two-point form, discussed next; another is barycentric coordinates, discussed in §8; and a third is the signed distance function, discussed in the exercises.