0. Review: Describing straight lines

You have in the past seen three ways to express a straight line in    R$^ 2$ algebraically:

Functionally,

as the graph $y = mx + b$ of a linear function.

Relationally,

as the graph of an equation $ax + b y + c = 0$ giving a linear relation between $x$ and $y$. (Not both of $a$ and $b$ are 0.)

Parametrically,

by an equation    x$(t) = P _ 0 + t$   v$$ giving the path of a moving point. (Here    v$\neq$   0$$.)

For the functional form, a disadvantage is that vertical lines cannot be expressed. In fact, any graph of this kind is closely tied to the orientation of the axes; it would be messy to rotate the graph and then re-express it in the same way, for example. Advantages of this form are that there is only one way to express each non-vertical line and that there is no restriction on the numbers $m$ and $b$ that can be used.

For the relational form, an advantage is that any line can be expressed. For example, $1 x + 0y + 2 = 0$ gives a vertical line. A disadvantage is that there is more than one way to express each line. For example, multiplying the equation through by $2$ gives the same line.

The parametric form can be viewed as the path of a moving point with velocity vector    v$$ and with $P(0) = P _ 0$. An advantage is that any line can be expressed. A disadvantage is that there are many ways to express each line: $P _ 0$ can be changed to any other point on the line, and    v$$ can be multiplied by any nonzero scalar, without changing the line represented.

In computer graphics, there are many applications for the relational and parametric forms, as you will see. The functional form is rarely used. It's easy to take a line written functionally and re-express it either relationally or parametrically. Therefore, in these notes we'll consider only these last two forms.