Equation of a line (idea)

So, you want to convey the idea of a straight line, but don't want to deal with the horrible imprecision of ASCII art?

   y
   | / 
   |/
   /
  /|
-/-------- x
/  |
   |

Rejoice! You can use math in a variety of flavors to express this concept abstractly.

In elementary algebra, we use the slope-intercept form of the equation of a line in the plane: y = mx+b. Pity the fool who does not recognize this formula.

In complex analysis, we can interpret x and y as linearly independent components of the complex variable z = x+iy, and reinterpret the slope-intercept form as:

0 = -y+mx+b
  = Re((m+i)(x+iy)+b)
  = Re((m+i)z+b)

If you want to generalize this to include vertical lines, you can use

0 = Re(az+b)

which more closely resembles the "standard" equation of a line from high school:

a1x - a2y + Re(b) = 0

with the slope being a₂/a₁.

In the wide world of Euclidean n-space, given two points in , or equivalently, the endpoints of two vectors emanating from the origin, we can describe the line passing through them in terms of vector addition and scalar multiplication. Say the vectors in question are u and v. Then, the line passing through them is the locus of points given by

u + t(v-u)

for all real t. Equivalently, of course, we can also write

v + t(u-v).