In 1965 Jack Elton Bresenham published the paper *Algorithm for computer control of a digital
plotter* in the IBM Systems Journal, volume 4, number 1.
It explained how a line could be approximated on an integer grid.
The algorithm is still used today as a rasterization
technique for rendering lines on video displays or printers.
As Bresenham's paper suggests, however, it was originally devised for a
plotter, capable of moving from one grid point to one of
the adjacent eight grid points.

We consider drawing a line from $(0, 0)$ to $(\Delta x, \Delta y)$ on an integer grid by choosing the points that lie close to the true line. We also assume that $\Delta x > 0$ and $\Delta x \geq \Delta y \geq 0$. The method is easily extended to the general case with arbitrary starting and ending points.

The line has the form

$y = f(x) = \frac{\Delta y}{\Delta x} x$

and we aim to sample the line at the grid points $(x_k, y_k)$ with $x_k = k$ for $k = 0, 1, \ldots, \Delta x$ and $y_0 = 0$.

We now associate with each step an *error* term,

$\nabla_k = 2\Delta x \left[f(x_k) - (y_{k-1} + \textstyle\frac{1}{2})\right] = 2x_k\Delta y - (2y_{k-1} + 1) \Delta x \; .$

The interpretation of this error term is important: It expresses the difference at $x = x_k$ between the line's true $y$-coordinate, $f(x_k)$, and the midpoint between the previously chosen $y$-coordinate, $y_{k-1}$, and its successor, $y_{k-1}+1$. The factor $2\Delta x$ is there to make all quantities integral. The sign of $\nabla_k$ is thus crucial:

- If $\nabla_k \leq 0$ then $y_k = y_{k-1}$ lies closest to the line (it is a tie for $\nabla_k = 0$).
- If $\nabla_k > 0$ then $y_k = y_{k-1}+1$ should be chosen.

Note how the magnitude of $\nabla_k$ also tells exactly how far from the true line the chosen grid point is. It is straightforward to show that $-2\Delta x < \nabla_k \leq 2\Delta x$ for $k = 1, 2, \ldots, \Delta x$.

To determine the $\nabla_k$'s we start with the base case:

$\nabla_1 = 2\cdot 1\cdot \Delta y - (2\cdot 0 + 1) \Delta x = 2\Delta y - \Delta x \; .$

For $k=1, 2, \ldots, \Delta x - 1$ we get

$\nabla_{k+1} - \nabla_k = 2(x_{k+1}-x_k)\Delta y - 2(y_k-y_{k-1})\Delta x \; .$

We now split into the two cases for $\nabla_k$ and get

$\nabla_{k+1} = \begin{cases} \nabla_k + 2\Delta y & \text{for } \nabla_k \leq 0 \; , \\ \nabla_k + 2\Delta y - 2\Delta x & \text{for } \nabla_k > 0 \; , \end{cases}$

for $k=1, 2, \ldots, \Delta x - 1$.

These expressions determine all the $\nabla_k$'s which, in turn, determines all the $y_k$'s.