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Line is one of the most basic geometric shapes, it is straight, continuous, infinitely long and infinitely thin. A finite continuous part of a line is called line segment, though in practice we sometimes call line segments also just lines. In flat, non-curved geometries shortest path between any two points always lies on a line.

Line is a one dimensional shape, i.e. any of its points can be directly identified by a single number -- the signed distance from a certain point on the line. But of course a line itself may exist in more than one dimensional spaces (just as a two dimensional sheet of paper can exist in our three dimensional space etc.).

   /               |     \            .'
  /   ________     |      \         .'
 /                 |       \      .'
/                  |        \   .'

some lines, in case you haven't seen one yet


Mathematically lines can be defined by equations with space coordinates (see analytic geometry) -- this is pretty important for example for programming as many times we need to compute intersections with lines; for example ray casting is a method of 3D rendering that "shoots lines from camera" and looks at which objects the lines intersect. Line equations can have different "formats", the two most important are:

As an equation for line segment we simply limit the equation for an infinite line, for example with the parametric equations we limit the possible values of t by an interval that corresponds to the two boundary points.

Example: let's try to find equations of a line in 2D that goes through points A = [1,2] and B = [4,3].

Point-slope equation is of form y = k * x + q. We want to find numbers k (slope) and q (offset). Slope says the line's direction (as dy/dx, just as in derivative of a function) and can be computed from points A and B as k = (By - Ay) / (Bx - Ax) = (3 - 2) / (4 - 1) = 1/3 (notice that this won't work for a vertical line as we'd be dividing by zero). Number q is an "offset" (different values will give a line with same direction but shifted differently), we can simply compute it by plugging in known values into the equation and working out q. We already know k and for x and y we can substitute coordinates of one of the points that lie on the line, for example A, i.e. q = y - k * x = Ay - k * Ax = 2 - 1/3 * 1 = 5/3. Now we can write the final equation of the line:

y = 1/3 * x + 5/3

This equation lets us compute any point on the line, for example if we plug in x = 3, we get y = 1/3 * 3 + 5/3 = 8/3, i.e. point [3,8/3] that lies on the line. We can verify that plugging in x = 1 and x = 4 gives us [1,2] (A) and [4,3] (B).

Now let's derive the parametric equations of the line. It will be of form:

x = Px + t * Dx

y = Py + t * Dy

Here P is a point that lies on the line, i.e. we may again use e.g. the point A, so Px = Ax = 1 and Py = Ay = 2. D is the direction vector of the line, we can compute it as B - A, i.e. Dx = Bx - Ax = 3 and Dy = By - Ay = 1. So the final parametric equations are:

x = 1 + t * 3

y = 2 + t * 1

Now for whatever t we plug into these equations we get the [x,y] coordinates of a point that lies on the line; for example for t = 0 we get x = 1 + 0 * 3 = 1 and y = 2 + 0 * 1 = 2, i.e. the point A itself. As an exercise you may try substituting other values of t, plotting the points and verifying they lie on a line.

Line Drawing Algorithms

Drawing lines with computers is a subject of computer graphics. On specific devices such as vector monitors this may be a trivial task, however as most display devices nowadays work with raster graphics, let's from now on focus only on such devices.

There are many algorithms for line rasterization. They differ in attributes such as:

             XXX               XX             .aXa
           XX                XX             lXa.
         XX                XX            .lXl
      XXX               XXX            .aal
    XX                XX             lXa.
  XX               XXX            .aXl
XX               XX               a.

      pixel         subpixel     subpixel accuracy
     accuracy       accuracy      + anti-aliasing

One of the most basic line rasterization algorithms is the DDA (Digital differential analyzer), however it is usually better to use at least the Bresenham's line algorithm which is still simple and considerably improves on DDA.

TODO: more algorithms, code example, general form (dealing with different direction etc.)

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