less_retarded_wiki/line.md

Line

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. Shortest path between any two points always lies on a line. { At least I hope :D ~drummyfish }

Line is a one dimensional shape, i.e. any of its points can be identified by a single straightforward number (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 see one yet

Equations

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:

• point-slope: This equation only works in 2D space (in 3D this kind of equation will not describe a line but rather a plane) and only for lines that aren't completely vertical (lines close to vertical may also pose problems in computers with limited precision numbers). The advantage is that we have a single, pretty simple equation. The equation is of form y = k * x + q where x and y are space coordinates, k is the slope of the line and q is an offset. See examples below for more details.
• parametric: This is a system of N equations, where N is the number of dimensions of the space the line is in. This way can describe any line in any dimensional space -- obviously the advantage here is that we can can use this form in any situation. The equations are of form Xn = Pn + t * Dn where Xn is nth coordinate (x, y, z, ...), Pn is nth coordinate of some point P that lies on the line, Dn is nth coordinate of the line's direction vector and t is a variable parameter (plugging in different numbers for t will yield different points that lie on the line). DON'T PANIC if you don't understand this, see the examples below :)

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

TODO