drummyfish/less_retarded_wiki
5
1
Fork 0
Code Issues 1 Pull requests Projects Releases Packages Wiki Activity
less_retarded_wiki/line.md
Miloslav Ciz 364e16e0e6 Update
2023-10-29 00:06:54 +02:00

8 KiB
Raw Blame History

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. 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

Representing Lines With 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.

Formulas

Here let be formulas for computing various things related to lines and line segments.

Consider two dimensional plane. Let L be a line (or line segment) going from point L1 = [L1x,L1y] to point L2 = [L2x,L2y]. Let dx = L2x - L1x and dy = L2y - L1y. Let K be another line (or line segment). Let P = [Px,Py] be a point.

  • line segment length: Use Pythagorean theorem: length(L) = sqrt(dx^2 + dy^2). The same goes for a line in 3D.
  • determine which side of line L point P lies on: A simple way is to use the simple formula for triangle winding, i.e. determine if triangle [L1,L2,P] goes clockwise or counterclockwise. This can also determine if the point lies exactly on the line (i.e. lies on neither side).
  • shortest distance of point P from line L: TODO
  • intersection of lines (or line segments) L and K: Represent the lines with their equations (see above), preferably parametric (allows any angle), put both points equal and solve the system of equations (watch out for the cases with no or many solutions). For line segments you also additionally have to check whether the intersection you find lies within BOTH line segments (with parametric representations this is easily done by checking if both parameters you get as a solution lie in the range 0 to 1).
  • angle between lines L and K: TODO
  • distance of lines P and L: TODO
  • project point P orthogonally to line L: TODO
  • TODO: more

TODO: 3D lines

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:

  • complexity of implementation
  • speed/efficiency (some algorithms avoid the use of floating point which requires special hardware)
  • support of antialiasing ("smooth" vs "pixelated" lines)
  • subpixel precision (whether start and end point of the line has to lie exactly on integer pixel coordinates; subpixel precision makes for smoother animation)
  • support for different width lines (and additionally e.g. the shape of line segment ends etc.)
  • ...
                                                 .
             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.)