Raycasting

In computer graphics raycasting refers to a rendering technique in which we determine which parts of the scene should be drawn according to which parts of the scene are hit by rays cast from the camera. This is based on the idea that we can trace rays of light that enter the camera by going backwards, i.e. starting from the camera towards the parts of the scene that reflected the light. The term raycasting specifically has two main meanings:

3D raycasting: Algorithm that works the same as raytracing but without recursion. I.e. raycasting is simpler than raytracing and only casts primary rays (those originating from the camera), hence, unlike in raytracing, there are no shadows, reflections and refractions. Raytracing is the extension of raycasting.
2D raycasting: Technique for rendering so called "pseudo3D" (primitive 3D) graphics, probably best known from the old game Wolf3D (predecessor of Doom). The principle of casting the rays is the same but we only limit ourselves to casting the rays within a single 2 dimensional plane and render the environemnt by columns (unlike the 3D variant that casts rays and renders by individual pixels).

2D Raycasting

{ We have an official LRS library for advanced 2D raycasting: raycastlib! And also a game built on top of it: Anarch. ~drummyfish }

{ Also there is a very cool oldschool book that goes through programming a whole raycasting game engine in C, called Tricks of The Game Programming Gurus, check it out! ~drummyfish }

2D raycasting can be used to relatively easily render "3Dish" looking environments (commonly labeled "pseudo 3D"), mostly some kind of right-angled labyrinth. There are limitations such as the inability for the camera to tilt up and down (which can nevertheless be faked with shearing). It used to be popular in very old games but can still be used nowadays for "retro" looking games, games for very weak hardware (e.g. embedded), in demos etc. It is pretty cool, very suckless rendering method.

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raycasted view, rendered by the example below

The method is called 2D because even though the rendered picture looks like a 3D view, the representation of the world we are rendering is 2 dimensional (usually a grid, a top-down plan of the environment with cells of either empty space or walls) and the casting of the rays is performed in this 2D space -- unlike with the 3D raycasting which really does cast rays in fully 3D environments. Also unlike with the 3D version which casts one ray per each rendered pixel (x * y rays per frame), 2D raycasting only casts one ray per rendered column (x rays per frame) which actually, compared to the 3D version, drastically reduces the number of rays cast and makes this method fast enough for real time rendering even using software_rendering (without a GPU).

The principle is following: for each column we want to render we cast a ray from the camera and find out which wall in our 2D world it hits first and at what distance -- according to the distance we use perspective to calculate how tall the wall columns should look from the camera's point of view, and we render the column. Tracing the ray through the 2D grid representing the environment can be done relatively efficiently with algorithms normally used for line rasterization. There is another advantage for weak-hardware computers: we can easily use 2D raycasting without a framebuffer (without double_buffering) because we can render each frame top-to-bottom left-to-right without overwriting any pixels (as we simply cast the rays from left to right and then draw each column top-to-bottom). And of course, it can be implemented using fixed point (integers only).

The classic version of 2D raycasting -- as seen in the early 90s games -- only renders walls with textures; floors and ceilings are untextured and have a solid color. The walls all have the same height, the floor and ceiling also have the same height in the whole environment. In the walls there can be sliding doors. 2D sprites (billboards) can be used with raycasting to add items or characters in the environment -- for correct rendering here we usually need a 1 dimensional z-buffer in which we write distances to walls to correctly draw sprites that are e.g. partially behind a corner. However we can extend raycasting to allow levels with different heights of walls, floor and ceiling, we can add floor and ceiling texturing and, in theory, probably also use different level geometry than a square grid (however at this point it would be worth considering if e.g. BSP rendering wouldn't be better).

Implementation

The core element to implement is the code for casting rays, i.e. given the square plan of the environment (e.g. game level), in which each square is either empty or a wall (which can possibly be of different types, to allow e.g. different textures), we want to write a function that for any ray (defined by its start position and direction) returns the information about the first wall it hits. This information most importantly includes the distance of the hit, but can also include additional things such as the type of the wall, texturing coordinate or its direction (so that we can shade differently facing walls with different brightness for better realism). The environment is normally represented as a 2 dimensional array, but instead of explicit data we can also use e.g. a function that procedurally generates infinite levels (i.e. we have a function that for given square coordinates computes what kind of square it is). As for the algorithm for tracing the ray in the grid we may actually use some kind of line rasterization algorithm, e.g. the DDA algorithm (tracing a line through a grid is analogous to drawing a line in a pixel grid). This can all be implemented with fixed point, i.e. integer only! No need for floating point.

Note on distance calculation and distortion: When computing the distance of ray hit from the camera, we usually DO NOT want to use the Euclidean distance of that point from the camera position (as is tempting) -- that would create a so called fish eye effect, i.e. looking straight into a perpendicular wall would make the wall look warped/bowled (as the part of the wall in the middle of the screen is actually closer to the camera position so it would, by perspective, look bigger). For non-distorted rendering we have to compute a distance that's perpendicular to the camera plane -- we can see the camera plane as a "canvas" onto which we project the scene, in 2D it is a line (unlike in 3D where it really is a plane) at a certain distance from the camera (usually conveniently chosen to be e.g. 1) whose direction is perpendicular to the direction the camera is facing. The good news is that with a little trick this distance can be computed even more efficiently than Euclidean distance, as we don't need to compute a square root! Instead we can utilize the similarity of triangles. Consider the following situation:

                I-_ 
               /   '-X
              /  r.'/|
      '-._   /  ,' / |
      p   '-C_.'  /  |
          1/,'|-./   |
          /'  | /    |
         V-._-A/-----B
             'J

In the above V is the position of the camera (viewer) which is facing towards the point I, p is the camera plane perpendicular to VI at the distance 1 from V. Ray r is cast from the camera and hits the point X. The length of the line r is the Euclidean distance, however we want to find out the distance JX = VI, which is perpendicular to p. There are two similar triangles: VCA and VIB; from this it follows that 1 / VA = VI / VB, from which we derive that JX = VB / VA. We can therefore calculate the perpendicular distance just from the ratio of the distances along one principal axis (X or Y). However watch out for the case when VA = VB = 0 to not divide by zero! In such case use the other principal axis (Y).

Here is a complete C example that uses only fixed point with the exception of the stdlib sin/cos functions, for simplicity's sake (these can easily be replaced by custom fixed point implementation):

#include <stdio.h>
#include <math.h>     // for simplicity we'll use float sin, cos from stdlib

#define U 1024        // fixed-point unit
#define LEVEL_SIZE 16 // level resolution
#define SCREEN_W 100
#define SCREEN_H 31

int wallHeight[SCREEN_W];
int wallDir[SCREEN_W];

int perspective(int distance)
{
  if (distance <= 0)
    distance = 1;

  return (SCREEN_H * U) / distance;
}

unsigned char level[LEVEL_SIZE * LEVEL_SIZE] =
{
#define E 1, // wall
#define l 0, // floor
 l l l l E l l l l l l l l l E E 
 l E l l E E E l l l l l E l l E 
 l l l l l l l l l l l l l l l l 
 l E l l E l E l E l E l E l l l 
 l l l l E l l l l l l l l l E l 
 l l l l E l l l l l l l l l E l 
 l E E l E l l l l l l l l l l l 
 l E E l E l l l l l l l l l l l 
 l E l l l l l l l l l l l l l E 
 l E l l E l l l l l l l l E l l 
 l E l l E l l l l l l l l E l l 
 l E l l l l E E E l l l l l l l 
 l E E l E l l l l l E E E l l E 
 l E E l E l l l l l E l l l E E 
 l l l l l l E E E E E l l E E E 
 l l E l l l l l l l l l E E E E 
#undef E
#undef l
};

unsigned char getTile(int x, int y)
{
  if (x < 0 || y < 0 || x >= LEVEL_SIZE || y >= LEVEL_SIZE)
    return 1;

  return level[y * LEVEL_SIZE + x];
}

// returns perpend. distance to hit and wall direction (0 or 1) in dir
int castRay(int rayX, int rayY, int rayDx, int rayDy, int *dir)
{
  int tileX = rayX / U, 
      tileY = rayY / U,
      addX = 1, addY = 1;

  // we'll convert all cases to tracing in +x, +y direction

  *dir = 0;

  if (rayDx == 0)
    rayDx = 1;
  else if (rayDx < 0)
  {
    rayDx *= -1;
    addX = -1;
    rayX = (tileX + 1) * U - rayX % U;
  }

  if (rayDy == 0)
    rayDy = 1;
  else if (rayDy < 0)
  {
    rayDy *= -1;
    addY = -1;
    rayY = (tileY + 1) * U - rayY % U;
  } 

  int origX = rayX, 
      origY = rayY;

  for (int i = 0; i < 20; ++i) // trace at most 20 squares
  {
    int px = rayX % U, // x pos. within current square
        py = rayY % U,
        tmp;

    if (py > ((rayDy * (px - U)) / rayDx) + U)
    {
      tileY += addY; // step up
      rayY = ((rayY / U) + 1) * U;

      tmp = rayX / U;
      rayX += (rayDx * (U - py)) / rayDy;

      if (rayX / U != tmp) // don't cross the border due to round. error
        rayX = (tmp + 1) * U - 1;

      *dir = 0;
    }
    else
    {
      tileX += addX; // step right
      rayX = ((rayX / U) + 1) * U;

      tmp = rayY / U;
      rayY += (rayDy * (U - px)) / rayDx;

      if (rayY / U != tmp)
        rayY = (tmp + 1) * U - 1;

      *dir = 1;
    }

    if (getTile(tileX,tileY)) // hit?
    {
      px = rayX - origX;
      py = rayY - origY;

      // get the perpend dist. to camera plane:
      return (px > py) ? ((px * U) / rayDx) : ((py * U) / rayDy);
      
      // the following would give the fish eye effect instead
      // return sqrt(px * px + py * py);
    }
  }

  return 100 * U; // no hit found
}

void drawScreen(void)
{
  for (int y = 0; y < SCREEN_H; ++y)
  {
    int lineY = y - SCREEN_H / 2;

    lineY = lineY >= 0 ? lineY : (-1 * lineY);

    for (int x = 0; x < SCREEN_W; ++x)
      putchar((lineY >= wallHeight[x]) ? '.' : (wallDir[x] ? '/' : '#'));

    putchar('\n');
  }
}

int main(void)
{
  int camX = 10 * U + U / 4,
      camY = 9 * U + U / 2,
      camAngle = 600, // U => full angle (2 * pi)
      quit = 0;

  while (!quit)
  {
    int forwX = cos(2 * 3.14 * camAngle) * U,
        forwY = sin(2 * 3.14 * camAngle) * U,
        vecFromX = forwX + forwY, // leftmost ray
        vecFromY = forwY - forwX,
        vecToX = forwX - forwY,   // rightmost ray
        vecToY = forwY + forwX;

    for (int i = 0; i < SCREEN_W; ++i) // process each screen column
    {
      // interpolate rays between vecFrom and vecTo
      int rayDx = (SCREEN_W - 1 - i) * vecFromX / SCREEN_W + (vecToX * i) / SCREEN_W,
          rayDy = (SCREEN_W - 1 - i) * vecFromY / SCREEN_W + (vecToY * i) / SCREEN_W,
          dir,
          dist = castRay(camX,camY,rayDx,rayDy,&dir);

      wallHeight[i] = perspective(dist);
      wallDir[i] = dir;
    }

    for (int i = 0; i < 10; ++i)
      putchar('\n');

    drawScreen();

    char c = getchar();

    switch (c) // movement
    {
      case 'a': camAngle += 30; break;
      case 'd': camAngle -= 30; break;
      case 'w': camX += forwX / 2; camY += forwY / 2; break;
      case 's': camX -= forwX / 2; camY -= forwY / 2; break;
      case 'q': quit = 1; break;
      default: break;
    }
  }

  return 0;
}

How to make this more advanced? Here are some hints and tips:

textured walls: This is pretty simply, the ray hit basically gives us a horizontal texturing coordinate, and we simply stretch the texture vertically to fit the wall. I.e. when the ray hits a wall, we take the hit coordinate along the principal axis of the wall (e.g. for vertical hit we take the Y coordinate) and mod it by the fixed point unit which will give us the texturing coordinate. This coordinate tells us the column of the texture that the rendered column shall have; we read this texture column and render it stretched vertically to fit the column height given by the perspective. Note that for cache friendliness (optimization) textures should be stored column-wide in memory as during rendering we'll be reading the texture by columns (row-wise stored textures would make us jump wide distances in the memory which CPU caches don't like).
textured floor/ceiling: Something aking mode7 rendering can be used.
sliding door: TODO
jumping: Camera can easily be shifted up and down. If we are to place the camera e.g. one fixed point unit above its original position, then for each column we render we compute, with perspective applied to this one fixed point unit (the same way with which we determine the column size on the screen) the vertical screen-space offset of the wall and render this wall column that many pixel lower.
looking up/down: Correct view of a camera that's slightly tilted up/down can't be achieved (at least not in a reasonably simple way), but there's a simple trick for faking it -- camera shearing. Shearing literally just shifts the rendered view vertically, i.e. if we're to look a bit up, we render that same way as usual but start higher up on the screen (in the part of the rendered image that's normally above the screen and not visible), so that the vertical center of the screen will be shifted downwards. For smaller angles this looks good enough.
multilevel floor/ceiling: This is a bit more difficult but it can be done e.g. like this: for each level square we store its floor and ceiling height. When casting a ray, we will consider any change in ceiling and/or floor height a hit, AND we'll need to return multiple of those hits (not just the first one). When we cast a ray and get a set of such hits, from each hit we'll know there are tiny walls on the floor and/or ceiling, and we'll know their distances. This can be used to correctly render everything.
different level geometry: In theory the level doesn't have to be a square grid but some kind of another representation, or we may keep it a square grid but allow placement of additional shapes in it such as cylinders etc. Here you simply have to figure out how to trace the rays so as to find the first thing it hits.
adding billboards (sprites): TODO
reflections: We can make our 2D raycaster a 2D raytracer, i.e. when we cast a camera ray and it hits a reflective wall (a mirror), we cast another, secondary reflected ray and trace it to see which wall it hits, i.e. which wall will get reflected in the reflective wall.
partly transparent walls: We can make some walls partially transparent, both with alpha blending or textures with transparent pixels. In both cases we'll have to look not just for the first hit of the ray, but also for the next.

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Raycasting

2D Raycasting

Implementation

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