Raycasting

The raycasting engine of Wolfenstein 3D was very limited, allowing it to run on a even a 286 computer: all the walls have the same height and are orthogonal squares on a 2D grid, as can be seen in this screenshot from a mapeditor for Wolf3D:

Things like stairs, jumping or height differences are impossible to make with this engine. Later games such as Doom and Duke Nukem 3D also used raycasting, but much more advanced engines that allowed sloped walls, different heights, textured floors and ceilings, transparent walls, etc... The sprites (enemies, objects and goodies) are 2D images, but sprites aren't discussed in this tutorial for now.

Raycasting is not the same as raytracing! Raycasting is a fast semi-3D technique that works in realtime even on 4MHz graphical calculators, while raytracing is a realistic rendering technique that supports reflections and shadows in true 3D scenes, and only recently computers became fast enough to do it in realtime for reasonably high resolutions and complex scenes.

The code of the untextured and textured raycasters is given in this document completely, but it's quite long, you can also download the code instead:

For every x of the screen (i.e. for every vertical stripe of the screen), send out a ray that starts at the player location and with a direction that depends on both the player's looking direction, and the x-coordinate of the screen. Then, let this ray move forward on the 2D map, until it hits a map square that is a wall. If it hit a wall, calculate the distance of this hit point to the player, and use this distance to calculate how high this wall has to be drawn on the screen: the further away the wall, the smaller it's on screen, and the closer, the higher it appears to be. These are all 2D calculations. This image shows a top down overview of two such rays (red) that start at the player (green dot) and hit blue walls:

To find the first wall that a ray encounters on its way, you have to let it start at the player's position, and then all the time, check whether or not the ray is inside a wall. If it's inside a wall (hit), then the loop can stop, calculate the distance, and draw the wall with the correct height. If the ray position is not in a wall, you have to trace it further: add a certain value to its position, in the direction of the direction of this ray, and for this new position, again check if it's inside a wall or not. Keep doing this until finally a wall is hit.

A human can immediatly see where the ray hits the wall, but it's impossible to find which square the ray hits immediatly with a single formula, because a computer can only check a finite number of positions on the ray. Many raycasters add a constant value to the ray each step, but then there's a chance that it may miss a wall! For example, with this red ray, its position was checked at every red spot:

As you can see, the ray goes straight through the blue wall, but the computer didn't detect this, because it only checked at the positions with the red dots. The more positions you check, the smaller the chance that the computer won't detect a wall, but the more calculations are needed. Here the step distance was halved, so now he detects that the ray went through a wall, though the position isn't completely correct:

For infinite precision with this method, an infinitely small step size, and thus an infinite number of calculations would be needed! That's pretty bad, but luckily, there's a better method that requires only very few calculations and yet will detect every wall: the idea is to check at every side of a wall the ray will encounter. We give each square width 1, so each side of a wall is an integer value and the places in between have a value after the point. Now the step size isn't constant, it depends on the distance to the next side:

As you can see on the image above, the ray hits the wall exactly where we want it. In the way presented in this tutorial, an algorithm is used that's based on DDA or "Digital Differential Analysis". DDA is a fast algorithm typically used on square grids to find which squares a line hits (for example to draw a line on a screen, which is a grid of square pixels). So we can also use it to find which squares of the map our ray hits, and stop the algorithm once a square that is a wall is hit.

Some raytracers work with Euclidean angles to represent the direction of the player and the rays, and determinate the Field Of View with another angle. I found however that it's much easier to work with vectors and a camera instead: the position of the player is always a vector (an x and a y coordinate), but now, we make the direction a vector as well: so the direction is now determinated by two values: the x and y coordinate of the direction. A direction vector can be seen as follows: if you draw a line in the direction the player looks, through the position of the player, then every point of the line is the sum of the position of the player, and a multiple of the direction vector. The length of a direction vector doesn't really matter, only its direction. Multiplying x and y by the same value changes the length but keeps the same direction.

This method with vectors also requires an extra vector, which is the camera plane vector. In a true 3D engine, there's also a camera plane, and there this plane is really a 3D plane so two vectors (u and v) are required to represent it. Raycasting happens in a 2D map however, so here the camera plane isn't really a plane, but a line, and is represented with a single vector. The camera plane should always be perpendicular on the direction vector. The camera plane represents the surface of the computer screen, while the direction vector is perpendicular on it and points inside the screen. The position of the player, which is a single point, is a point in front of the camera plane. A certain ray of a certain x-coordinate of the screen, is then the ray that starts at this player position, and goes through that position on the screen or thus the camera plane.

The image above represents such a 2D camera. The green spot is the position (vector "pos"). The black line, ending in the black spot, represents the direction vector (vector "dir"), so the position of the black dot is pos+dir. The blue line represents the full camera plane, the vector from the black dot to the right blue dot represents the vector "plane", so the position of the right blue point is pos+dir+plane, and the posistion of the left blue dot is pos+dir-plane (these are all vector additions).

The red lines in the image are a few rays. The direction of these rays is easily calculated out of the camera: it's the sum of the direction vector of the camear, and a part of the plane vector of the camera: for example the third red ray on the image, goes through the right part of the camera plane at the point about 1/3th of its length. So the direction of this ray is dir + plane*1/3. This ray direction is the vector rayDir, and the X and Y component of this vector are then used by the DDA algorithm.

The two outer lines, are the left and right border of the screen, and the angle between those two lines is called the Field Of Vision or FOV. The FOV is determinated by the ratio of the length of the direction vector, and the length of the plane. Here are a few examples of different FOV's:

If the direction vector and the camera plane vector have the same length, the FOV will be 90°:

If the direction vector is much longer than the camera plane, the FOV will be much smaller than 90°, and you'll have a very narrow vision. You'll see everything more detailed though and there will be less depth, so this is the same as zooming in:

If the direction vector is shorter than the camera plane, the FOV will be larger than 90° (180° is the maximum, if the direction vector is close to 0), and you'll have a much wider vision, like zooming out:

When the player rotates, the camera has to rotate, so both the direction vector and the plane vector have to be rotated. Then, the rays will all automaticly rotate as well.

If you don't know about vectors and matrices, try to find a tutorial with google, an appendix about those is planned for this tutorial later.

There's nothing that forbids you to use a camera plane that isn't perpendicular to the direction, but the result will look like a "skewed" world.

To start with the basics, we'll begin with an untextured raycaster. This example also includes an fps counter (frames per second), and input keys with collision detection to move and rotate.

The map of the world is a 2D array, where each value represents a square. If the value is 0, that square represents an empty, walkthroughable square, and if the value is higher than 0, it represents a wall with  a certain color or texture. The map declared here is very small, only 24 by 24 squares, and is defined directly in the code. For a real game, like Wolfenstein 3D, you use a bigger map and load it from a file instead. All the zero's in the grid are empty space, so basicly you see a very big room, with a wall around it (the values 1), a small room inside it (the values 2), a few pilars (the values 3), and a corridor with a room (the values 4). Note that this code isn't inside any function yet, put it before the main function starts.

The variables time and oldTime will be used to store the time of the current and the previous frame, the time difference between these two can be used to determinate how much you should move when a certain key is pressed (to move a constant speed no matter how long the calculation of the frames takes), and for the FPS counter.

The ray starts at the position of the player (posX, posY).

cameraX is the x-coordinate on the camera plane that the current x-coordinate of the screen represents, done this way so that the right side of the screen will get coordinate 1, the center of the screen gets coordinate 0, and the left side of the screen gets coordinate -1. Out of this, the direction of the ray can be calculated as was explained earlier: as the sum of the direction vector, and a part of the plane vector. This has to be done both for the x and y coordinate of the vector (since adding two vectors is adding their x-coordinates, and adding their y-coordinates).

mapX and mapY represent the current square of the map the ray is in. The ray position itself is a floating point number and contains both info about in which square of the map we are, and where in that square we are, but mapX and mapY are only the coordinates of that square.

sideDistX and sideDistY are initially the distance the ray has to travel from its start position to the first x-side and the first y-side. Later in the code they will be incremented while steps are taken.

deltaDistX and deltaDistY are the distance the ray has to travel to go from 1 x-side to the next x-side, or from 1 y-side to the next y-side. The following image shows the initial sideDistX, sideDistY and deltaDistX and deltaDistY:

When deriving deltaDistX geometrically you get, with Pythagoras, the formulas below. For the blue triangle (deltaDistX), one side has length 1 (as it is exactly one cell) and the other has length raydirY / raydirX because it is exaclty the amount of units the ray goes in the y-direction when taking 1 step in the X-direction. For the green triangle (deltaDistY), the formula is similar.

The DDA algorithm will always jump exactly one square each loop, either a square in the x-direction, or a square in the y-direction. If it has to go in the negative or positive x-direction, and the negative or positive y-direction will depend on the direction of the ray, and this fact will be stored in stepX and stepY. Those variables are always either -1 or +1.

Finally, hit is used to determinate whether or not the coming loop may be ended, and side will contain if an x-side or a y-side of a wall was hit. If an x-side was hit, side is set to 0, if an y-side was hit, side will be 1. By x-side and y-side, I mean the lines of the grid that are the borders between two squares.

If the ray direction has a negative x-component, stepX is -1, if the ray direciton has a positive x-component it's +1. If the x-component is 0, it doesn't matter what value stepX has since it'll then be unused.
The same goes for the y-component.

If the ray direction has a negative x-component, sideDistX is the distance from the ray starting position to the first side to the left, if the ray direciton has a positive x-component the first side to the right is used instead.
The same goes for the y-component, but now with the first side above or below the position.
For these values, the integer value mapX is used and the real position subtracted from it, and 1.0 is added in some of the cases depending if the side to the left or right, of the top or the bottom is used. Then you get the perpendicular distance to this side, so multiply it with deltaDistX or deltaDistY to get the real Euclidean distance.

sideDistX and sideDistY get incremented with deltaDistX with every jump in their direction, and mapX and mapY get incremented with stepX and stepY respectively.

When the ray has hit a wall, the loop ends, and then we'll know whether an x-side or y-side of a wall was hit in the variable "side", and what wall was hit with mapX and mapY. We won't know exactly where the wall was hit however, but that's not needed in this case because we won't use textured walls for now.

Depending on whether the ray hit an X side or Y side, the formula is computed using sideDistX, or sideDistY.

Then out of this lineHeight (which is thus the height of the vertical line that should be drawn), the start and end position of where we should really draw are calculated. The center of the wall should be at the center of the screen, and if these points lie outside the screen, they're capped to 0 or h-1.

The speed modifiers use frameTime, and a constant value, to determinate the speed of the moving and rotating of the input keys. Thanks to using the frameTime, we can make sure that the moving and rotating speed is independent of the processor speed.

If the up arrow is pressed, the player will move forward: add dirX to posX, and dirY to posY. This assumes that dirX and dirY are normalized vectors (their length is 1), but they were initially set like this, so it's ok. There's also a simple collision detection built in, namely if the new position will be inside a wall, you won't move. This collision detection can be improved however, for example by checking if a circle around the player won't go inside the wall instead of just a single point.

The same is done if you press the down arrow, but then the direction is subtracted instead.

To rotate, if the left or right arrow is pressed, both the direction vector and plane vector are rotated by using the formulas of multiplication with the rotation matrix (and over the angle rotSpeed).

Here's an example of what happens if the camera plane isn't perpendicular to the direction vector, the world appears skewed:

The core of the textured version of the raycaster is almost the same, only at the end some extra calculations need to be done for the textures, and a loop in the y-direction is required to go through every pixel to determinate which texel (texture pixel) of the texture should be used for it.

The vertical stripes can't be drawn with the vertical line command anymore, instead every pixel has to be drawn seperately. The best way is to use a 2D array as screen buffer this time, and copy it to the screen at once, that goes a lot faster than using pset.

Of course we now also need an extra array for the textures, and since the "drawbuffer" function works with single integer values for colors (instead of 3 separate bytes for R, G and B), the textures are stored in this format as well. Normally, you'd load the textures from a texture file, but for this simple example some dumb textures are generated instead.

The code is mostly the same as the previous example, the bold parts are new. Only new parts are explained.

The screenWidth and screenHeight are now defined in the beginning because we need the same value for the screen function, and to create the screen buffer. Also new are the texture width and height that are defined here. These are obviously the width and height in texels of the textures.

The world map is changed too, this is a more complex map with corridors and rooms to show the different textures. Again, the 0's are empty walkthrougable spaces, and each positive number corresponds to a different texture.

The value wallX represents the exact value where the wall was hit, not just the integer coordinates of the wall. This is required to know which x-coordinate of the texture we have to use. This is calculated by first calculating the exact x or y coordinate in the world, and then substracting the integer value of the wall off it. Note that even if it's called wallX, it's actually an y-coordinate of the wall if side==1, but it's always the x-coordinate of the texture.

Finally, texX is the x-coordinate of the texture, and this is calculated out of wallX.

The value of texY is calculated by increasing by a precomputed step size (which is possible because this is constant in the vertical stripe) for each pixel. The step size tells how much to increase in the texture coordinates (in floating point) for every pixel in vertical screen coordinates. It then needs to cast the floating point value to integer to select the actual texture pixel.

NOTE: The stepping being done here is affine texture mapping, which means we can interpolate linearly between two points rather than have to compute a different division for each pixel. This is not perspective correct in general, but for perfectly vertical walls (and also perfectly horizontal floors/ceilings) it is, so we can use it for raycasting.

The color of the pixel to be drawn is then simply gotten from texture[texNum][texX][texY], which is the correct texel of the correct texture.

Like the untextured raycaster, here too we'll make the color value darker if an y-side of the wall was hit, because that looks a little bit better (like there is a sort of lighting). However, because the color value doesn't exist out of a separate R, G and B value, but these 3 bytes sticked together in a single integer, a not so intuitive calculation is used.

The color is made darker by dividing R, G and B through 2. Dividing a decimal number through 10, can be done by removing the last digit (e.g. 300/10 is 30: the last zero is removed). Similarly, dividing a binary number through 2, which is what is done here, is the same as removing the last bit. This can be done by bitshifting it to the right with >>1. But, here we're bitshifting a 24-bit integer (actually 32-bit, but the first 8 bits aren't used). Because of this, the last bit of one byte will become the first bit of the next byte, and that screws up the color values! So after the bitshift, the first bit of every byte has to be set to zero, and that can be done by binary "AND-ing" the value with the binary value 011111110111111101111111, which is 8355711 in decimal. So the result of this is indeed a darker color.

Finally, the current buffer pixel is set to this color, and we move on to the next y.

Note: Usually images are stored by horizontal scanlines, but for a raycaster the textures are drawn as vertical stripes. Therefore, to optimally use the cache of the CPU and avoid page misses, it might be more efficient to store the textures in memory vertical stripe by vertical stripe, instead of per horizontal scanline. To do this, after generating the textures, swap their X and Y by (this code only works if texWidth and texHeight are the same):

Or just swap X and Y where the textures are generated, but in many cases after loading an image or getting a texture from other formats you'll have it in scanlines anyway and have to swap it this way.

Just replace the part of the code that generates the texture patterns with the following (and make sure those textures are in the correct path). You can download the textures here.

In the original Wolfenstein 3D, the colors of one side was also made darker than the color of the other side of a wall to create the shadow effect, but they used a seperate texture every time, a dark and a light one. Here however, only one texture is used for each wall and the line of code that divided R, G and B through 2 is what makes the y-sides darker.

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