### Introduction

I recently got interested in 3D game engines for realistic images. I am currently reading through

a book describing a complete ray tracing algorithm [3]. However, since there is a lot of coding to do and a lot of complex geometry, I searched for a simpler side project. I then started to look into 2D

ray casting to generate 3D environments, as used in the Wolfenstein 3D engine. There are several

descriptions on how to build one [1,2]. However, the Wolfenstein black book [1] talks a lot about memory optimisations and execution speed. Lots of the chapters include assembly code and c snippets dealing with the old hardware. Implementing something like this is way easier, now.

Computers got way faster and programming languages way more convenient. So I decided to

build a clone in Scala. The result can be seen in the video above. The left side is the 3D view, and the right side shows a 2D map view. The red lines are the rays from the ray casting algorithm which represent what the player sees and the green line is the camera. The code can be found here [:github:]. If you want to move in the map use "WASD".

a book describing a complete ray tracing algorithm [3]. However, since there is a lot of coding to do and a lot of complex geometry, I searched for a simpler side project. I then started to look into 2D

ray casting to generate 3D environments, as used in the Wolfenstein 3D engine. There are several

descriptions on how to build one [1,2]. However, the Wolfenstein black book [1] talks a lot about memory optimisations and execution speed. Lots of the chapters include assembly code and c snippets dealing with the old hardware. Implementing something like this is way easier, now.

Computers got way faster and programming languages way more convenient. So I decided to

build a clone in Scala. The result can be seen in the video above. The left side is the 3D view, and the right side shows a 2D map view. The red lines are the rays from the ray casting algorithm which represent what the player sees and the green line is the camera. The code can be found here [:github:]. If you want to move in the map use "WASD".

#### Assumptions About The World

The world in Wolfenstein 3D is represented by a flat grid. Different heights in the map are
not possible. All objects stand on the same plane and have to be boxes. This assumption makes
a lot of things easier. In a 3D engine, the main task is to determine what is seen from a specific
view (the player's camera). The nice thing is that we do not have to determine what is visible in any possible direction, because we are moving on a plane, which restricts us to a 2D problem. The second
assumption of a world on a grid will simplify the calculation of ray to box intersection later.

#### Implementing The Player And Camera

The basic idea is to describe a player in 2D by it's position: $p = (p_x, p_y)$.
Furthermore, we save the position the player is looking at: $d = (d_x, d_y)$.
We also save the image plane: $i = (i_x, i_y)$ which is perpendicular to the player.

In the image above the image plane is marked in green, and the rays for collision are marked in green. In order to
calculate the direction for each ray, we first calculate the x coordinate in camera space as $c_x = 2 * x / w$ where $x$
is the row in the image and $w$ is the width of the image, then we compute the ray from the player that goes through that
position as:
- $r_{dx} = d_x + i_x * c_x$
- $r_{dy} = d_y + i_y * c_x$

#### Digital Differential Analysis Ray Casting

In order to find what wall we see at the image row $c_x$, we follow the ray until it hits a wall. We initialize the search by calculating the distance to the first time the ray hits the x-axis (in the image this is the segment $a$ to $b$) and the first time we hit the y axis (in the image this is the segment $a$ to $c$). From there we estimate the slope $\delta = (\delta_x, \delta_y)$ in x and y which tells us how far we have to travel to hit each axis again. We then travel in the closest next grid position and check if the position is a wall hit.

@tailrec private def traceRay( grid: Vector2D[Int], sideDist: Vector2D[Double], side: Int, deltaDist: Vector2D[Double], step: Vector2D[Int], hit: Boolean, ): Hit = if(hit) Hit(grid, side) else sideDist match { case Vector2D(x, y) if x < y => traceRay( Vector2D[Int](grid.x + step.x, grid.y), Vector2D[Double](sideDist.x + deltaDist.x, sideDist.y), 0, deltaDist, step, world.collids(grid.x + step.x, grid.y) ) case Vector2D(x, y) if x >= y => traceRay( Vector2D[Int](grid.x, grid.y + step.y), Vector2D[Double](sideDist.x, sideDist.y + deltaDist.y), 1, deltaDist, step, world.collids(grid.x, grid.y + step.y) ) }In the implementation we move through the grid bu the delta distance and the current side distance (distance traveled along the $x$ and $y$ component). We also save which side ($x$ or $y$) we hit the wall at. In the recursion we follow the closest next map square until we find a hit and then return the position of the hit $h = (h_x, h_y)$ as well as the side.

#### Rendering A Scene

So for each image row, we run the ray tracer until we hit a wall. We then compute the distance to the wall in order to calculate how high the wall appears in the image $w_d$ from the player position, the hit position, the step size and the direction:

private def projectedDistance( hit: Hit, pos: Vector2D[Double], step: Vector2D[Int], dir: Vector2D[Double], ): Double = if (hit.side == 0) (hit.pos.x - pos.x + (1 - step.x) / 2.0) / dir.x else (hit.pos.y - pos.y + (1 - step.y) / 2.0) / dir.yThe height of the wall is then simply $l_h = \frac{h}{w_d}$. We then compress the slice of the texture we hit to that height and paint it in the image at the row position we send the ray for.

### References

[1] GAME ENGINE BLACK BOOK: WOLFENSTEIN 3D, 2ND EDITION, 2018