raycastlib/raycastlib.h

#ifndef RAYCASTLIB_H
#define RAYCASTLIB_H

/**
  raycastlib - Small C header-only raycasting library for embedded and low
  performance computers, such as Arduino. Only uses integer math and stdint
  standard library.

  author: Miloslav "drummyfish" Ciz
  license: CC0

  - Game field's bottom left corner is at [0,0].
  - X axis goes right.
  - Y axis goes up.
  - Each game square is UNITS_PER_SQUARE * UNITS_PER_SQUARE.
  - Angles are in Units, 0 means pointing right (x+) and positively rotates
    clockwise, a full angle has UNITS_PER_SQUARE Units.
  */

#include <stdint.h>

#ifndef RAYCAST_TINY

#define UNITS_PER_SQUARE 1024 ///< No. of Units in a side of a spatial square.
typedef int32_t Unit;   /**< Smallest spatial unit, there is UNITS_PER_SQUARE
                             units in a square's length. This effectively
                             serves the purpose of a fixed-point arithmetic. */
typedef int32_t int_maybe32_t;
typedef uint32_t uint_maybe32_t;

#else

#define UNITS_PER_SQUARE 64
typedef int16_t Unit;
typedef int16_t int_maybe32_t;
typedef uint16_t uint_maybe32_t;

#endif

#define VERTICAL_FOV (UNITS_PER_SQUARE / 2)

#define logVector2D(v)\
  printf("[%d,%d]\n",v.x,v.y);

#define logRay(r){\
  printf("ray:\n");\
  printf("  start: ");\
  logVector2D(r.start);\
  printf("  dir: ");\
  logVector2D(r.direction);}\

#define logHitResult(h){\
  printf("hit:\n");\
  printf("  sqaure: ");\
  logVector2D(h.square);\
  printf("  pos: ");\
  logVector2D(h.position);\
  printf("  dist: %d\n", h.distance);\
  printf("  texcoord: %d\n", h.textureCoord);}\

#define logPixelInfo(p){\
  printf("pixel:\n");\
  printf("  position: ");\
  logVector2D(p.position);\
  printf("  depth: %d\n", p.depth);\
  printf("  wall: %d\n", p.isWall);\
  printf("  textCoordY: %d\n", p.textureCoordY);\
  printf("  hit: ");\
  logHitResult(p.hit);\
  }\

/// Position in 2D space.
typedef struct
{
  Unit y;
  Unit x;
} Vector2D;

typedef struct
{
  Vector2D start;
  Vector2D direction;
} Ray;

typedef struct
{
  Vector2D square;     ///< Collided square coordinates.
  Vector2D position;   ///< Exact collision position in Units.
  Unit     distance;   /**< Euclidean distance to the hit position, or -1 if
                            no collision happened. */
  Unit     textureCoord; /// Normalized (0 to UNITS_PER_SQUARE - 1) tex coord.
  uint8_t  direction;    ///< Direction of hit.
} HitResult;

// TODO: things like FOV could be constants to make them precomp. and faster?

typedef struct
{
  Vector2D position;
  Unit direction;
  Vector2D resolution;
  Unit fovAngle;
  Unit height;
} Camera;

typedef struct
{
  Vector2D position;  ///< On-screen position.
  int8_t isWall;      ///< Whether the pixel is a wall or a floor(/ceiling).
  Unit depth;         ///< Corrected depth.
  HitResult hit;      ///< Corresponding ray hit.
  Unit textureCoordY; ///< Normalized (0 to UNITS_PER_SQUARE - 1) tex coord.
} PixelInfo;

typedef struct
{
  uint16_t maxHits;
  uint16_t maxSteps;
} RayConstraints;

/**
 Function used to retrieve the cells of the rendered scene. It should return
 a "type" of given square as an integer (e.g. square height) - between squares
 that return different numbers there is considered to be a collision.
*/ 
typedef Unit (*ArrayFunction)(int16_t x, int16_t y);

typedef void (*PixelFunction)(PixelInfo info);

typedef void
  (*ColumnFunction)(HitResult *hits, uint16_t hitCount, uint16_t x, Ray ray);

/**
 Simple-interface function to cast a single ray.

 @return          The first collision result.
 */
HitResult castRay(Ray ray, ArrayFunction arrayFunc);

/**
 Maps a single point in the world to the screen (2D position + depth).
 */
PixelInfo mapToScreen(Vector2D worldPosition, Unit height, Camera camera);

/**
 Casts a single ray and returns a list of collisions.
 */
void castRayMultiHit(Ray ray, ArrayFunction arrayFunc, HitResult *hitResults,
  uint16_t *hitResultsLen, RayConstraints constraints);

Vector2D angleToDirection(Unit angle);
Unit cosInt(Unit input);
Unit sinInt(Unit input);

/// Normalizes given vector to have UNITS_PER_SQUARE length.
Vector2D normalize(Vector2D v);

/// Computes a cos of an angle between two vectors.
Unit vectorsAngleCos(Vector2D v1, Vector2D v2);

uint16_t sqrtInt(uint_maybe32_t value);
Unit dist(Vector2D p1, Vector2D p2);
Unit len(Vector2D v);

/**
 Converts an angle in whole degrees to an angle in Units that this library
 uses.
 */   
Unit degreesToUnitsAngle(int16_t degrees);

///< Computes the change in size of an object due to perspective.
Unit perspectiveScale(Unit originalSize, Unit distance);

/**
 Casts rays for given camera view and for each hit calls a user provided
 function.
 */
void castRaysMultiHit(Camera cam, ArrayFunction arrayFunc,
  ColumnFunction columnFunc, RayConstraints constraints);

void render(Camera cam, ArrayFunction arrayFunc, PixelFunction pixelFunc,
  RayConstraints constraints);

//=============================================================================
// privates

#ifdef RAYCASTLIB_PROFILE
  // function call counters for profiling
  uint32_t profile_sqrtInt = 0;
  uint32_t profile_clamp = 0;
  uint32_t profile_cosInt = 0;
  uint32_t profile_angleToDirection = 0;
  uint32_t profile_dist = 0;
  uint32_t profile_len = 0;
  uint32_t profile_pointIsLeftOfRay = 0;
  uint32_t profile_castRaySquare = 0;
  uint32_t profile_castRayMultiHit = 0; 
  uint32_t profile_castRay = 0;
  uint32_t profile_absVal = 0;
  uint32_t profile_normalize = 0;
  uint32_t profile_vectorsAngleCos = 0;
  uint32_t profile_perspectiveScale = 0;
  uint32_t profile_wrap = 0;
  uint32_t profile_divRoundDown = 0;
  #define profileCall(c) profile_##c += 1

  #define printProfile() {\
    printf("profile:\n");\
    printf("  sqrtInt: %d\n",profile_sqrtInt);\
    printf("  clamp: %d\n",profile_clamp);\
    printf("  cosInt: %d\n",profile_cosInt);\
    printf("  angleToDirection: %d\n",profile_angleToDirection);\
    printf("  dist: %d\n",profile_dist);\
    printf("  len: %d\n",profile_len);\
    printf("  pointIsLeftOfRay: %d\n",profile_pointIsLeftOfRay);\
    printf("  castRaySquare: %d\n",profile_castRaySquare);\
    printf("  castRayMultiHit : %d\n",profile_castRayMultiHit);\
    printf("  castRay: %d\n",profile_castRay);\
    printf("  normalize: %d\n",profile_normalize);\
    printf("  vectorsAngleCos: %d\n",profile_vectorsAngleCos);\
    printf("  absVal: %d\n",profile_absVal);\
    printf("  perspectiveScale: %d\n",profile_perspectiveScale);\
    printf("  wrap: %d\n",profile_wrap);\
    printf("  divRoundDown: %d\n",profile_divRoundDown); }
#else
  #define profileCall(c)
#endif

Unit clamp(Unit value, Unit valueMin, Unit valueMax)
{
  profileCall(clamp);

  if (value >= valueMin)
  {
    if (value <= valueMax)
      return value;
    else
      return valueMax;
  }
  else
    return valueMin;
}

inline Unit absVal(Unit value)
{
  profileCall(absVal);

  return value >= 0 ? value: -1 * value;
}

/// Like mod, but behaves differently for negative values.
inline Unit wrap(Unit value, Unit mod)
{
  profileCall(wrap);

  return value >= 0 ? (value % mod) : (mod + (value % mod) - 1);
}

/// Performs division, rounding down, NOT towards zero.
inline Unit divRoundDown(Unit value, Unit divisor)
{
  profileCall(divRoundDown);

  return value / divisor - ((value >= 0) ? 0 : 1);
}

// Bhaskara's cosine approximation formula
#define trigHelper(x) (((Unit) UNITS_PER_SQUARE) *\
  (UNITS_PER_SQUARE / 2 * UNITS_PER_SQUARE / 2 - 4 * (x) * (x)) /\
  (UNITS_PER_SQUARE / 2 * UNITS_PER_SQUARE / 2 + (x) * (x)))

Unit cosInt(Unit input)
{
  profileCall(cosInt);

  // TODO: could be optimized with LUT

  input = wrap(input,UNITS_PER_SQUARE);

  if (input < UNITS_PER_SQUARE / 4)
    return trigHelper(input);
  else if (input < UNITS_PER_SQUARE / 2)
    return -1 * trigHelper(UNITS_PER_SQUARE / 2 - input);
  else if (input < 3 * UNITS_PER_SQUARE / 4)
    return -1 * trigHelper(input - UNITS_PER_SQUARE / 2);
  else
    return trigHelper(UNITS_PER_SQUARE - input);
}

#undef trigHelper

Unit sinInt(Unit input)
{
  return cosInt(input - UNITS_PER_SQUARE / 4);
}

Vector2D angleToDirection(Unit angle)
{
  profileCall(angleToDirection);

  Vector2D result;

  result.x = cosInt(angle);
  result.y = -1 * sinInt(angle);

  return result;
}

uint16_t sqrtInt(uint_maybe32_t value)
{
  profileCall(sqrtInt);

  uint_maybe32_t result = 0;
  uint_maybe32_t a = value;

#ifdef RAYCAST_TINY
  uint_maybe32_t b = 1u << 14;
#else
  uint_maybe32_t b = 1u << 30;
#endif

  while (b > a)
    b >>= 2;

  while (b != 0)
  {
    if (a >= result + b)
    {
      a -= result + b;
      result = result +  2 * b;
    }

    b >>= 2;
    result >>= 1;
  }

  return result;
}

Unit dist(Vector2D p1, Vector2D p2)
{
  profileCall(dist);

  int_maybe32_t dx = p2.x - p1.x;
  int_maybe32_t dy = p2.y - p1.y;

  dx = dx * dx;
  dy = dy * dy;

  return sqrtInt((uint_maybe32_t) (dx + dy));
}

Unit len(Vector2D v)
{
  profileCall(len);

  v.x *= v.x;
  v.y *= v.y;

  return sqrtInt(((uint_maybe32_t) v.x) + ((uint_maybe32_t) v.y));
}

int8_t pointIsLeftOfRay(Vector2D point, Ray ray)
{
  profileCall(pointIsLeftOfRay);

  Unit dX    = point.x - ray.start.x;
  Unit dY    = point.y - ray.start.y;
  return (ray.direction.x * dY - ray.direction.y * dX) > 0;
         // ^ Z component of cross-product
}

/**
  Casts a ray within a single square, to collide with the square borders.  
 */
void castRaySquare(Ray localRay, Vector2D *nextCellOff, Vector2D *collOff)
{
  profileCall(castRaySquare);

  nextCellOff->x = 0;
  nextCellOff->y = 0;

  Ray criticalLine = localRay;

  #define helper(c1,c2,n)\
    {\
      nextCellOff->c1 = n;\
      collOff->c1 = criticalLine.start.c1 - localRay.start.c1;\
      collOff->c2 = \
        (((int_maybe32_t) collOff->c1) * localRay.direction.c2) /\
        ((localRay.direction.c1 == 0) ? 1 : localRay.direction.c1);\
    }

  #define helper2(n1,n2,c)\
    if (pointIsLeftOfRay(localRay.start,criticalLine) == c)\
      helper(y,x,n1)\
    else\
      helper(x,y,n2)
      
  if (localRay.direction.x > 0)
  {
    criticalLine.start.x = UNITS_PER_SQUARE - 1;

    if (localRay.direction.y > 0)
    {
      // top right
      criticalLine.start.y = UNITS_PER_SQUARE - 1;
      helper2(1,1,1)
    }
    else
    {
      // bottom right
      criticalLine.start.y = 0;
      helper2(-1,1,0)
    }
  }
  else
  {
    criticalLine.start.x = 0;

    if (localRay.direction.y > 0)
    {
      // top left
      criticalLine.start.y = UNITS_PER_SQUARE - 1;
      helper2(1,-1,0)
    }
    else
    {
      // bottom left
      criticalLine.start.y = 0;
      helper2(-1,-1,1)
    }
  }

  #undef helper2
  #undef helper

  collOff->x += nextCellOff->x;
  collOff->y += nextCellOff->y;
}

void castRayMultiHit(Ray ray, ArrayFunction arrayFunc, HitResult *hitResults,
  uint16_t *hitResultsLen, RayConstraints constraints)
{
  profileCall(castRayMultiHit);

  Vector2D initialPos = ray.start;
  Vector2D currentPos = ray.start;

  Vector2D currentSquare;

  currentSquare.x = divRoundDown(ray.start.x, UNITS_PER_SQUARE);
  currentSquare.y = divRoundDown(ray.start.y,UNITS_PER_SQUARE);

  *hitResultsLen = 0;

  int16_t squareType = arrayFunc(currentSquare.x,currentSquare.y);

  Vector2D no, co; // next cell offset, collision offset

  no.x = 0;        // just to supress a warning
  no.y = 0;
  co.x = 0;
  co.y = 0;

  for (uint16_t i = 0; i < constraints.maxSteps; ++i)
  {
    int16_t currentType = arrayFunc(currentSquare.x,currentSquare.y);

    if (currentType != squareType)
    {
      // collision

      HitResult h;

      h.position = currentPos;
      h.square   = currentSquare;
      h.distance = dist(initialPos,currentPos);

      if (no.y > 0)
      {
        h.direction = 0;
        h.textureCoord = co.x;
      }
      else if (no.x > 0)
      {
        h.direction = 1;
        h.textureCoord = co.y;
      }
      else if (no.y < 0)
      {
        h.direction = 2;
        h.textureCoord = co.x;
      }
      else
      {
        h.direction = 3;
        h.textureCoord = co.y;
      }

      hitResults[*hitResultsLen] = h;

      *hitResultsLen += 1;

      squareType = currentType;

      if (*hitResultsLen >= constraints.maxHits)
        break;
    }

    ray.start.x = wrap(currentPos.x,UNITS_PER_SQUARE);
    ray.start.y = wrap(currentPos.y,UNITS_PER_SQUARE);

    castRaySquare(ray,&no,&co);

    currentSquare.x += no.x;
    currentSquare.y += no.y;

    // offset into the next cell
    currentPos.x += co.x;
    currentPos.y += co.y;
  }
}

HitResult castRay(Ray ray, ArrayFunction arrayFunc)
{
  profileCall(castRay);

  HitResult result;
  uint16_t  len;
  RayConstraints c;

  c.maxSteps = 1000;
  c.maxHits = 1;

  castRayMultiHit(ray,arrayFunc,&result,&len,c);

  if (len == 0)
    result.distance = -1;

  return result;
}

void castRaysMultiHit(Camera cam, ArrayFunction arrayFunc,
  ColumnFunction columnFunc, RayConstraints constraints)
{
  uint16_t fovHalf = cam.fovAngle / 2;

  Vector2D dir1 = angleToDirection(cam.direction - fovHalf);
  Vector2D dir2 = angleToDirection(cam.direction + fovHalf);

  Unit dX = dir2.x - dir1.x;
  Unit dY = dir2.y - dir1.y;

  HitResult hits[constraints.maxHits];
  uint16_t hitCount;

  Ray r;
  r.start = cam.position;

  for (uint16_t i = 0; i < cam.resolution.x; ++i)
  {
    r.direction.x = dir1.x + (dX * i) / cam.resolution.x;
    r.direction.y = dir1.y + (dY * i) / cam.resolution.x;

    castRayMultiHit(r,arrayFunc,hits,&hitCount,constraints);

    columnFunc(hits,hitCount,i,r);
  }
}

PixelFunction _pixelFunction = 0;
ArrayFunction _arrayFunction = 0;
Camera _camera;
Unit _floorDepthStep = 0; 
Unit _startHeight = 0;
int_maybe32_t _camResYLimit = 0;
Unit _middleRow = 0;

void _columnFunction(HitResult *hits, uint16_t hitCount, uint16_t x, Ray ray)
{
  int_maybe32_t y = _camResYLimit; // screen y (for floor), will only go up
  int_maybe32_t y2 = 0;            // screen y (for ceil), will only fo down

  Unit worldZPrev = _startHeight;
  Unit worldZPrevCeil =
    UNITS_PER_SQUARE * 5 - _startHeight - 2 * _camera.height;

  PixelInfo p;
  p.position.x = x;

  #define VERTICAL_DEPTH_MULTIPLY 2

  //  we'll be simulatenously drawing the floor and the ceiling now  
  for (uint_maybe32_t j = 0; j < hitCount; ++j)
  {
    HitResult hit = hits[j];

    /* FIXME/TODO: The adjusted (=orthogonal, camera-space) distance could
       possibly be computed more efficiently by not computing Euclidean
       distance at all, but rather compute the distance of the collision
       point from the projection plane (line). */

    Unit dist = // adjusted distance
      (hit.distance * vectorsAngleCos(angleToDirection(_camera.direction),
      ray.direction)) / UNITS_PER_SQUARE;

    dist = dist == 0 ? 1 : dist; // prevent division by zero

    Unit wallHeight = _arrayFunction(hit.square.x,hit.square.y);

    Unit worldZ2 = wallHeight - _camera.height;

    Unit worldZ2Ceil = (UNITS_PER_SQUARE * 5 - wallHeight) - _camera.height;

    int_maybe32_t z1Screen = _middleRow - perspectiveScale(
      (worldZPrev * _camera.resolution.y) / UNITS_PER_SQUARE,dist);

    int_maybe32_t z1ScreenNoClamp = z1Screen;

    z1Screen = clamp(z1Screen,0,_camResYLimit);

    int_maybe32_t z1ScreenCeil = _middleRow - perspectiveScale(
      (worldZPrevCeil * _camera.resolution.y) / UNITS_PER_SQUARE,dist);

    int_maybe32_t z1ScreenCeilNoClamp = z1ScreenCeil;

    z1ScreenCeil = clamp(z1ScreenCeil,0,_camResYLimit);

    int_maybe32_t z2Screen = _middleRow - perspectiveScale(
      (worldZ2 * _camera.resolution.y) / UNITS_PER_SQUARE,dist);

    int_maybe32_t wallScreenHeightNoClamp = z2Screen - z1ScreenNoClamp;

    z2Screen = clamp(z2Screen,0,_camResYLimit);

    int_maybe32_t z2ScreenCeil = _middleRow - perspectiveScale(
      (worldZ2Ceil * _camera.resolution.y) / UNITS_PER_SQUARE,dist);

    int_maybe32_t wallScreenHeightCeilNoClamp = z2ScreenCeil - z1ScreenCeilNoClamp;

    z2ScreenCeil = clamp(z2ScreenCeil,0,_camResYLimit);

    int_maybe32_t zTop = z1Screen < z2Screen ? z1Screen : z2Screen;

    int_maybe32_t zBottomCeil = z1ScreenCeil > z2ScreenCeil ?
      z1ScreenCeil : z2ScreenCeil;

    if (zTop <= zBottomCeil)
      zBottomCeil = zTop; // walls on ceiling and floor met

    // draw floor until wall

    p.isWall = 0;

    Unit floorCameraDiff = absVal(worldZPrev) * VERTICAL_DEPTH_MULTIPLY;

    for (int_maybe32_t i = y; i > z1Screen; --i)
    {
      p.position.y = i;
      p.depth = (_camera.resolution.y - i) * _floorDepthStep + floorCameraDiff;
      _pixelFunction(p);  
    }

    // draw ceiling until wall

    Unit ceilCameraDiff = absVal(worldZPrevCeil) * VERTICAL_DEPTH_MULTIPLY;

    for (int_maybe32_t i = y2; i < z1ScreenCeil; ++i)
    {
      p.position.y = i;
      p.depth = i * _floorDepthStep + ceilCameraDiff;
      _pixelFunction(p);  
    }

    // draw wall

    p.isWall = 1;
    p.depth = dist;

    for (int_maybe32_t i = z1Screen < y ? z1Screen : y; i > z2Screen; --i)
    {
      p.position.y = i;
      p.hit = hit;
      p.textureCoordY = ((i - z1ScreenNoClamp) * UNITS_PER_SQUARE) / wallScreenHeightNoClamp;
      _pixelFunction(p);
    }

    // draw ceiling wall

    for (int_maybe32_t i = z1ScreenCeil > y2 ? z1ScreenCeil : y2;
      i < z2ScreenCeil; ++i)
    {
      p.position.y = i;
      p.hit = hit;
      p.textureCoordY = ((i - z1ScreenCeilNoClamp) * UNITS_PER_SQUARE) / wallScreenHeightCeilNoClamp;
      _pixelFunction(p);
    }

    y = y > zTop ? zTop : y;
    worldZPrev = worldZ2;

    y2 = y2 < zBottomCeil ? zBottomCeil : y2;

    worldZPrevCeil = worldZ2Ceil;

    if (y <= y2)
      break; // walls on ceiling and floor met
  }

  // draw floor until horizon

  p.isWall = 0;

  Unit floorCameraDiff = absVal(worldZPrev) * VERTICAL_DEPTH_MULTIPLY;
  Unit horizon = y2 < _middleRow ? _middleRow : y2;

  for (int_maybe32_t i = y; i >= horizon; --i)
  {
    p.position.y = i;
    p.depth = (_camera.resolution.y - i) * _floorDepthStep + floorCameraDiff;

    _pixelFunction(p);
  }

  // draw ceiling until horizon

  Unit ceilCameraDiff = absVal(worldZPrevCeil) * VERTICAL_DEPTH_MULTIPLY;
  horizon = y > _middleRow ? _middleRow : y;

  for (int_maybe32_t i = y2; i < horizon; ++i)
  {
    p.position.y = i;
    p.depth = i * _floorDepthStep + ceilCameraDiff;

    _pixelFunction(p);
  }

  #undef VERTICAL_DEPTH_MULTIPLY
}

void render(Camera cam, ArrayFunction arrayFunc, PixelFunction pixelFunc,
  RayConstraints constraints)
{
  _pixelFunction = pixelFunc;
  _arrayFunction = arrayFunc;
  _camera = cam;
  _camResYLimit = cam.resolution.y - 1;
  _middleRow = cam.resolution.y / 2;

  _startHeight = arrayFunc(
    divRoundDown(cam.position.x,UNITS_PER_SQUARE),
    divRoundDown(cam.position.y,UNITS_PER_SQUARE)) -1 * cam.height;

  // TODO
  _floorDepthStep = (12 * UNITS_PER_SQUARE) / cam.resolution.y; 

  castRaysMultiHit(cam,arrayFunc,_columnFunction,constraints);
}

Vector2D normalize(Vector2D v)
{
  profileCall(normalize);

  Vector2D result;

  Unit l = len(v);

  l = l != 0 ? l : 1;

  result.x = (v.x * UNITS_PER_SQUARE) / l;
  result.y = (v.y * UNITS_PER_SQUARE) / l;

  return result;
}

Unit vectorsAngleCos(Vector2D v1, Vector2D v2)
{
  profileCall(vectorsAngleCos);

  v1 = normalize(v1);
  v2 = normalize(v2);

  return (v1.x * v2.x + v1.y * v2.y) / UNITS_PER_SQUARE;
}

PixelInfo mapToScreen(Vector2D worldPosition, Unit height, Camera camera)
{
  // TODO: precompute some stuff that's constant in the frame

  PixelInfo result;

  Unit d = dist(worldPosition,camera.position);

  Vector2D toPoint;

  toPoint.x = worldPosition.x - camera.position.x;
  toPoint.y = worldPosition.y - camera.position.y;

  Vector2D cameraDir = angleToDirection(camera.direction);

  result.depth = // adjusted distance
    (d * vectorsAngleCos(cameraDir,toPoint)) / UNITS_PER_SQUARE;

  result.position.y = camera.resolution.y / 2 -
    (camera.resolution.y *
     perspectiveScale(height - camera.height,result.depth)) / UNITS_PER_SQUARE;

  Unit middleColumn = camera.resolution.x / 2;

  Unit a = sqrtInt(d * d - result.depth * result.depth);

  Ray r;
  r.start = camera.position;
  r.direction = cameraDir;

  if (!pointIsLeftOfRay(worldPosition,r))
    a *= -1;

  Unit alpha = camera.fovAngle / 2;

  Unit cos = cosInt(alpha);

  Unit b =
    (result.depth * sinInt(alpha)) / (cos == 0 ? 1 : cos); // sin/cos = tan

  result.position.x = (a * middleColumn) / b;
  result.position.x = middleColumn - result.position.x;

  return result;
}

Unit degreesToUnitsAngle(int16_t degrees)
{
  return (degrees * UNITS_PER_SQUARE) / 360;
}

Unit perspectiveScale(Unit originalSize, Unit distance)
{
  profileCall(perspectiveScale);

  return distance != 0 ?
   (originalSize * UNITS_PER_SQUARE) /
      ((VERTICAL_FOV * 2 * distance) / UNITS_PER_SQUARE)
   : 0;
}

#endif