游戏中的数学 14：世界点怎样变成屏幕点

// $WORK_DIR/.tmp/game-math-lab/Program.cs
// GameMathLab.Project
static Vector2 Project(
    Vector3 p, double width, double height,
    double fov, double aspect,
    double near = .1, double far = 100)
{
    var depth = -p.Z;
    if (depth < near || depth > far)
        return new(float.NaN, float.NaN);

    var tan = Math.Tan(DegToRad(fov / 2));
    var nx = p.X / (depth * tan * aspect);
    var ny = p.Y / (depth * tan);
    return new(
        (float)((nx + 1) * .5 * width),
        (float)((1 - ny) * .5 * height));
}

// Program.cs · GameMathLab.Unproject
static Vector3 Unproject(
    Vector2 s, double width, double height,
    double fov, double aspect)
{
    var nx = s.X / width * 2 - 1;
    var ny = 1 - s.Y / height * 2;
    var tan = Math.Tan(DegToRad(fov / 2));

    return Vector3.Normalize(new(
        (float)(nx * tan * aspect),
        (float)(ny * tan),
        -1));
}

// Program.cs · Chapter14 回代主链
var screen = Project(p, width, height, fov, aspect);
var ray = Unproject(screen, width, height, fov, aspect);

var along = Vector3.Dot(p, ray);
var closest = ray * along;
var rayError = Vector3.Distance(p, closest);
var screen2 = Project(closest, width, height, fov, aspect);

maxRay = Math.Max(maxRay, rayError);
maxPixel = Math.Max(
    maxPixel,
    Vector2.Distance(screen, screen2));

if (i % 100 == 0)
    depthErrors.Add(new(depth, rayError));

// Program.cs · Chapter14 输入与断言
const double width = 1920, height = 1080;
const double fov = 60, aspect = width / height;
const double near = .1, far = 100;
var cameraPosition = new Vector3(3, -2, 5);
var view = Matrix4x4.CreateTranslation(-cameraPosition);
var random = new Random(1414);

for (var i = 0; i < 10_000; i++)
{
    var depth = .2 + random.NextDouble() * 90;
    var halfY = depth * Math.Tan(DegToRad(fov / 2));
    var cameraPoint = new Vector3(
        (float)((random.NextDouble() * 1.8 - .9) * halfY * aspect),
        (float)((random.NextDouble() * 1.8 - .9) * halfY),
        (float)-depth);
    var worldPoint = cameraPoint + cameraPosition;
    var p = Vector3.Transform(worldPoint, view);
    // 执行上一段回代主链
}

Check(maxPixel < 1e-3,
    "project-unproject ray roundtrip stays subpixel");
Check(double.IsNaN(Project(new Vector3(0, 0, 1),
        width, height, fov, aspect, near, far).X),
    "behind-camera point is rejected");
Check(double.IsNaN(Project(new Vector3(0, 0, -.05f),
        width, height, fov, aspect, near, far).X),
    "near-clipped point is rejected");
Check(double.IsNaN(Project(new Vector3(0, 0, -101),
        width, height, fov, aspect, near, far).X),
    "far-clipped point is rejected");

cd "$WORK_DIR"
/usr/bin/time -p dotnet run \
  --project .tmp/game-math-lab/GameMathLab.csproj \
  -- --out .tmp/game-math-lab/out

CH14 PASS samples=10000 max_pixel_error=1.373E-004px max_ray_error=2.480E-005m behind=rejected near=rejected far=rejected
TOTAL pass=15 fail=0 skip=0 chapters=15 elapsed_ms=112.8
exit_code=0

{
  "samples": 10000,
  "width_px": 1920,
  "height_px": 1080,
  "fov_deg": 60,
  "near_m": 0.1,
  "far_m": 100,
  "camera_position": { "x": 3, "y": -2, "z": 5 },
  "max_pixel_error": 0.0001373291015625,
  "max_point_to_ray_error_m": 2.47955322265625E-05,
  "behind_camera": "rejected",
  "near_clip": "rejected",
  "far_clip": "rejected"
}