游戏中的数学 12：射线求交与首个有效命中

// $WORK_DIR/.tmp/game-math-lab/Program.cs
// GameMathLab.RaySphere
static bool RaySphere(
    Vector3 o, Vector3 d, Vector3 c,
    double radius, out Hit hit)
{
    var oc = o - c;
    var b = Vector3.Dot(oc, d);
    var cc = Vector3.Dot(oc, oc) - (float)(radius * radius);
    var disc = b * b - cc;
    if (disc < 0) { hit = default; return false; }

    var s = Math.Sqrt(disc);
    var near = -b - s;
    var far = -b + s;
    var t = near >= 0 ? near : far;
    if (t < 0) { hit = default; return false; }
    var point = o + d * (float)t;
    hit = new(t, point, Vector3.Normalize(point - c));
    return true;
}

// Program.cs · Chapter12 固定种子样本
var random = new Random(1212);
int hit = 0, miss = 0;
double maxResidual = 0;

for (var i = 0; i < 10_000; i++)
{
    var origin = new Vector3(
        random.NextSingle() * 8 - 4,
        random.NextSingle() * 8 - 4,
        -8);
    var target = new Vector3(
        random.NextSingle() * 8 - 4,
        random.NextSingle() * 8 - 4,
        0);
    var dir = Vector3.Normalize(target - origin);

    if (RaySphere(origin, dir, Vector3.Zero, 2, out var sphereHit))
        hit++;
    else
        miss++;
}

// Program.cs · Chapter12 命中点回代
if (RaySphere(origin, dir, Vector3.Zero, 2, out var sphereHit))
{
    hit++;
    var residual = Math.Abs(
        Vector3.Distance(sphereHit.Point, Vector3.Zero) - 2);
    maxResidual = Math.Max(maxResidual, residual);

    residualByAngle.Add(new(
        Math.Acos(Math.Clamp(
            Vector3.Dot(dir, Vector3.UnitZ), -1, 1))
            * 180 / Math.PI,
        residual));
}

Check(maxResidual < 1e-4,
    "ray-sphere hit satisfies surface equation");

// Program.cs · RayAabb 的单轴核心
if (Math.Abs(direction) < 1e-8)
{
    if (origin < min || origin > max)
        return false; // 平行且位于 slab 外
    continue;         // 平行且位于 slab 内，不做除法
}
var t1 = (min - origin) / direction;
var t2 = (max - origin) / direction;
if (t1 > t2) (t1, t2) = (t2, t1);
enter = Math.Max(enter, t1);
exit = Math.Min(exit, t2);

// Program.cs · RayPlane
var length = normal.Length();
if (!float.IsFinite(length) || length < 1e-8 ||
    !double.IsFinite(offset))
{
    hit = default;
    return false;
}

var unitNormal = normal / length;
var unitOffset = offset / length;
var denom = Vector3.Dot(d, unitNormal);
if (Math.Abs(denom) < 1e-8)
{
    hit = default;
    return false;
}

var t = (unitOffset - Vector3.Dot(o, unitNormal)) / denom;
if (!double.IsFinite(t) || t < 0)
{
    hit = default;
    return false;
}

hit = new(t, o + d * (float)t,
    denom < 0 ? unitNormal : -unitNormal);
return true;

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

CH12 PASS rays=10000 hit=2316 miss=7684 parallel=1 plane_t=2.000 aabb_t=2.000 inside_exit=1.000 max_residual=6.199E-006m
TOTAL pass=15 fail=0 skip=0 chapters=15 elapsed_ms=148.7
exit_code=0

{
  "rays": 10000,
  "hit": 2316,
  "miss": 7684,
  "parallel_cases": 1,
  "scaled_plane": {
    "t": 2,
    "point": {"x": 0, "y": 2, "z": 0},
    "normal": {"x": 0, "y": -1, "z": 0}
  },
  "aabb_entry": {
    "t": 2,
    "point": {"x": -1, "y": 0, "z": 0},
    "normal": {"x": -1, "y": 0, "z": 0}
  },
  "aabb_inside_exit": {
    "t": 1,
    "point": {"x": 0, "y": 1, "z": 0},
    "normal": {"x": 0, "y": 1, "z": 0}
  },
  "max_surface_residual_m": 6.198883056640625E-06
}