游戏中的数学 08：曲线参数为什么不等于路程

// $WORK_DIR/.tmp/game-math-lab/Program.cs · Chapter08
Vector2 p0 = new(0, 0);
Vector2 p1 = new(2, 6);
Vector2 p2 = new(8, -2);
Vector2 p3 = new(10, 4);

var table = BuildArcTable(p0, p1, p2, p3, 4000);
var uniformT = Enumerable.Range(0, 101)
    .Select(i => Bezier(p0, p1, p2, p3, i / 100f))
    .ToArray();
var uniformS = Enumerable.Range(0, 101)
    .Select(i => Bezier(p0, p1, p2, p3,
        (float)ArcToT(table, i / 100.0)))
    .ToArray();

// Program.cs · Bezier
static Vector2 Bezier(
    Vector2 p0, Vector2 p1,
    Vector2 p2, Vector2 p3, float t)
{
    var u = 1 - t;
    return u * u * u * p0
         + 3 * u * u * t * p1
         + 3 * u * t * t * p2
         + t * t * t * p3;
}

// Program.cs · BuildArcTable
static List<Arc> BuildArcTable(
    Vector2 p0, Vector2 p1,
    Vector2 p2, Vector2 p3, int segments)
{
    var result = new List<Arc> { new(0, 0) };
    var previous = p0;
    double sum = 0;

    for (var i = 1; i <= segments; i++)
    {
        var t = i / (double)segments;
        var point = Bezier(p0, p1, p2, p3, (float)t);
        sum += Vector2.Distance(previous, point);
        result.Add(new Arc(t, sum));
        previous = point;
    }
    return result;
}

// Program.cs · ArcToT
static double ArcToT(IReadOnlyList<Arc> table, double fraction)
{
    if (table.Count < 2 || table[^1].S < 1e-9)
        return 0;
    var target = table[^1].S * Math.Clamp(fraction, 0, 1);
    var lo = 0;
    var hi = table.Count - 1;

    while (lo + 1 < hi)
    {
        var mid = (lo + hi) / 2;
        if (table[mid].S < target) lo = mid;
        else hi = mid;
    }

    var a = table[lo];
    var b = table[hi];
    var span = b.S - a.S;
    if (span < 1e-9) return a.T;
    var f = (target - a.S) / span;
    return a.T + (b.T - a.T) * f;
}

// Program.cs · Chapter08：统计与断言
var dt = SegmentLengths(uniformT);
var ds = SegmentLengths(uniformS);
var degenerate = BuildArcTable(
    Vector2.One, Vector2.One,
    Vector2.One, Vector2.One, 8);

Check(
    Std(ds) < Std(dt) * .05,
    "arc-length lookup equalizes segment lengths");
Check(ArcToT(degenerate, .5) == 0,
    "zero-length curve maps to start parameter");

WriteData(8, new
{
    samples = 101,
    lookup_samples = 4001,
    parameter_mean = dt.Average(),
    parameter_std = Std(dt),
    arclength_mean = ds.Average(),
    arclength_std = Std(ds),
    total_length = table[^1].S,
    degenerate_length = degenerate[^1].S,
    degenerate_t = 0
});

$ cd $WORK_DIR
$ dotnet run --project .tmp/game-math-lab/GameMathLab.csproj -- --out .tmp/game-math-lab/out

CH08 PASS segments=100 t_std=0.019488m arc_std=0.000005m length=12.433628m
TOTAL pass=15 fail=0 skip=0 chapters=15 elapsed_ms=112.8
exit_code=0

{
  "samples": 101,
  "lookup_samples": 4001,
  "parameter_mean": 0.12433057606220245,
  "parameter_std": 0.019488149363260005,
  "arclength_mean": 0.12433023974299431,
  "arclength_std": 0.000005271764799195839,
  "total_length": 12.433628018479794,
  "degenerate_length": 0,
  "degenerate_t": 0
}