贝塞尔曲线细分：中点落在弦上，仍可能漏掉整个弯曲

B(t) = (1-t)^3 P0 + 3(1-t)^2 t P1 + 3(1-t)t^2 P2 + t^3 P3
x(t) = 3t
y(t) = 6t(1-t)(1-2t),  0 <= t <= 1
B(0.5) = (1.5, 0)
B(0.25) = (0.75, 0.5625)

public readonly record struct Point(double X, double Y)
{
    public static Point operator +(Point a, Point b) => new(a.X + b.X, a.Y + b.Y);
    public static Point operator -(Point a, Point b) => new(a.X - b.X, a.Y - b.Y);
    public static Point operator *(Point a, double s) => new(a.X * s, a.Y * s);
    public double Length => Math.Sqrt(X * X + Y * Y);
}
public readonly record struct Cubic(Point A, Point B, Point C, Point D);
public readonly record struct Leaf(double T0, double T1, Point A, Point D, double Bound);
public sealed record Tessellation(Leaf[] Leaves, bool Complete);

public static class Bezier
{
    public static double Distance(Point p, Point a, Point b)
    {
        Point v = b - a, w = p - a;
        double vv = v.X * v.X + v.Y * v.Y;
        double t = vv == 0 ? 0 : Math.Clamp((w.X * v.X + w.Y * v.Y) / vv, 0, 1);
        return (p - (a + v * t)).Length;
    }
    public static Tessellation Flatten(Cubic curve, double tolerance, int maxDepth)
    {
        foreach (Point p in new[] { curve.A, curve.B, curve.C, curve.D })
            if (!double.IsFinite(p.X) || !double.IsFinite(p.Y) ||
                Math.Abs(p.X) > 1e4 || Math.Abs(p.Y) > 1e4)
                throw new ArgumentOutOfRangeException(nameof(curve));
        if (!double.IsFinite(tolerance) || tolerance < 1e-6 || tolerance > 1e4)
            throw new ArgumentOutOfRangeException(nameof(tolerance));
        if (maxDepth < 0 || maxDepth > 12)
            throw new ArgumentOutOfRangeException(nameof(maxDepth));
        var leaves = new List<Leaf>();
        bool complete = Visit(curve, 0, 1, tolerance, maxDepth, leaves);
        return new(leaves.ToArray(), complete);
    }
    private static bool Visit(Cubic c, double t0, double t1, double tolerance,
        int depth, List<Leaf> leaves)
    {
        double bound = Math.Max(Distance(c.B, c.A, c.D), Distance(c.C, c.A, c.D));
        if (bound <= tolerance || depth == 0)
        {
            leaves.Add(new(t0, t1, c.A, c.D, bound));
            return bound <= tolerance;
        }
        // de Casteljau 二分，两个子段共享同一个中点。
        Point ab = (c.A + c.B) * .5, bc = (c.B + c.C) * .5, cd = (c.C + c.D) * .5;
        Point abc = (ab + bc) * .5, bcd = (bc + cd) * .5;
        Point middle = (abc + bcd) * .5;
        double tm = (t0 + t1) * .5;
        bool left = Visit(new(c.A, ab, abc, middle), t0, tm, tolerance, depth - 1, leaves);
        bool right = Visit(new(middle, bcd, cd, c.D), tm, t1, tolerance, depth - 1, leaves);
        return left && right;
    }
}

Require(Bezier.Distance(Evaluate(S, .5), S.A, S.D) == 0);
Require(ReferenceDistance(Evaluate(S, .25), S.A, S.D) == .5625);
Require(Bezier.Flatten(S, .05, 12).Leaves.Length > 1);

dotnet build "$LAB_DIR/BezierLab.csproj" -c Release
dotnet run --project "$LAB_DIR/BezierLab.csproj" -c Release --no-build

build exit=0 warnings=0 errors=0 elapsed=00:00:09.10
run exit=0 passed=14 failed=0 skipped=0
assertion_duration_ms=100.629
sampled_checks=99385 observed_bound_excess=0
midpoint_error=0 quarter_error=0.5625
normal_segments=12 max_bound=0.03404591941892772
normal_sampled_deviation=0.023051471641495985
depth_1_complete=false