问在3D中计算点到三角形距离的最快方法？

EN

    public void ClosestPointToShouldWork()
    {
        var r = new Random(0);
        double next() => r.NextDouble() * 5 - 1;
        var t = new Triangle(new Vector3(0,0,0), new Vector3(3.5,2,0), new Vector3(3,0.0,0));

        DrawTriangle(t);

        var hash = new Vector3( 0, 0, 0 );
        for (int i = 0; i < 800; i++)
        {
            var pt = new Vector3( next(), next(), 0 );
            var pc = t.ClosestPointTo( pt );
            hash += pc;

            DrawLine(pc,pt);
        }

        // Test the hash
        // If it doesn't match then eyeball the visualization
        // and see what has gone wrong

        hash.ShouldBeApproximately( new Vector3(1496.28118561104,618.196568578824,0),1e-5  );

    }

public class Triangle
{
    public Vector3 A => EdgeAb.A;
    public Vector3 B => EdgeBc.A;
    public Vector3 C => EdgeCa.A;

    public readonly Edge3 EdgeAb;
    public readonly Edge3 EdgeBc;
    public readonly Edge3 EdgeCa;

    public Triangle(Vector3 a, Vector3 b, Vector3 c)
    {
        EdgeAb = new Edge3( a, b );
        EdgeBc = new Edge3( b, c );
        EdgeCa = new Edge3( c, a );
        TriNorm = Vector3.Cross(a - b, a - c);
    }

    public Vector3[] Verticies => new[] {A, B, C};

    public readonly Vector3 TriNorm;

    private static readonly RangeDouble ZeroToOne = new RangeDouble(0,1);

    public Plane TriPlane => new Plane(A, TriNorm);

    // The below three could be pre-calculated to
    // trade off space vs time

    public Plane PlaneAb => new Plane(EdgeAb.A, Vector3.Cross(TriNorm, EdgeAb.Delta  ));
    public Plane PlaneBc => new Plane(EdgeBc.A, Vector3.Cross(TriNorm, EdgeBc.Delta  ));
    public Plane PlaneCa => new Plane(EdgeCa.A, Vector3.Cross(TriNorm, EdgeCa.Delta  ));

    public static readonly  RangeDouble Zero1 = new RangeDouble(0,1);

    public Vector3 ClosestPointTo(Vector3 p)
    {
        // Find the projection of the point onto the edge

        var uab = EdgeAb.Project( p );
        var uca = EdgeCa.Project( p );

        if (uca > 1 && uab < 0)
            return A;

        var ubc = EdgeBc.Project( p );

        if (uab > 1 && ubc < 0)
            return B;

        if (ubc > 1 && uca < 0)
            return C;

        if (ZeroToOne.Contains( uab ) && !PlaneAb.IsAbove( p ))
            return EdgeAb.PointAt( uab );

        if (ZeroToOne.Contains( ubc ) && !PlaneBc.IsAbove( p ))
            return EdgeBc.PointAt( ubc );

        if (ZeroToOne.Contains( uca ) && !PlaneCa.IsAbove( p ))
            return EdgeCa.PointAt( uca );

        // The closest point is in the triangle so 
        // project to the plane to find it
        return TriPlane.Project( p );

    }
}

public struct Edge3
{

    public readonly Vector3 A;
    public readonly Vector3 B;
    public readonly Vector3 Delta;

    public Edge3(Vector3 a, Vector3 b)
    {
        A = a;
        B = b;
        Delta = b -a;
    }

    public Vector3 PointAt(double t) => A + t * Delta;
    public double LengthSquared => Delta.LengthSquared();

    public double Project(Vector3 p) => (p - A).Dot( Delta ) / LengthSquared;

}

public struct Plane
{
    public Vector3 Point;
    public Vector3 Direction;

    public Plane(Vector3 point, Vector3 direction )
    {
            Point = point;
            Direction = direction;
    }

    public bool IsAbove(Vector3 q) => Direction.Dot(q - Point) > 0;

}