空间中两个3D磁盘之间的碰撞检测

我想开发一种算法来执行3D空间中两个3D磁盘之间的碰撞检测。有没有可能建议一种分析方法，这样我就可以开发这个程序

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谢谢，

所以每个磁盘都位于一个无限平面的顶部，每个磁盘都有一个法向量和到原点的距离。在两个平面相交的地方，它们形成一条线

这条线将中心c1和c2投影为线p1和p2上的两个点，垂直距离为h1和h2。投影点之间的距离为l

如果磁盘接触，公共线将包含接触点。如果它们不接触，或相互渗透，则该点pC是同时距离两个磁盘中心最近的线上的点

要找到从p1到pC的距离t1，请使用类似的三角形来求解（t1）/h1=（l-t1）/h2

从距离t1和公共线方向，计算点pC，然后如果两个圆盘到其中心的距离等于或小于其半径，则两个圆盘相交

圆盘 上面的数学是基于

更新

• 将厚度添加到磁盘属性，并使用它检查磁盘是否包含点
• 链接到我很久以前写的MATALB代码。可能会遗漏一些东西，但这可能会给你一个良好的起点。关于磁盘没有什么特别的，只是具有齐次坐标的几何体的框架

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请详细描述在任何时候以何种形式提供的信息？@jAlex非常感谢您提供如此详细的答案。我所说的3D磁盘是指，如果磁盘有厚度会怎样。这使得碰撞检测非常困难。我开发了一个代码，用很多球体填充3D磁盘，并执行球体之间的碰撞检测，这需要很长的时间才能完成。非常感谢您对这一点的思考。您可以很容易地修改我的答案，使其包含厚度。修改

Disk.Containts（）

函数以检查

DistanceTo（point）@jAlex再次感谢您的回答。我将在MATLAB中重写这段代码，因为我在其中生成了随机数。请在我的答案中查看我的更新，并链接到
MATLAB
code以了解一般几何知识。
t_1 = h_1*ell/(h_1+h_2)

intersect = ( distance(c_1-p_C)<=R_1 ) && ( distance(c_2-p_C)<=R_2 )

    static void Main(string[] args)
    {
        var disk_1 = new Disk(Point.Origin, Vector3.UnitZ, 1);
        var disk_2 = new Disk(
            Point.Origin + Vector3.UnitY,
            Vector3.UnitY,
            0.6f);
        if (Disk.Intersect(disk_1, disk_2))
        {
            Debug.WriteLine($"Crash");
        }
    }

using System.Numerics;
public class Disk : Plane
{
    public Disk(Point center, Vector3 normal, float radius, float thickness)
        : base(center, normal)
    {
        this.Center=center;
        this.Radius=radius;
        this.Thickness = thickness;
    }

    public Point Center { get; }
    public float Radius { get; }
    public float Thickness { get; }

    public bool Containts(Point point)
    {
        if (DistanceTo(point)<= Thickness/2)
        {
            var d = (point - Center).Length();
            return d<=Radius;
        }
        return false;
    }

    public static bool Intersect(Disk disk_1, Disk disk_2)
    {
        var commonLine = Line.Meet(disk_1, disk_2);
        var p_1 = commonLine.Project(disk_1.Center);
        var p_2 = commonLine.Project(disk_2.Center);
        var ell = p_1.DistanceTo(p_2);
        var h_1 = commonLine.DistanceTo(disk_1.Center);
        var h_2 = commonLine.DistanceTo(disk_2.Center);
        if (Math.Abs(h_1+h_2)>1e-8)
        {
            var t_1 = h_1*ell/(h_1+h_2);
            var contact = p_1 - commonLine.Direction * t_1;
            return disk_1.Containts(contact) && disk_2.Containts(contact);
        }
        else
        {
            return ell <= disk_1.Radius + disk_2.Radius;
        }
    }
}

using System.Numerics;
public class Point 
{
    public Vector3 Vector { get; }
    public float Scalar { get; }

    public Point(Vector3 position)
        : this(position, 1) { }
    public Point(Vector3 vector, float scalar)
    {
        this.Vector = vector;
        this.Scalar=scalar;
    }
    public Point(Vector4 coordinates)
        : this(new Vector3(coordinates.X, coordinates.Y, coordinates.Z), coordinates.W)
    { }
    public Point(Plane plane)
        : this(-plane.Scalar*plane.Vector, plane.Vector.LengthSquared())
    { }
    public Point(Line line)
        : this(Vector3.Cross(line.Vector, line.Moment), line.Vector.LengthSquared())
    { }

    public static implicit operator Point(Vector3 position)
        => new Point(position, 1);
    public static implicit operator Point(Vector4 coordinates)
        => new Point(coordinates);

    public static readonly Point Origin = new Point(Vector3.Zero, 1);

    public static Point Meet(Plane plane, Line line)
    {
        return new Point(
            Vector3.Cross(line.Moment, plane.Vector)+plane.Scalar*line.Vector,
            -Vector3.Dot(plane.Vector, line.Vector));
    }
    public static Point Meet(Plane plane_1, Plane plane_2, Plane plane_3)
    {
        return Meet(plane_1, Line.Meet(plane_2, plane_3));
    }

    public float Magnitude { get => Math.Abs(Scalar); }
    public Vector3 Position { get => Vector/Scalar; }
    public float DistanceTo(Point point)
        => (Scalar*point.Vector - point.Scalar*Vector).Length()/(Scalar*point.Scalar);
    public float DistanceTo(Plane plane)
        => (Vector3.Dot(plane.Vector, Vector) + Scalar*plane.Scalar)/(Scalar*plane.Vector.Length());
    public float DistanceTo(Line line)
        => (Vector3.Cross(line.Vector, Vector) + Scalar * line.Moment).Length()/(Scalar*line.Vector.Length());
    public static Point operator +(Point point, Vector3 delta)
        => new Point(point.Vector + point.Scalar*delta, point.Scalar);
    public static Vector3 operator -(Point point, Point @base)
        => point.Position - @base.Position;

}
public class Plane 
{
    public Vector3 Vector { get; }
    public float Scalar { get; }

    public Plane(Vector3 vector, float scalar)
    {
        this.Vector=vector;
        this.Scalar=scalar;
    }
    public Plane(Vector4 coordinates)
        : this(new Vector3(coordinates.X, coordinates.Y, coordinates.Z), coordinates.W)
    { }
    public Plane(Point point, Vector3 normal)
        : this(normal, -Vector3.Dot(point.Position, normal))
    { }

    public Plane(Point point)
        : this(-point.Scalar*point.Vector, point.Vector.LengthSquared())
    { }
    public Plane(Line line)
        : this(Vector3.Cross(line.Moment, line.Vector), line.Moment.LengthSquared())
    { }
    public static implicit operator Plane(Vector4 coordinates)
        => new Plane(coordinates);

    public static Plane Join(Point point, Line line)
    {
        return new Plane(
            Vector3.Cross(line.Vector, point.Position) + line.Moment,
            -Vector3.Dot(point.Position, line.Moment));
    }
    public static Plane Join(Point point_1, Point point_2, Point point_3)
    {
        return Join(point_1, Line.Join(point_2, point_3));
    }
    public float Magnitude { get => Vector.Length(); }
    public Vector3 Normal { get => Vector3.Normalize(Vector); }
    public float Offset { get => -Scalar/Magnitude; }
    public Vector3 Position
    {
        get => Normal*Offset;
    }
    public float DistanceTo(Point point)
        => point.DistanceTo(this);

    public Point Project(Point point)
    {
        float t = Vector3.Dot(Normal, point.Position)-Offset;
        return point.Position - Normal*t;
    }
}
public class Line 
{
    public Vector3 Vector { get; }
    public Vector3 Moment { get; }

    public Line(Vector3 vector, Vector3 moment)
    {
        this.Vector=vector;
        this.Moment=moment;
    }

    public static Line Ray(Point point, Vector3 direction)
    {
        return new Line(direction,
            Vector3.Cross(point.Position, direction));
    }

    public static Line Join(Point point_1, Point point_2)
    {
        return new Line(
            point_2.Position-point_1.Position,
            Vector3.Cross(point_1.Position, point_2.Position));
    }
    public static Line Meet(Plane plane_1, Plane plane_2)
    {
        return new Line(
            Vector3.Cross(plane_1.Vector, plane_2.Vector),
            plane_2.Vector*plane_1.Scalar-plane_1.Vector*plane_2.Scalar);
    }
    public Point Along(float travel)
        => Position.Position + Direction * travel;
    public float Magnitude { get => Vector.Length(); }
    public Vector3 Direction { get => Vector3.Normalize(Vector); }
    public Point Position
    {
        get => new Point(this);
    }
    public float DistanceTo(Point point)
        => point.DistanceTo(this);
    public float DistanceTo(Line line)
        => (Vector3.Dot(Vector, line.Moment) + Vector3.Dot(line.Vector, Moment))/Vector3.Cross(Vector, line.Vector).Length();

    public Point Project(Point point)
    {
        return Along(Vector3.Dot(Direction, point.Position-Position.Position));
    }
}