C# 对于点位置的判断

1.判断点是否在一群点内部

要判断一个点是否在一个由多个点围成的多边形内部（例如一圈点），可以使用射线法（Ray Casting Algorithm）来实现。以下是一个简单的 C# 实现示例

using System;

public class Point
{
    public double X { get; set; }
    public double Y { get; set; }

    public Point(double x, double y)
    {
        X = x;
        Y = y;
    }
}

public class Program
{
    public static bool IsPointInPolygon(Point testPoint, Point[] polygon)
    {
        bool inside = false;
        int count = polygon.Length;
        for (int i = 0, j = count - 1; i < count; j = i++)
        {
            if (((polygon[i].Y > testPoint.Y) != (polygon[j].Y > testPoint.Y)) &&
                (testPoint.X < (polygon[j].X - polygon[i].X) * (testPoint.Y - polygon[i].Y) / (polygon[j].Y - polygon[i].Y) + polygon[i].X))
            {
                inside = !inside;
            }
        }
        return inside;
    }

    public static void Main()
    {
        Point[] polygon = new Point[]
        {
            new Point(0, 0),
            new Point(0, 4),
            new Point(4, 4),
            new Point(4, 0)
        };

        Point testPoint = new Point(2, 2);
        bool isInside = IsPointInPolygon(testPoint, polygon);

        Console.WriteLine($"Point ({testPoint.X}, {testPoint.Y}) is inside the polygon: {isInside}");
    }
}

 2.判断点在直线左侧还是右侧

要在C#中判断一个点在一条直线的左侧还是右侧，可以使用点与直线方程的方法。具体来说，对于直线上的两个点A和B，以及要测试的点P，可以通过计算点P相对于直线AB的位置来确定其是否在直线的左侧还是右侧。

以下是一个简单的C#示例：

using System;

public class Point
{
    public double X { get; set; }
    public double Y { get; set; }

    public Point(double x, double y)
    {
        X = x;
        Y = y;
    }
}

public class Line
{
    public Point A { get; set; }
    public Point B { get; set; }

    public Line(Point a, Point b)
    {
        A = a;
        B = b;
    }

    // 计算点P相对于直线AB的位置
    public double PointRelativeToLine(Point P)
    {
        return (B.X - A.X) * (P.Y - A.Y) - (B.Y - A.Y) * (P.X - A.X);
    }
}

public class Program
{
    public static void Main()
    {
        Point pointA = new Point(1, 1);
        Point pointB = new Point(4, 5);
        Line lineAB = new Line(pointA, pointB);

        Point testPoint = new Point(2, 3);
        double position = lineAB.PointRelativeToLine(testPoint);

        if (position > 0)
        {
            Console.WriteLine("Point is on the left side of the line.");
        }
        else if (position < 0)
        {
            Console.WriteLine("Point is on the right side of the line.");
        }
        else
        {
            Console.WriteLine("Point is on the line.");
        }
    }
}

3.判断两条直线的交点 

要判断两条直线的交点，可以使用直线的参数方程来求解。两条直线的参数方程可以表示为：

直线1： (x = x_1 + t_1 \cdot (x_2 - x_1)) 和 (y = y_1 + t_1 \cdot (y_2 - y_1))

直线2： (x = x_3 + t_2 \cdot (x_4 - x_3)) 和 (y = y_3 + t_2 \cdot (y_4 - y_3))

要求两条直线的交点，需要解方程组，即求解 (t_1) 和 (t_2)，然后代入其中一个直线的参数方程中即可求得交点的坐标。

以下是一个C#示例：

using System;

public class Point
{
    public double X { get; set; }
    public double Y { get; set; }

    public Point(double x, double y)
    {
        X = x;
        Y = y;
    }
}

public class Line
{
    public Point A { get; set; }
    public Point B { get; set; }

    public Line(Point a, Point b)
    {
        A = a;
        B = b;
    }

    // 计算两条直线的交点
    public Point IntersectionPoint(Line otherLine)
    {
        double x1 = A.X;
        double y1 = A.Y;
        double x2 = B.X;
        double y2 = B.Y;
        
        double x3 = otherLine.A.X;
        double y3 = otherLine.A.Y;
        double x4 = otherLine.B.X;
        double y4 = otherLine.B.Y;

        double denominator = (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4);

        if (denominator == 0)
        {
            throw new InvalidOperationException("Lines are parallel. No intersection point exists.");
        }

        double t1 = ((x1 - x3) * (y3 - y4) - (y1 - y3) * (x3 - x4)) / denominator;
        double t2 = -((x1 - x2) * (y1 - y3) - (y1 - y2) * (x1 - x3)) / denominator;

        double intersectionX = x1 + t1 * (x2 - x1);
        double intersectionY = y1 + t1 * (y2 - y1);

        return new Point(intersectionX, intersectionY);
    }
}

public class Program
{
    public static void Main()
    {
        Point pointA1 = new Point(1, 1);
        Point pointB1 = new Point(4, 5);
        Line line1 = new Line(pointA1, pointB1);

        Point pointA2 = new Point(2, 3);
        Point pointB2 = new Point(6, 1);
        Line line2 = new Line(pointA2, pointB2);

        try
        {
            Point intersectionPoint = line1.IntersectionPoint(line2);
            Console.WriteLine($"Intersection Point: ({intersectionPoint.X}, {intersectionPoint.Y})");
        }
        catch (InvalidOperationException ex)
        {
            Console.WriteLine(ex.Message);
        }
    }
}