计算距离A的给定距离处的AB线上的点

Question

我很疯狂地计算沿着给定线AB的点,在离A的给定距离处,这样我就可以"绘制"两个给定点之间的线.一开始听起来很简单,但我似乎无法做到这一点.更糟糕的是,我不明白我哪里出错了.几何(和一般的数学)不是我强大的套件.

我已经阅读了类似的问题,并在那里有答案.事实上,我直接从Mads Elvheim的回答中解除了我对CalculatePoint函数的当前实现:给定一个起点和终点,以及一个距离,计算沿线的一个点(加上后面的注释中的修正 - 如果我理解正确的话)因为我的独立尝试解决问题让我无处可去,除了一流的快递票务frusterpationland.

这是我的更新代码(请参阅EDIT备注帖子的底部):

using System;
using System.Drawing;
using System.Windows.Forms;

namespace DrawLines
{
    public class MainForm : Form
    {
        // =====================================================================
        // Here's the part I'm having trouble with. I don't really understand
        // how this is suposed to work, so I can't seem to get it right!
        // ---------------------------------------------------------------------

        // A "local indirector" - Just so I don't have go down and edit the 
        // actual call everytime this bluddy thing changes names.
        private Point CalculatePoint(Point a, Point b, int distance) {
            return CalculatePoint_ByAgentFire(a, b, distance);
        }

        #region CalculatePoint_ByAgentFire
        //AgentFire: Better approach (you can rename the struct if you need):
        struct Vector2
        {
            public readonly double X;
            public readonly double Y;
            public Vector2(double x, double y) {
                this.X = x;
                this.Y = y;
            }
            public static Vector2 operator -(Vector2 a, Vector2 b) {
                return new Vector2(b.X - a.X, b.Y - a.Y);
            }
            public static Vector2 operator *(Vector2 a, double d) {
                return new Vector2(a.X * d, a.Y * d);
            }
            public override string ToString() {
                return string.Format("[{0}, {1}]", X, Y);
            }
        }
        // For getting the midpoint you just need to do the (a - b) * d action:
        //static void Main(string[] args)
        //{
        //    Vector2 a = new Vector2(1, 1);
        //    Vector2 b = new Vector2(3, 1);
        //    float distance = 0.5f; // From 0.0 to 1.0.
        //    Vector2 c = (a - b) * distance;
        //    Console.WriteLine(c);
        //}
        private Point CalculatePoint_ByAgentFire(Point a, Point b, int distance) {
            var vA = new Vector2(a.X, a.Y);
            var vB = new Vector2(b.X, b.Y);
            double lengthOfHypotenuse = LengthOfHypotenuseAsDouble(a,b);
            double portionOfDistanceFromAtoB = distance / lengthOfHypotenuse;
            var vC = (vA - vB) * portionOfDistanceFromAtoB;
            Console.WriteLine("vC="+vC);
            return new Point((int)(vC.X+0.5), (int)(vC.Y+0.5));
        }
        // Returns the length of the hypotenuse rounded to an integer, using
        // Pythagoras' Theorem for right angle triangles: The length of the
        // hypotenuse equals the sum of the square of the other two sides.
        // Ergo: h = Sqrt(a*a + b*b)
        private double LengthOfHypotenuseAsDouble(Point a, Point b) {
            double aSq = Math.Pow(Math.Abs(a.X - b.X), 2); // horizontal length squared
            double bSq = Math.Pow(Math.Abs(b.Y - b.Y), 2); // vertical length  squared
            return Math.Sqrt(aSq + bSq); // length of the hypotenuse
        }

        #endregion

        //dbaseman: I thought something looked strange about the formula ... the question 
        //you linked was how to get the point at a distance after B, whereas you want the
        //distance after A. This should give you the right answer, the start point plus 
        //distance in the vector direction.
        //
        // Didn't work as per: http://s1264.photobucket.com/albums/jj496/corlettk/?action=view&current=DrawLinesAB-broken_zps069161e9.jpg
        //
        private Point CalculatePoint_ByDbaseman(Point a, Point b, int distance) {
            // a. calculate the vector from a to b:
            double vectorX = b.X - a.X;
            double vectorY = b.Y - a.Y;
            // b. calculate the length:
            double magnitude = Math.Sqrt(vectorX * vectorX + vectorY * vectorY);
            // c. normalize the vector to unit length:
            vectorX /= magnitude;
            vectorY /= magnitude;
            // d. calculate and Draw the new vector, which is x1y1 + vxvy * (mag + distance).
            return new Point(
                (int)((double)a.X + vectorX * distance)     // x = col
              , (int)((double)a.Y + vectorY * distance)     // y = row
            );
        }

        // MBo: Try to remove 'magnitude' term in the parentheses both for X and for Y expressions.
        //
        // Didn't work as per: http://s1264.photobucket.com/albums/jj496/corlettk/?action=view&current=DrawLinesAB-broken_zps069161e9.jpg
        //
        //private Point CalculatePoint_ByMBo(Point a, Point b, int distance) {
        //    // a. calculate the vector from a to b:
        //    double vectorX = b.X - a.X;
        //    double vectorY = b.Y - a.Y;
        //    // b. calculate the length:
        //    double magnitude = Math.Sqrt(vectorX * vectorX + vectorY * vectorY);
        //    // c. normalize the vector to unit length:
        //    vectorX /= magnitude;
        //    vectorY /= magnitude;
        //    // d. calculate and Draw the new vector, which is x1y1 + vxvy * (mag + distance).
        //    return new Point(
        //        (int)(  ((double)a.X + vectorX * distance)  +  0.5  )
        //      , (int)(  ((double)a.X + vectorX * distance)  +  0.5  )
        //    );
        //}

        // Didn't work
        //private Point CalculatePoint_ByUser1556110(Point a, Point b, int distance) {
        //    Double magnitude = Math.Sqrt(Math.Pow(b.Y - a.Y, 2) + Math.Pow(b.X - a.X, 2));
        //    return new Point(
        //        (int)(a.X + distance * (b.X - a.X) / magnitude + 0.5)
        //      , (int)(a.Y + distance * (b.Y - a.Y) / magnitude + 0.5)
        //    );
        //}

        // didn't work
        //private static Point CalculatePoint_ByCadairIdris(Point a, Point b, int distance) {
        //    // a. calculate the vector from a to b:
        //    double vectorX = b.X - a.X;
        //    double vectorY = b.Y - a.Y;
        //    // b. calculate the proportion of hypotenuse
        //    double factor = distance / Math.Sqrt(vectorX*vectorX + vectorY*vectorY);
        //    // c. factor the lengths
        //    vectorX *= factor;
        //    vectorY *= factor;
        //    // d. calculate and Draw the new vector,
        //    return new Point((int)(a.X + vectorX), (int)(a.Y + vectorY));
        //}

        // Returns a point along the line A-B at the given distance from A
        // based on Mads Elvheim's answer to:
        // https://stackoverflow.com/questions/1800138/given-a-start-and-end-point-and-a-distance-calculate-a-point-along-a-line
        private Point MyCalculatePoint(Point a, Point b, int distance) {
            // a. calculate the vector from o to g:
            double vectorX = b.X - a.X;
            double vectorY = b.Y - a.Y;
            // b. calculate the length:
            double magnitude = Math.Sqrt(vectorX * vectorX + vectorY * vectorY);
            // c. normalize the vector to unit length:
            vectorX /= magnitude;
            vectorY /= magnitude;
            // d. calculate and Draw the new vector, which is x1y1 + vxvy * (mag + distance).
            return new Point(
                (int)(((double)a.X + vectorX * (magnitude + distance)) + 0.5) // x = col
              , (int)(((double)a.Y + vectorY * (magnitude + distance)) + 0.5) // y = row
            );
        }

        // =====================================================================

        private const int CELL_SIZE = 4; // width and height of each "cell" in the bitmap.

        private readonly Bitmap _bitmap; // to draw on (displayed in picBox1).
        private readonly Graphics _graphics; // to draw with.

        // actual points on _theLineString are painted red.
        private static readonly SolidBrush _thePointBrush = new SolidBrush(Color.Red);
        // ... and are labeled in Red, Courier New, 12 point, Bold
        private static readonly SolidBrush _theLabelBrush = new SolidBrush(Color.Red);
        private static readonly Font _theLabelFont = new Font("Courier New", 12, FontStyle.Bold);

        // the interveening calculated cells on the lines between actaul points are painted Black.
        private static readonly SolidBrush _theLineBrush = new SolidBrush(Color.Black);

        // the points in my line-string.
        private static readonly Point[] _theLineString = new Point[] {
            //          x,   y
            new Point(170,  85), // A
            new Point( 85,  70), // B
            //new Point(209,  66), // C
            //new Point( 98, 120), // D
            //new Point(158,  19), // E
            //new Point(  2,  61), // F
            //new Point( 42, 177), // G
            //new Point(191, 146), // H
            //new Point( 25, 128), // I
            //new Point( 95,  24)  // J
        };

        public MainForm() {
            InitializeComponent();
            // initialise "the graphics system".
            _bitmap = new Bitmap(picBox1.Width, picBox1.Height);
            _graphics = Graphics.FromImage(_bitmap);
            picBox1.Image = _bitmap;
        }

        #region actual drawing on the Grpahics

        private void DrawCell(int x, int y, Brush brush) {
            _graphics.FillRectangle(
                brush
              , x * CELL_SIZE, y * CELL_SIZE    // x, y
              , CELL_SIZE, CELL_SIZE        // width, heigth
            );
        }

        private void DrawLabel(int x, int y, char c) {
            string s = c.ToString();
            _graphics.DrawString(
                s, _theLabelFont, _theLabelBrush
              , x * CELL_SIZE + 5   // x
              , y * CELL_SIZE - 8   // y
            );
        }

        // ... there should be no mention of _graphics or CELL_SIZE below here ...

        #endregion

        #region draw points on form load

        private void MainForm_Load(object sender, EventArgs e) {
            DrawPoints();
        }

        // draws and labels each point in _theLineString
        private void DrawPoints() {
            char c = 'A'; // label text, as a char so we can increment it for each point.
            foreach ( Point p in _theLineString ) {
                DrawCell(p.X, p.Y, _thePointBrush);
                DrawLabel(p.X, p.Y, c++);
            }
        }

        #endregion

        #region DrawLines on button click

        private void btnDrawLines_Click(object sender, EventArgs e) {
            DrawLinesBetweenPointsInTheString();
        }

        // Draws "the lines" between the points in _theLineString.
        private void DrawLinesBetweenPointsInTheString() {
            int n = _theLineString.Length - 1; // one less line-segment than points
            for ( int i = 0; i < n; ++i )
                Draw(_theLineString[i], _theLineString[i + 1]);
            picBox1.Invalidate(); // tell the graphics system that the picture box needs to be repainted.
        }

        // Draws all the cells along the line from Point "a" to Point "b".
        private void Draw(Point a, Point b) {
            int maxDistance = LengthOfHypotenuse(a, b);
            for ( int distance = 1; distance < maxDistance; ++distance ) {
                var point = CalculatePoint(a, b, distance);
                DrawCell(point.X, point.X, _theLineBrush);
            }
        }

        // Returns the length of the hypotenuse rounded to an integer, using
        // Pythagoras' Theorem for right angle triangles: The length of the
        // hypotenuse equals the sum of the square of the other two sides.
        // Ergo: h = Sqrt(a*a + b*b)
        private int LengthOfHypotenuse(Point a, Point b) {
            double aSq = Math.Pow(Math.Abs(a.X - b.X), 2); // horizontal length squared
            double bSq = Math.Pow(Math.Abs(b.Y - b.Y), 2); // vertical length  squared
            return (int)(Math.Sqrt(aSq + bSq) + 0.5); // length of the hypotenuse
        }

        #endregion

        #region Windows Form Designer generated code
        /// <summary>
        /// Required method for Designer support - do not modify
        /// the contents of this method with the code editor.
        /// </summary>
        private void InitializeComponent() {
            this.picBox1 = new System.Windows.Forms.PictureBox();
            this.btnDrawLines = new System.Windows.Forms.Button();
            ((System.ComponentModel.ISupportInitialize)(this.picBox1)).BeginInit();
            this.SuspendLayout();
            // 
            // picBox1
            // 
            this.picBox1.Dock = System.Windows.Forms.DockStyle.Fill;
            this.picBox1.Location = new System.Drawing.Point(0, 0);
            this.picBox1.Name = "picBox1";
            this.picBox1.Size = new System.Drawing.Size(1000, 719);
            this.picBox1.TabIndex = 0;
            this.picBox1.TabStop = false;
            // 
            // btnDrawLines
            // 
            this.btnDrawLines.Location = new System.Drawing.Point(23, 24);
            this.btnDrawLines.Name = "btnDrawLines";
            this.btnDrawLines.Size = new System.Drawing.Size(77, 23);
            this.btnDrawLines.TabIndex = 1;
            this.btnDrawLines.Text = "Draw Lines";
            this.btnDrawLines.UseVisualStyleBackColor = true;
            this.btnDrawLines.Click += new System.EventHandler(this.btnDrawLines_Click);
            // 
            // MainForm
            // 
            this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
            this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
            this.ClientSize = new System.Drawing.Size(1000, 719);
            this.Controls.Add(this.btnDrawLines);
            this.Controls.Add(this.picBox1);
            this.Location = new System.Drawing.Point(10, 10);
            this.MinimumSize = new System.Drawing.Size(1016, 755);
            this.Name = "MainForm";
            this.SizeGripStyle = System.Windows.Forms.SizeGripStyle.Hide;
            this.StartPosition = System.Windows.Forms.FormStartPosition.Manual;
            this.Text = "Draw Lines on a Matrix.";
            this.Load += new System.EventHandler(this.MainForm_Load);
            ((System.ComponentModel.ISupportInitialize)(this.picBox1)).EndInit();
            this.ResumeLayout(false);
        }

        private System.Windows.Forms.PictureBox picBox1;
        private System.Windows.Forms.Button btnDrawLines;
        #endregion
    }

}

对不起,如果它有点长,但这是从我的真实项目挖掘的SSCCE,这是一个A*最短路线算法的实现来运行MazeOfBolton ...即一个迷宫跑步者.

我实际想要做的是在迷宫(矩阵)中围绕两个给定点(原点和目标)预先计算"围栏"(即缓冲的MBR),使得"围栏"内的所有点都在给定的范围内距离"两点之间的直线"的距离,以便快速消除远离目标的数十万条可能的路径.

请注意,这个编程挑战在几年前已经结束,所以这里的"竞争性plagerism"没有问题.不,这不是功课,事实上我是一名专业的程序员......我只是在这里度过了我的舒适区,即使是相对简单的几何形状.叹.

所以...请任何人都可以给我任何指示,以帮助我正确地获得CalculatePoint函数:计算距离A的给定距离的AB线上的点？

提前感谢您的慷慨......即使在阅读这篇文章时也是如此.

干杯.基思.

---

编辑:我刚刚更新了发布的源代码,因为:

(1)我刚刚意识到它不是自包含的.我忘记了单独的MainForm.Designer .cs文件,我已将其附加到已发布代码的底部.

(2)最新版本包括我到目前为止所尝试的内容,并附有一张照片,链接到每张故障的图片......并且它们都是一样的.伊？WTF？

我想我的问题可能在其他地方,就像其他人以前错过的一些时髦的窗体设置,因为我忘了发布设计师生成的代码......除了everythingelse(在我的实际项目中)描绘了我期望它的确切位置,那么为什么计算出的点应该是不同的.我不知道!？!？!？我非常沮丧,而且我变得暴躁,所以我想我会留下这一天;-)

去表明我们经常低估了计算机做多少努力所做的事情......甚至只画一条简单的线条......它甚至不是曲线,更不用说一个大圆圈或一个横向的墨卡托或任何东西看上去......只是一条简单的布鲁迪线!？!？!？;-)

非常感谢所有发帖的人!

再次干杯.基思.

Answer 1

计算向量AB

首先定义从A点(1,-1)到B点(2,4)的矢量,从B中减去A.矢量将是Vab(1,5).

计算AB的长度

使用毕达哥拉斯定理来计算向量AB的长度.

|Vab| = SQRT(1²+5²)

长度为(圆形)5.1

计算单位矢量

将矢量除以其长度以获得单位矢量(长度为1的矢量).

V1(1/5.1,5/5.1) = V1(0.2, 0.98)

计算长度为4的向量

现在将V1乘以你想要的长度,例如4,得到Vt.

Vt(0.2*4,0.98*4) = Vt(0.8,3.92)

计算目标点

将向量Vt添加到A点以获得点T(目标).

T = A + Vt = T(1.8,2.92)

编辑:回答你的编辑

LengthOfHypotenuse方法看起来应该是这样的

• 修正了计算bSq时的错误
• 并删除了多余的Math.Abs​​调用,因为2的pow为始终为正
• 删除了0.5的添加,不知道为什么你需要那个

• 你应该至少使用一个浮点值作为返回值(双倍或小数也可以)

  //You should work with Vector2 class instead of Point and use their Length property
  private double LengthOfHypotenuse(Point a, Point b) {
      double aSq = Math.Pow(a.X - b.X, 2); // horizontal length squared
      double bSq = Math.Pow(a.Y - b.Y, 2); // vertical length  squared
      return Math.Sqrt(aSq + bSq); // length of the hypotenuse
  }
  

Draw(Point a,Point b)方法应如下所示:

• 更正了DrawCell()调用

  private void Draw(Point a, Point b) {
      double maxDistance = LengthOfHypotenuse(a, b);
      for (int distance = 0; distance < maxDistance; ++distance) {
          var point = CalculatePoint(new Vector2(a), new Vector2(b), distance);
          DrawCell(point.X, point.Y, _theLineBrush);
      }
  }
  

您的CalculatePoint(Point a,Point b,int distance)方法:

• 将一些计算移到Vector2类中

  private Point CalculatePoint(Vector2 a, Vector2 b, int distance) {
      Vector2 vectorAB = a - b;
  
      return a + vectorAB.UnitVector * distance;
  }
  

我已经为你扩展了Vector类,以添加缺少的运算符(积分给AgentFire)

    //AgentFire: Better approach (you can rename the struct if you need):
    struct Vector2 {
        public readonly double X;
        public readonly double Y;
        public Vector2(Point p) : this(p.X,p.Y) { 
        }

        public Vector2(double x, double y) {
            this.X = x;
            this.Y = y;
        }
        public static Vector2 operator -(Vector2 a, Vector2 b) {
            return new Vector2(b.X - a.X, b.Y - a.Y);
        }
        public static Vector2 operator +(Vector2 a, Vector2 b) {
            return new Vector2(b.X + a.X, b.Y + a.Y);
        }
        public static Vector2 operator *(Vector2 a, double d) {
            return new Vector2(a.X * d, a.Y * d);
        }
        public static Vector2 operator /(Vector2 a, double d) {
            return new Vector2(a.X / d, a.Y / d);
        }

        public static implicit operator Point(Vector2 a) {
            return new Point((int)a.X, (int)a.Y);
        }

        public Vector2 UnitVector {
            get { return this / Length; }
        }

        public double Length {
            get {
                double aSq = Math.Pow(X, 2);
                double bSq = Math.Pow(Y, 2);
                return Math.Sqrt(aSq + bSq);
            }
        }

        public override string ToString() {
            return string.Format("[{0}, {1}]", X, Y);
        }
    }

Answer 2

更好的方法(如果需要,可以重命名结构):

struct Vector2
{
    public readonly float X;
    public readonly float Y;

    public Vector2(float x, float y)
    {
        this.X = x;
        this.Y = y;
    }

    public static Vector2 operator -(Vector2 a, Vector2 b)
    {
        return new Vector2(b.X - a.X, b.Y - a.Y);
    }
    public static Vector2 operator +(Vector2 a, Vector2 b)
    {
        return new Vector2(a.X + b.X, a.Y + b.Y);
    }
    public static Vector2 operator *(Vector2 a, float d)
    {
        return new Vector2(a.X * d, a.Y * d);
    }

    public override string ToString()
    {
        return string.Format("[{0}, {1}]", X, Y);
    }
}

要获得中点,您只需要执行(a - b) * d + a操作:

class Program
{
    static void Main(string[] args)
    {
        Vector2 a = new Vector2(1, 1);
        Vector2 b = new Vector2(3, 1);
        float distance = 0.5f; // From 0.0 to 1.0.
        Vector2 c = (a - b) * distance + a;
        Console.WriteLine(c);
    }
}

这会给你一点:

output:\> [2, 1]

之后您需要的是从0.0到1.0并绘制像素.for(the distance; up toone; d += step)

Answer 3

尝试删除 X 和 Y 表达式括号中的“幅度”项：

(int)(  ((double)a.X + vectorX * distance)  +  0.5  )