处理布冯针头模拟

Question

嘿，伙计们，我们今天开始在我的计算机课上编写布冯针头模拟的代码。我的老师给我们看了下面的代码，但无法解释为什么模拟不能得到精确的近似。噪声很大，近似值在3.6到2.8之间变化很大。我认为有些东西是有缺陷的，但我只是看不清哪一部分。另外，我也不明白程序的最后一部分(如何绘制图表上的点)。18岁是从哪里来的？

如果你有一些空闲时间，我会感激的答案，因为我正在努力计算的时候。事先谢谢你的回答。

//set up and define all the variables that we will use
int numTrials = 0;
int intersects = 0;
int lineWidth = 30;
int needleWidth = 0.5*lineWidth;
int boardHeight;
float piApprox;
final float PI = 3.14159265359;

//Defines our initial enviroment properties and creates our grid
void setup(){
  size(600,800);
  background(255);
  stroke(0);
  boardHeight = int (height - 200);
  for (int i = 0; i<=width; i+=lineWidth){ 
    line(i,0,i,boardHeight);
  }
}

//Continously executes the same command
void draw(){
  //generates a random x and y co-ordinate. This becomes the first point
    int a = round(random(width));
  int b = round(random(boardHeight));
  float theta = random(PI);
  //Generates a random x and y co-ordinate which is one needlelength away from first point. This becomes the second point
    int c = round(needleWidth*cos(theta)+a);
  int d = round(needleWidth*sin(theta)+b);
  numTrials++;

  //checks for intersections
  boolean touching = false;
  for (int i = 0; i<=width; i+=lineWidth){
    if ((min(a,c) <= i) && (max(a,c) > i)){
      intersects++;
      touching = true;
    }
  }
  //changes colour of line
    if (touching){
    stroke(0,50,155);
  }
  else{
    stroke(0,155,0);
  }
  line(a,b,c,d);
  //Calculates PI and then calls upon the GUI and the graph functions which are updated after every new line)
    piApprox =((numTrials)/( intersects));
  printData();
  graph();
}

void printData(){
  PFont f;
  f = createFont("LetterGothicStd.ttf",32,true);
  textFont(f,12);
  String e = "Number of Trials:" + numTrials + "     ";
  String f = "PI approximation: " + piApprox;
  fill(255);
  stroke(255);
  rect(0,height-20,400,20);
  fill(0);
  text(e,3,height-8);
  text(f,150,height-8);
}

void graph(){
  //draw PI line
  int piLine = height - 20 - round(18 * PI);
  stroke(255,0,0);
  line(0,piLine,width,piLine);

    //Speed determines how often a point is drawn
  int speed = 5;
  //Clears graph when it reaches the end of the screen
  if (round(numTrials/speed) % width == 0){
    fill(255);
    stroke(255);
    rect(0,boardHeight,width,180);
  }

  //plots points
  if(numTrials % speed == 0){
    int pointW = round(numTrials/speed) % width;
        int pointH = height - 20 - round(18 * piApprox);
    stroke(0,55,55);
    point(pointW,pointH);
  }
}

Answer 1

我运行了代码，在移除字体、更改颜色(供我使用)并将intersects和numTrials更改为float类型之后，它很好地收敛，并且看起来像预期的那样工作。也许你没有通过足够的试验，这导致你的PI估计不合理地波动(你可以看到这种情况发生在我之前的测试中，如下面屏幕截图上的行所示)。

我添加了frameRate(600)，使模拟运行速度快了10倍。

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选择的数字18似乎提供了一个合理的边界之间的底部的舞台和位置的线是画。改变这个数字会影响到红线所在的y坐标，以及近似值线收敛到的坐标。