[从头学数学] 第191节 空间向量与立体几何

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了[九转金丹]之第五转的修炼。
这次要研究的是[空间向量与立体几何]。

正剧开始：

星历2016年04月23日 11:00:22, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究[空间向量与立体几何]。

<span style="font-size:18px;">#例5
def dot(a, b):
    if (len(a) >= 3):
        return a[0]*b[0] +a[1]*b[1]+a[2]*b[2];
    elif (len(a) >= 2):
        return a[0]*b[0] +a[1]*b[1];
    else:
        return a*b;

#向量的绝对值
def rAbs(a):
    if (len(a) >= 3):
        return math.sqrt(a[0]**2 +a[1]**2 + a[2]**2);
    elif (len(a) >= 2):
        return math.sqrt(a[0]**2 +a[1]**2);
    else:
        return a;

def angle(a, b):
    return math.acos(dot(a,b)/(rAbs(a)*rAbs(b)))*180/math.pi;
    
def tmp():
    a = [0,-0.25,1];
    b = [0, 0.25, 1];
    print(angle(a, b));
	</span>

<span style="font-size:18px;">#例6
>>> 
90.0

def tmp():
    a = [-0.5, -0.5, 0.5];
    b = [1, 0, 1];
    print(angle(a, b));</span>

<span style="font-size:18px;">//平面向量
	if (1) {
		var r = 20;          
        config.setSector(1,1,1,1);            
        config.graphPaper2D(0, 0, r);          
        config.axis2D(0, 0,190);            
              
        //坐标轴设定      
        var scaleX = 2*r, scaleY = 2*r;        
        var spaceX = 2, spaceY = 2;         
        var xS = -10, xE = 10;        
        var yS = -10, yE = 10;        
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');          
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');       
            
  
                
        var array = [[1, 2], [4, 5], [7,2], [-8, -5], [-5, 3], [-8, 3], [-8, 5], [-2, 5], [-2, -1], [7, -3], [3, 1]];  
  
  
        var size = array.length;  
                
        var transform = new Transform();    
        var tmp = [];    
            
        array = transform.scale(transform.translate(array, 0, 0), scaleX/spaceX, scaleY/spaceY);        
          
        tmp = [].concat(array);          
		vectorDraw(tmp, 'red');   
         
        tmp = [].concat(array);          
        shape.pointDraw(tmp, 'orange', 1, 1);       
  
  
        plot.setFillStyle('blue');  
        plot.fillText('向量图', -270, -170, 300);  
        
    }</span>

<span style="font-size:18px;">	if (1) {
		var r = 20;          
        config.setSector(1,1,1,1);            
        config.graphPaper2D(0, 0, r);          
        config.axis2D(0, 0,190);            
              
        //坐标轴设定      
        var scaleX = 4*r, scaleY = 4*r;        
        var spaceX = 2, spaceY = 2;         
        var xS = -10, xE = 10;        
        var yS = -10, yE = 10;        
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');          
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');       
            
  
                
        var array = [];

		for (var x = -2; x <= 2; x+=4) {
			for (var y = -2; y <= 2; y+=4) {
				for (var z = -2; z<=2; z+=4) {
					array.push([x,y,z]);
				}
			}
		}  
  
		array = shape.xyzSort(array);
        var size = array.length;  
		
		var array2D = [];
		for (var i = 0; i < size; i++) {
			array2D.push(shape.point3D(array[i][0], array[i][1], array[i][2]));
		}
                
        var transform = new Transform();    
        var tmp = [];    
            
        array2D = transform.scale(transform.translate(array2D, 0, 0), scaleX/spaceX, scaleY/spaceY);        
          
        tmp = [].concat(array2D);          
		vectorDraw(tmp, 'red');   
         
        tmp = [].concat(array2D);          
        shape.pointDraw(tmp, 'orange', 1, 1);       
  
  
        plot.setFillStyle('blue');  
        plot.fillText('向量图', -270, -170, 300);  
        
    }</span>

<span style="font-size:18px;">//点阵中任意两点的距离排序集
function distanceSort(pointArray) {
	//传入点阵列pointArray
	//格式为[[px1, py1], [px2, py2], ...]

	//document.write(pointArray.join(' , ')+'<br/>');
	
	//对于pointArray中的每个点，求它与所有其它点的距离
	//结果放入distanceArray
	//格式为[[点1序号，点2序号, 距离值]]
	var distanceArray = [];
	
	//点的数量
	var size = pointArray.length;	
	
	//临时变量
	var distance = x1 = y1 = z1 = x2 = y2 = z2 = 0;
	
	var dimension = pointArray[0].length;
	

		
	//计算并压入距离
	for (var i = 0; i < size; i++) {
		for (var j = i+1; j < size; j++) {
			x1 = pointArray[i][0];
			y1 = pointArray[i][1];			
			
			x2 = pointArray[j][0];
			y2 = pointArray[j][1];
			
			if (dimension > 2) {
				//三维点的处理
				z1 = pointArray[i][2];
				z2 = pointArray[j][2];
				distance = Math.sqrt(Math.pow(x1-x2, 2)+Math.pow(y1-y2, 2)+Math.pow(z1-z2, 2));
				
			}
			else {
				distance = Math.sqrt(Math.pow(x1-x2, 2)+Math.pow(y1-y2, 2));
			}
			
			//注意这里已经保证i < j
			//所以起始点序号必须要小于终点序号
			//这是为了连接起始点和终点的直线不会重复
			distanceArray.push([i, j, distance]);
		}
	}
	
	//对距离阵列排序
	//排序权重：起始点序号 >  距离 > 终点序号 
	distanceArray.sort(function(a, b) {
		if (a[0] == b[0]) {
			if (Math.abs(a[2] - b[2]) < 0.000001) {
				return a[1]-b[1];
			}
			else {
				return a[2]-b[2];
			}
		
		}
		else {
			return a[0] - b[0];
		}
	});
	
	//document.write(distanceArray.join(' , ')+'<br/>');
	
	return distanceArray;
}


	if (1) {
		var r = 20;          
        config.setSector(1,1,1,1);            
        config.graphPaper2D(0, 0, r);          
        config.axis2D(0, 0,190);            
              
        //坐标轴设定      
        var scaleX = 4*r, scaleY = 4*r;        
        var spaceX = 2, spaceY = 2;         
        var xS = -10, xE = 10;        
        var yS = -10, yE = 10;        
        config.axisSpacing(xS, xE, spaceX, scaleX, 'X');          
        config.axisSpacing(yS, yE, spaceY, scaleY, 'Y');       
            
  
                
        var array = [];

		for (var x = -2; x <= 2; x+=4) {
			for (var y = -2; y <= 2; y+=4) {
				for (var z = -2; z<=2; z+=4) {
					array.push([x,y,z]);
				}
			}
		}  
  
		//array = shape.xyzSort(array);
        var size = array.length;  
		
		var array2D = [];
		for (var i = 0; i < size; i++) {
			array2D.push(shape.point3D(array[i][0], array[i][1], array[i][2]));
		}
		
		
		        //去除重复点  
        var pointArray = removeDuplicatedPoint(array2D);  
        //无重复的点的数量  
        var points = pointArray.length;  
          
        //得到距离阵列  
        //格式为[[点1序号，点2序号, 距离值], ...]  
        var distanceArray = distanceSort(pointArray);  
        //边的数量  
        var edges = distanceArray.length;  
          
        //存放需要连通的边  
        var linkedArray = [];  
        //连通的边的数量  
        var links = 0;  
          
        //每个顶点相关的边的集合  
        var edgeOfVertex = [];  
          
        for (var i = 0; i < points; i++) {  
              
              
            //获得顶点相关的边的集合  
            edgeOfVertex = [];  
            for (var j = 0; j < edges; j++) {  
                if (distanceArray[j][0] == i ||  
                    distanceArray[j][1] == i) {  
                    edgeOfVertex.push(distanceArray[j]);  
                }  
            }  
              
            //根据起始点寻找最短长度的两条边  
            edgeOfVertex.sort(function(a, b) {  
                return a[2] - b[2];  
            });  
              
            var choice = 2;  
            if (edgeOfVertex.length > choice) {  
                edgeOfVertex = edgeOfVertex.slice(0, choice);  
            }  
              
            linkedArray = linkedArray.concat(edgeOfVertex);  
        }  
          
          
        //document.write(linkedArray.join(' , ')+'<br/>');  
        linkedArray = removeDuplicatedPoint(linkedArray);  
        links = linkedArray.length;  
          
        //document.write(linkedArray.join(' , ')+'<br/>');      
          
        var startPoint, endPoint, x1, y1, x2, y2;  
        //比例缩放  
        var scale = 40;  
          
        for (var i = 0; i < links; i++) {  
            startPoint = linkedArray[i][0];  
            endPoint = linkedArray[i][1];  
            x1 = pointArray[startPoint][0];  
            y1 = pointArray[startPoint][1];  
            x2 = pointArray[endPoint][0];  
            y2 = pointArray[endPoint][1];  
              
            shape.vectorDraw([[x1,y1], [x2, y2]], 'red', scale);  
        }  
		
		shape.pointDraw(pointArray, 'blue', scale, 1);
		
		
		
        /*      
		//array2D = shape.angularSort(array2D);
        var transform = new Transform();    
        var tmp = [];    
            
        array2D = transform.scale(transform.translate(array2D, 0, 0), scaleX/spaceX, scaleY/spaceY);        
          
        tmp = [].concat(array2D);          
		vectorDraw(tmp, 'red');   
         
        tmp = [].concat(array2D);          
        shape.pointDraw(tmp, 'orange', 1, 1);       
		*/
  
        plot.setFillStyle('blue');  
        plot.fillText('向量图', -270, -170, 300);  
        
    }</span>

<span style="font-size:18px;">array = [[0,0,0],  [0, 0, 5], [5,0,5], [0, 5, 0], [0, 5, 5]];</span>

<span style="font-size:18px;">array = [[0,0,0],  [0, 0, 5], [5,0,0], [0, 5, 0]];</span>

<span style="font-size:18px;">array = [[0,0,0],  [0, 0, 5], [5,0,0], [0, 5, 0], [2, 2, 2], [2, 0, 2]];</span>

<span style="font-size:18px;">array = [[0,0,0],  [0, 0, 5], [5,0,0], [0, 5, 0], [2, 2, 2], [2, 0, 2], [-7,3, -7]];</span>

本节到此结束，欲知后事如何，请看下回分解。