[从头学数学] 第190节 圆锥曲线与方程

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[圆锥曲线与方程]。

正剧开始：

星历2016年04月22日 17:18:56, 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[圆锥曲线与方程]。

<span style="font-size:18px;">        //存放函数图像上的点        
        var a = [], b = [], c = [], d = [];       
                  
        //需要显示的函数说明        
        var f1 = '曲线 xy = ±8';    
		var epsilon = 0.000001;
		var k = 8;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.5) {   
			for (var y = yS; y <= yE; y+=0.5) {
				if (Math.abs(x*y - k)<epsilon || Math.abs(x*y + k)<epsilon) {  
					
					a.push([x, y]);   
				} 
			}
        }   </span>

<span style="font-size:18px;">		var pointA = [-1, -1], pointB = [3, 7];
        //函数描点        
        for (var x = xS; x <= xE; x+=0.5) {   
			for (var y = yS; y <= yE; y+=0.5) {
				if (floatEqual(distanceCalc([x, y], pointA), distanceCalc([x,y], pointB))){  
					
					a.push([x, y]);
				} 
			}
        }  </span>

<span style="font-size:18px;">       //存放函数图像上的点        
        var a = [], b = [], c = [], d = [];       
                  
        //需要显示的函数说明        
        var f1 = '曲线 例3', f2 = '';    
		var epsilon = 0.000001;
		var pointF = [0, 2];
        //函数描点        
        for (var x = xS; x <= xE; x+=0.5) {   
			for (var y = yS; y <= yE; y+=0.5) {
				if (floatEqual(distanceCalc([x, y], pointF)-2,y)){  
					
					a.push([x, y]);     
				} 
			}
        }  </span>

<span style="font-size:18px;">        //需要显示的函数说明        
        var f1 = '曲线 题2', f2 = '';    
		var epsilon = 0.000001;
		var k = 5;
		var pointO = [0, 0], pointA = [k, 0];
		
		var d1 = d2 = 0;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.5) {   
			for (var y = yS; y <= yE; y+=0.5) {
				d1 = distanceCalc([x, y], pointO);
				d2 = distanceCalc([x, y], pointA);
				
				
				if (floatEqual(d1*d1-d2*d2, k)){  
					
					a.push([x, y]);   
				} 
			}
        }  </span>

<span style="font-size:18px;">        //需要显示的函数说明        
        var f1 = '曲线 题3', f2 = '';    
		var epsilon = 0.000001;

		var pointA = [-3, 0], pointB = [3, 0];
		
		var d1 = d2 = 0;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.2) {   
			for (var y = yS; y <= yE; y+=0.2) {
				d1 = distanceCalc([x, y], pointA);
				d2 = distanceCalc([x, y], pointB);
				
				
				if (floatEqual(d1*d1+d2*d2, 26)){  
					
					a.push([x, y]);    
				} 
			}
        } </span>

<span style="font-size:18px;">        //需要显示的函数说明        
        var f1 = '曲线 题1', f2 = '';    
		var epsilon = 0.000001;
		
		var k;
		//y = k(x-3)+4
		//[0, -3*k+4], [-4/k+3, 0]

<span style="white-space:pre">		</span>for (var k = -10; k <= 10; k+=0.2) {
<span style="white-space:pre">			</span>if (k != 0) {
<span style="white-space:pre">			</span>a.push([-4/k+3, -3*k+4])
<span style="white-space:pre">			</span>}
<span style="white-space:pre">		</span>}
</span>

如果用x^2/a^2+y^2/b^2这种公式去描点，得到的点会非常稀疏。

所以一般用到某两点距离和为定值这种表述来描点：

<span style="font-size:18px;">if (1) {      
        var r = 20;            
        config.setSector(1,1,1,1);              
        config.graphPaper2D(0, 0, r);            
        config.axis2D(0, 0,180);              
                
        //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 transform = new Transform();            
        //存放函数图像上的点        
        var a = [], b = [], c = [], d = [];       
                  
        //需要显示的函数说明        
        var f1 = '曲线 椭圆 x^2/16+y^2/9 = 1', f2 = '';    
		var epsilon = 0.000001;
		
		//椭圆的焦距等 a = 4, b = 3, c= 5
		var pointA = [-4, 0], pointB = [4, 0];
		
		var d1 = d2 = 0;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.03) {   
			for (var y = yS; y <= yE; y+=0.03) {
				
				
				if (floatEqual(distanceCalc([x, y], pointA)+distanceCalc([x,y], pointB), 10)){  					
					a.push([x, y]);        
		  
				} 
			}
        }     
		
 
        a = shape.angularSort(a);        
        //存放临时数组        
        var tmp = [];        
                  
        //显示变换        
        if (a.length > 0) {        
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);         
            //函数1        
            tmp = [].concat(a);            
            shape.pointDraw(tmp, 'red');            
            tmp = [].concat(a);            
            shape.strokeDraw(tmp, 'pink');          
                      
            plot.setFillStyle('red');        
            plot.fillText(f1, 100, -90, 200);          
        }      

		b = [pointA, pointB];
		
		if (b.length > 0) {    
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);                
 
            tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue', 1, 1);    
    
        }    
                
        
    }
	
	
	

}</span>

<span style="font-size:18px;">       //需要显示的函数说明        
        var f1 = '曲线 例3', f2 = '';    
		var epsilon = 0.000001;
		
		//椭圆的焦距等 a = 4, b = 3, c= 5
		var pointA = [-5, 0], pointB = [5, 0];
		
		var d1 = d2 = 0;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.03) {   
			for (var y = yS; y <= yE; y+=0.03) {
				
				
				if (floatEqual(slopeCalc([x,y], pointA)*slopeCalc([x,y], pointB), -4/9)){  					
					a.push([x, y]);        
		  
				} 
			}
        }     
		</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, 0,180);              
                
        //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 transform = new Transform();            
        //存放函数图像上的点        
        var a = [],  b = [], c = [], d = [];       
                  
        //需要显示的函数说明        
        var f1 = '曲线 双曲线 x^2/9-y^2/16 = 1', f2 = '';    
		var epsilon = 0.000001;
		

		var pointA = [-5, 0], pointB = [5, 0];
		
		var d1 = d2 = 0;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.03) {   
			for (var y = yS; y <= yE; y+=0.03) {
				
				
				
				if (floatEqual(Math.abs(distanceCalc([x,y], pointA)- distanceCalc([x,y], pointB)), 6)){  
					//按象限划分
					if (x<=0) {
					
						a.push([x, y]);   
					}
					else {
						b.push([x,y]);
					}	  
		  
				} 
			}
        }     
		
 
           
		
        //存放临时数组        
        var tmp = [];        
                  
        //显示变换        
        if (a.length > 0) { 
			a = shape.angularSort(a);  
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);         
            //函数1        
            tmp = [].concat(a);            
            shape.pointDraw(tmp, 'red');            
            tmp = [].concat(a);            
            shape.multiLineDraw(tmp, 'pink');          
                      
            plot.setFillStyle('red');        
            plot.fillText(f1, 100, -90, 200);          
        }      

		
		
		if (b.length > 0) { 
			b = shape.angularSort(b);
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);                
            //函数2    
            tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');    

            tmp = [].concat(b);        
            shape.multiLineDraw(tmp, '#22CCFF'); 

                
            plot.setFillStyle('blue');    
            plot.fillText(f2, 100, -120, 200);     
    
        }    
                
		//点的标注
        var p = [pointA, pointB];
		
		if (p.length > 0) {    
            p = transform.scale(transform.translate(p, 0, 0), scaleX/spaceX, scaleY/spaceY);                
            //函数2    
            tmp = [].concat(p);        
            shape.pointDraw(tmp, 'blue', 1, 1);    
    
        } 
		
    }</span>

<span style="font-size:18px;">        //需要显示的函数说明        
        var f1 = '曲线 双曲线 x^2/9-y^2/4 = 1', f2 = '';    
		var epsilon = 0.000001;
		

		var pointA = [-Math.sqrt(13), 0], pointB = [Math.sqrt(13), 0];
		
		var d1 = d2 = 0;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.03) {   
			for (var y = yS; y <= yE; y+=0.03) {
				
				
				
				if (floatEqual(Math.abs(distanceCalc([x,y], pointA)- distanceCalc([x,y], pointB)), 6)){  
					//按象限划分
					if (x<=0) {
					
						a.push([x, y]);   
					}
					else {
						b.push([x,y]);
					}	  
		  
				} 
			}
        }     </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, 0,180);              
                
        //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 transform = new Transform();            
        //存放函数图像上的点        
        var a = [],  b = [], c = [], d = [];       
                  
        //需要显示的函数说明        
        var f1 = '曲线 抛物线y^2 = 2(x-2.5)', f2 = '';    
		var epsilon = 0.000001;
		

		var pointA = [Math.sqrt(25), 0], pointB = [Math.sqrt(25), 0];
		
		var d1 = d2 = 0;
        //函数描点        
        for (var x = xS; x <= xE; x+=0.05) {   
			for (var y = yS; y <= yE; y+=0.05) {
				if (floatEqual(Math.abs(distanceCalc([x,y], pointA)-(x-0)), 0)){  
				
						a.push([x, y]);   
		  
				} 
			}
        }   
		
        //存放临时数组        
        var tmp = [];        
                  
        //显示变换        
        if (a.length > 0) {    
			a = shape.angularSort(a); 
            a = transform.scale(transform.translate(a, 0, 0), scaleX/spaceX, scaleY/spaceY);         
            //函数1        
            tmp = [].concat(a);            
            shape.pointDraw(tmp, 'red');            
            tmp = [].concat(a);            
            shape.multiLineDraw(tmp, 'pink');          
                      
            plot.setFillStyle('red');        
            plot.fillText(f1, -100, -120, 200);          
        }      

		
		
		if (b.length > 0) {    
			b = shape.angularSort(b);
            b = transform.scale(transform.translate(b, 0, 0), scaleX/spaceX, scaleY/spaceY);                
            //函数2    
            tmp = [].concat(b);        
            shape.pointDraw(tmp, 'blue');    

            tmp = [].concat(b);        
            shape.multiLineDraw(tmp, '#22CCFF'); 

                
            plot.setFillStyle('blue');    
            plot.fillText(f2, 100, -120, 200);     
    
        }    
                
		//点的标注
        var p = [pointA];
		
		if (p.length > 0) {    
            p = transform.scale(transform.translate(p, 0, 0), scaleX/spaceX, scaleY/spaceY);                
            //函数2    
            tmp = [].concat(p);        
            shape.pointDraw(tmp, 'blue', 1, 1);    
    
        } 
		
    }</span>

几个方便函数：

<span style="font-size:18px;">//平面点之间的距离
function distanceCalc(pointA, pointB) {
	var x1 = pointA[0], y1 = pointA[1],
		x2 = pointB[0], y2 = pointB[1];
	return Math.sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));

}

//平面点之间的斜率
function slopeCalc(pointA, pointB) {
	var x1 = pointA[0], y1 = pointA[1],
		x2 = pointB[0], y2 = pointB[1];
	if (floatEqual(x1, x2)) {
		return 100000;
	}
	else {
		return (y1-y2)/(x1-x2);
	}

}

//浮点数的相等判断
function floatEqual(value1, value2) {
	var epsilon = 0.001;
	if (Math.abs(value1 - value2)<=epsilon) {
		return true;
	}
	else {
		return false;
	}
}
</span>

本节到此结束，欲知后事如何，请看下回分解。