[从头学数学] 第140节 勾股定理

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入了筑基中期的修炼，
这次要修炼的目标是[勾股定理]。

正剧开始：

星历2016年03月19日 10:03:20, 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[勾股定理]。

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 20;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  
	
	
	var triangle = new Triangle();
	var array = triangle.know3edges([3, 4, 5]);
	var tmp = [];
	
	plot.translate(r, -r)
		.save();
	for (var i = 0; i < 4; i++) {
		tmp = [].concat(array);
		shape.angleDraw(tmp, 'red', 2*r, 'ABC');
		
		plot.rotate(Math.PI/2);
		plot.translate(2*r, 0);
	}
	plot.restore();

	
	
}</span>

这里是小伟用到的工具：

<span style="font-size:18px;">function Triangle() {
	this.edges = [];
	this.angles = [];
	
	//已知三条边
	this.know3edges = function(edges) {
		this.edges = [];
		this.angles = [];
		
		this.edges = edges;
		//角度为弧度单位
		//a边对应角
		this.angles.push(Math.acos((edges[1]*edges[1] + edges[2]*edges[2]-edges[0]*edges[0])/(2*edges[1]*edges[2])));
		
		//b边对应角
		this.angles.push(Math.acos((edges[0]*edges[0] + edges[2]*edges[2]-edges[1]*edges[1])/(2*edges[0]*edges[2])));
		
		//c边对应角
		this.angles.push(Math.acos((edges[0]*edges[0] + edges[1]*edges[1]-edges[2]*edges[2])/(2*edges[0]*edges[1])));	
		
		var x0 = 0, y0 = 0;
		var x1 = x0 + this.edges[0], y1 = y0;
		var x2 = x0 + this.edges[1] * Math.cos(-this.angles[2]),
			y2 = y0 + this.edges[1] * Math.sin(-this.angles[2]);
			
		var retArray = new Array();
		retArray.push([x0, y0]);
		retArray.push([x1, y1]);
		retArray.push([x2, y2]);
		
		return retArray;
	
	}
	
	//已知两个角
	this.know2angles = function(angles, r) {
		this.edges = [];
		this.angles = [];
		
		this.angles = [angles[0]/180*Math.PI, angles[1]/180*Math.PI, 
					  (180-(angles[0]+angles[1]))/180*Math.PI];
		
		//设其中一边长度为10
		r = r > 0 ? r : 10;
		//A边
		this.edges.push(r);
		
		var angleA = this.angles[0];
			angleB = this.angles[1];
			angleC = this.angles[2];
			
		//B边
		this.edges.push(Math.sin(angleB)/Math.sin(angleA)*r);
		//C边
		this.edges.push(Math.sin(angleC)/Math.sin(angleA)*r);
		
		var x0 = 0, y0 = 0;
		var x1 = x0 + this.edges[0], y1 = y0;
		var x2 = x0 + this.edges[1] * Math.cos(-this.angles[2]),
			y2 = y0 + this.edges[1] * Math.sin(-this.angles[2]);
			
		var retArray = new Array();
		retArray.push([x0, y0]);
		retArray.push([x1, y1]);
		retArray.push([x2, y2]);
		
		return retArray;
	}
	
	//已知2条边
	this.know2edges = function(edges, angle) {
		this.edges = [];
		this.angles = [];
		
		//如果没有指定两边的夹角，默认为90度
		angle = angle ? angle/180*Math.PI : Math.PI/2;
		
		var edgeC = Math.sqrt(edges[0]*edges[0]+edges[1]*edges[1]-2*edges[0]*edges[1]*Math.cos(angle));
		this.edges = [edges[0], edges[1], edgeC];
		
		var edgeA = this.edges[0],
			edgeB = this.edges[1];
		
		//角度为弧度单位
		//a边对应角
		this.angles.push(Math.acos((edgeB*edgeB + edgeC*edgeC-edgeA*edgeA)/(2*edgeB*edgeC)));
		
		//b边对应角
		this.angles.push(Math.acos((edgeA*edgeA + edgeC*edgeC-edgeB*edgeB)/(2*edgeA*edgeC)));
		
		//c边对应角
		this.angles.push(angle);	
		
		var x0 = 0, y0 = 0;
		var x1 = x0 + this.edges[0], y1 = y0;
		var x2 = x0 + this.edges[1] * Math.cos(-this.angles[2]),
			y2 = y0 + this.edges[1] * Math.sin(-this.angles[2]);
			
		var retArray = new Array();
		retArray.push([x0, y0]);
		retArray.push([x1, y1]);
		retArray.push([x2, y2]);
		
		return retArray;
	
	}
		
	//返回角度和边信息的字符串
	this.info = function() {
		var angleLabel = ['C', 'A', 'B'];
		var edgeLabel = ['ab', 'bc', 'ac'];
		var s = '';
		
		for (var i = 0; i < 3; i++) {
			s += edgeLabel[i]+' : ';
			s += this.edges[i].toFixed(2)+' ; ';
		}
		
		for (var i = 0; i < 3; i++) {
			s += angleLabel[i]+' : ';
			s += (this.angles[i]*180/Math.PI).toFixed(2) + ' ; ';
		}
		
		return s;
	}

}</span>

还有：

<span style="font-size:18px;">this.angleDraw = function(array, style, scale, vertexLabel) {
	//vertexLabel是顶点编号顺序字符串 ABC,...
	style = style ? style : 'black';
	//array是一个存放二维坐标点序列的数组
	var a0 = new Array();
	a0 = [].concat(array);
	
	scale = scale ? scale : 1;
	var len = a0.length;
	
	if (scale != 1 && scale > 0) {
		for (var i = 0; i < len; i++) {
			for (var j = 0; j < 2; j++) {
				a0[i][j]*=scale;
			}
		}
	}
	
	//进行环状排序，这样传入的array就可以任意顺序放置坐标点。
	var a = this.angularSort(a0);	
	
	//分两次绘点和连线
	var tmp = [].concat(a);	
	this.pointDraw(tmp, style);
	tmp = [].concat(a);
	this.strokeDraw(tmp, style);
	
	
	var d1, d2, d3, angle;
	var x1,y1, x2, y2, x3, y3;
	var s;
	//坐标点编号
	var s0 = vertexLabel ? vertexLabel : 'ABCDEFGHIJKLMNOPQRSTUVWXYZ';
	
	//标记边的长度
	var edgeLong = 0;
	var measure = 0;
	//为每个点利用余弦定理求角
	for (var i = 0; i < len; i++) {
		if (i == 0) {
			x1 = a[len-1][0];
			y1 = a[len-1][1];
			x3 = a[i+1][0];
			y3 = a[i+1][1];
		}
		else if (i == len-1) {
			x1 = a[i-1][0];
			y1 = a[i-1][1];
			x3 = a[0][0];
			y3 = a[0][1];
		}
		else {
			x1 = a[i-1][0];
			y1 = a[i-1][1];
			x3 = a[i+1][0];
			y3 = a[i+1][1];
		}
		x2 = a[i][0];
		y2 = a[i][1];
		
		d1 = (x1-x2)*(x1-x2)+(y1-y2)*(y1-y2);
		d2 = (x2-x3)*(x2-x3)+(y2-y3)*(y2-y3);
		d3 = (x1-x3)*(x1-x3)+(y1-y3)*(y1-y3);
		
		angle = Math.acos((d1+d2-d3)/(2*Math.sqrt(d1*d2)))/Math.PI*180;

		s = angle.toFixed(2)+'°';

		//document.write(s+'<p>');
		//标注角度和顶点编号
		plot.setFillStyle('purple');
		plot.fillText(s, x2, -y2-5, 100);
		plot.setFillStyle(style);
		plot.fillText(s0[i], x2, -y2+20, 20);
		
		edgeLong = (Math.sqrt(d1)/scale).toFixed(2);
		measure = plot.measureText(edgeLong);
		
		plot.setFillStyle('blue');
		plot.fillText(edgeLong, (x1+x2-measure)/2, -(y1+y2)/2+20, measure);
		
	}	
	
	//由于处理会改变原矩阵，看以下操作能否恢复原矩阵
	if (scale != 1 && scale > 0) {
		for (var i = 0; i < len; i++) {
			for (var j = 0; j < 2; j++) {
				a0[i][j]/=scale;
			}
		}
	}

}</span>

看到这个题，阿伟就想笑，这个对角线就比木板多了4个毫米，如果这块板太薄，过门时很容易折坏，

如果稍微厚那么一点，就等着撞门框吧，这理论的数据可当不的真的哦。

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 80;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r/4);
	config.axis2D(0, 0,180);  
	
	
	var triangle = new Triangle();
	var array = triangle.know2edges([Math.sqrt(2.6*2.6-2.4*2.4), 2.4]);
	var tmp = [];
	
	config.setSector(1,3,1,1); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', r);
	
	array = triangle.know2edges([Math.sqrt(2.6*2.6-1.9*1.9), 1.9]);
	config.setSector(1,3,1,2); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'green', r);


	
	
}</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 25;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r);
	config.axis2D(0, 0,180);  
	
	/*
	var triangle = new Triangle();
	var array = triangle.know2edges([Math.sqrt(2.6*2.6-2.4*2.4), 2.4]);
	var tmp = [];
	
	config.setSector(2,3,1,1); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', r);
	
	array = triangle.know2edges([Math.sqrt(2.6*2.6-1.9*1.9), 1.9]);
	config.setSector(2,3,1,2); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'green', r);

	*/
	
	var array = [];
	var thita = 0;
	var rho = 0;
	var x, y;
	
	for (var i = 0; i < 36; i++) {
		rho = Math.sqrt(i);
		x = rho*Math.cos(thita);
		y = rho*Math.sin(thita);
		
		array.push([x, -y]);
		
		thita += Math.PI/12;
	}
	
	var tmp = [];
	tmp = [].concat(array);
	shape.pointDraw(tmp, 'red', r);
	tmp = [].concat(array);
	shape.multiLineDraw(tmp, 'pink', r);
	
	
}</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 50;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r/2);
	config.axis2D(0, 0,180);  
	
	
	var triangle = new Triangle();
	var array = triangle.know2edges([3, 4]);
	var tmp = [];
	
	config.setSector(2,3,1,1); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', r);	
	
	array = triangle.know2edges([2.4, 0.7]);

	config.setSector(2,3,1,2); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', 2*r, 'AOB');	
	
}</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 50;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r/2);
	config.axis2D(0, 0,180);  
	
	
	var triangle = new Triangle();
	var array = triangle.know2edges([19, 39]);
	var tmp = [];
	
	config.setSector(2,3,1,1); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', 6);	
	
	array = triangle.know2edges([Math.sqrt(7*7-5*5), 5]);

	config.setSector(2,3,1,2); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', 20);	
	
}</span>

<span style="font-size:18px;">function myDraw(xGlobal, yGlobal) {  
	var config = new PlotConfiguration();      
	config.init();  
	
	config.setPreference();  
	var r = 50;
	config.setSector(1,1,1,1);  
	config.graphPaper2D(0, 0, r/2);
	config.axis2D(0, 0,180);  
	
	
	var triangle = new Triangle();
	var array = triangle.know2edges([Math.sqrt(88*88-32*32), 32]);
	var tmp = [];
	
	config.setSector(2,3,1,1); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', 2);	
	
	array = triangle.know2edges([5, 12]);

	config.setSector(2,3,1,2); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'green', 50);	
	
}</span>

<span style="font-size:18px;">	var mathText = new MathText();
	
	var s = [
		'AE = DB',
		'AD^[2]+DB^[2] = AB^[2]',
		'AB^[2] = AC^[2]+BC^[2] = 2*AC^[2]'
	];
	
	var x =40, y=40;
		var r1 = 40;
		
		var len = s.length;
		for (var i = 0; i < len; i++) {
		
			if (s[i] == '') {
				if (x < 100) {
					x += 300;
					y-=r1*3;
				}
				else {
					x = 20;
					y += r1;
				}
			}
			else {			
				mathText.print(s[i], x, y);
				y+=r1;
			}
		}		</span>

<span style="font-size:18px;">	var triangle = new Triangle();
	var array = [];
	var tmp = [];
	
	
	array = triangle.know3edges([15, 8, 17]);
	config.setSector(2,3,1,1); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', 10);	
	
	array = triangle.know3edges([13, 14, 15]);

	config.setSector(2,3,1,2); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'green', 10);	</span>

<span style="font-size:18px;">	var triangle = new Triangle();
	var array = [];
	var tmp = [];
	
	
	array = triangle.know3edges([18, 24, 30]);
	config.setSector(2,3,1,1); 
	tmp = [].concat(array);
	shape.angleDraw(tmp, 'red', 10);	</span>

本节到此结束，欲知后事如何，请看下回分解。