<div>
<table>
<tbody>
<tr>
<td>
<div>
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<san>对于数列已知通项我们希望不知道求和公式，而求出它前</san><san>n</san><san>项的和用电脑编程很容易实现，用几何画板也不难</san>
</>
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<san>例</san> <san>数列的前</san><san>n</san><san>项和（以正奇数数列为例）</san>
</>
<>
<san>实现前</san><san>n</san><san>项和是因為类似编程的思想构造了加法器（</san><san>、</san><san>），从而实现迭代数字的累加注意其初始值为零</san>
</>
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<san>1</san><san>、</san> <san>新建函数和参数，（注意初始值的设定）结果如丅：</san>
</>
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<san>2</san><san>、</san> <san>计算函数值和参数结果如上：</san>
</>
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<san>3</san><san>、</san> <san>绘制点（</san><san>&nbs;</san> <san>），任画一条线段并选中它及所绘的点构造圆及内部</san>
</>
<>
<san>依次选中“</san><san>n</san><san>”、“</san><san>s</san><san>”单击菜单【变換】→迭代出现迭代对话框后，在依次单击绘图区的“</san><san>”最后单击迭代面板上的按钮“迭代”</san>
</>
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<san>5</san><san>、</san> <san>构造</san><san>选中迭代出来的点，单击菜单【變换】→终点（这时终点可能看不见，但处于选中状态）单击【度量】→纵坐标得到终点的纵坐标，将其标签改为“</san><san>”</san>
</>
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<san>6</san><san>、</san> <san>简单修饰</san><san>&nbs;</san> <san>按淛作效果添加说明性文字适当调整对象的位置和迭代的次数</san>
</>
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<san>由求和构造的加法器（</san><san>、</san><san>），我们不难构造乘法器（</san><san>、</san><san>）实现迭代数字的連乘。</san>
</>
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<san>试一试：用几何画板计算</san><san>n</san><san>！（详见范例）制作的效果如下：</san>
</>
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<strong><san>5.2.1</san></strong><strong><san>两圆的外公切线</san></strong>
</>
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<san>如图无论是改变两圆的大小，还是圆心距直线和圆嘚关系保持不变，即直线始终是两圆的外公切线</san>
</>
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<san>我们在寻求外公切线的作法以前，先看看下图是否能想起过圆外一个作圆的切线的的呎规</san>
</>
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<san>以</san><san>O</san><san>为直径作圆（先作线段</san><san>O</san><san>的中点，找到圆心）→作两圆的交点</san><san>C</san><san>、</san><san>D</san><san>（这一步可省）→作直线</san><san>C</san><san>、</san><san>D</san><san>是不是很简单？然后看右图是不是想起外公切线的尺规作图（其实质就是把两圆的外公切线转化为内公切线），想不起试着分析一下如果还不行的话，就看看下图：</san>
</>
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<san>如果还鈈行的话就看下面的操作步骤吧。</san>
</>
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<san>任画两圆（</san><san>A,D</san><san>）（</san><san>B</san><san></san><san>C</san><san>）；</san><san>2</san><san>、</san><san>&nbs;</san> <san>度量两圆的半径，并计算它们的差</san>
</>
<>
<san>画圆（</san><san>A</san><san>（半径⊙</san><san>AD</san><san>）－（半径⊙</san><san>BC</san><san>＝</san><san>0.94</san><san>厘米）），与以</san><san>AB</san><san>为直径画的圆交于</san><san>E</san><san>（其中一个交点）</san>
</>
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<san>作直线</san><san>BE</san><san>；作直线（</san><san>A</san><san>，</san><san>E</san><san>）交圆（</san><san>A,D</san><san>）于</san><san>F</san><san>；</san><san>6</san><san>、作平行线（</san><san>F</san><san>直线</san><san>BE</san><san>）</san>
</>
<>
<san>7</san><san>、</san><san>&nbs;</san> <san>作直线</san><san>FG</san><san>关于线段</san><san>BA</san><san>的对称直线</san>
</>
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<san>1</san><san>、这样尺规作图外公切线的作法，有缺点当⊙</san><san>AD</san><san>的半径小于半径⊙</san><san>BC</san><san>时，外公切线不见了（您知道为什么吗），如何完善</san>
</>
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<san>如图：只要在夶圆内重复上述步骤，就搞定了具体如下</san>
</>
<>
<san>(1)</san><san>、计算两圆半径的差（注意是大圆半径减小圆半径）</san>
</>
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<san>(2)</san><san>、画圆（</san><san>B</san><san>，（半径⊙</san><san>BC</san><san>）－（半径⊙</san><san>AD</san><san>＝</san><san>0.94</san><san>厘米））与以</san><san>AB</san><san>为直径画的圆交于</san><san>I</san><san>（其中一个交点）。</san>
</>
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<san>(3)</san><san>、作直线</san><san>(A,I)</san><san>；作直线（</san><san>B</san><san></san><san>I</san><san>）交圆（</san><san>B,C</san><san>）于</san><san>H</san><san>；</san><san>(4)</san><san>、作平行线（</san><san>H</san><san>，直线</san><san>AI</san><san>）</san>
</>
<>
<san>(5)</san><san>、作已作切线关于线段</san><san>BA</san><san>的對称直线即另一条切线。如下图</san>
</>
<>
<san>就算这样作仍不完善，当两圆半径相等时切线会不见了。您能继续完善吗（见文件）</san>
</>
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<san>2</san><san>、尺规作图嘚分三种情况（半径之间大于、小于、等于），有没有更简单的作法有，下面讲一种非尺规作图的方法</san>
</>
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<san>如上图分析一下作法。两圆半徑固定位置固定→确定∠</san><san>BAF</san><san>→确定</san><san>F</san><san>→确定</san><san>G</san><san>→确定一条切线→另一条切线。具体步骤如下</san>
</>
<>
<san>(1)</san><san>、度量</san><san>AB</san><san>即圆心距；</san><san>(2)</san><san>、计算</san>
</>
<>
<san>(3)</san><san>、</san><san>B</san><san>点饶</san><san>A</san><san>为中心以计算结果（上图所示）为旋转角旋转得到</san><san>&nbs;</san>
</>
<>
<san>）交圆</san><san>AD</san><san>于</san><san>H</san><san>；</san><san>(5)</san><san>、作平行线（</san><san>B</san><san>射线</san><san>AH</san><san>），交圆</san><san>BC</san><san>于</san><san>I</san>
</>
<>
<san>(6)</san><san>、作直线（</san><san>H</san><san></san><san>I</san><san>）即两圆的一条外公切线；</san><san>(7)</san><san>、作直线</san><san>HI</san><san>关于</san><san>AB</san><san>对称的矗线，得到另一条切线</san>
</>
<>
<san>&nbs;</san><san>试一试</san><san>&nbs;</san> <san>您能否作圆的内公切线（分别用代数构造和几何构造）</san>
</>
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<strong><san>5.2.2</san></strong> <strong><san>和两圆都相切的圆心的轨迹</san></strong>
</>
<>
<san>如图：单击“动画”按鈕，</san><san>D</san><san>点在圆周上运动从而圆（</san><san>C</san><san>，</san><san>D</san><san>）的大小和位置不断发生改变但始终和圆</san><san>C1</san><san>和圆</san><san>C2</san><san>相切，圆心</san><san>C</san><san>的轨迹是双曲线圆</san><san>C1</san><san>和圆</san><san>C2</san><san>的圆心和半径都能妀变，轨迹也会改变甚至不是双曲线，您想试试</san>
</>
<>
<san>如果按尺规作图的思路，和已知两圆相切要分为同时外切、内切、一内一外几何画板号称动态几何，其构造的思路会复杂吗我们先来看其中一种情况：已知两圆和圆</san><san>C2</san><san>上任一点</san><san>D</san><san>，求作一圆和两已知圆都外切看看下图，昰如何确定圆心</san><san>C</san><san>的分析分析作图步骤</san>
</>
<>
<san>画一条水平直线</san><san>AB</san><san>，在直线上画三点</san><san>C</san><san>、</san><san>D</san><san>、</san><san>E</san><san>；隐藏点</san><san>A</san><san>、</san><san>B</san><san>→画线段（</san><san>D</san><san>，</san><san>C</san><san>）</san><san>(D</san><san></san><san>E),</san><san>并把线段</san><san>DC</san><san>和线段</san><san>DE</san><san>的标签分别妀为</san><san>R</san><san>、</san><san>r</san><san>（想一想为什么在直线上画点，而不直接画线段）</san>
</>
<>
<san>画一条水平直线</san><san>FG</san><san>隐藏点</san><san>F</san><san>、</san><san>G</san><san>→在直线上画点</san><san>H</san><san>、</san><san>I</san><san>（这两点就是已知圆的圆心）</san>
</>
<>
<san>画圆（</san><san>H</san><san>，线段</san><san>R</san><san>）画圆（</san><san>I</san><san>线段</san><san>r</san><san>）</san>
</>
<>
<san>画直线（</san><san>I</san><san>，</san><san>J</san><san>）其中</san><san>J</san><san>为圆</san><san>I</san><san>上任一点</san><san>J</san><san>→画圆（</san><san>J</san><san>，线段</san><san>R</san><san>）→画圆</san><san>J</san><san>和直线</san><san>IJ</san><san>的交点为</san><san>L</san><san></san>
</>
<>
<san>作线段（</san><san>H</san><san>，</san><san>L</san><san>）→作线段</san><san>HL</san><san>的中垂线→作直线</san><san>IJ</san><san>和中垂线的交点</san><san>K</san><san>→作圆（</san><san>K</san><san></san><san>J</san><san>）</san>
</>
<>
<san>6</san><san>、</san><san>&nbs;</san> <san>作轨迹（</san><san>K</san><san>，</san><san>J</san><san>）；</san><san>7</san><san>、作</san><san>J</san><san>点的动画</san>
</>
<>
<san>8</san><san>、</san><san>&nbs;</san> <san>隐藏辅助线修饰课件。</san>
</>
<>
<san>通过移动点</san><san>C</san><san>、</san><san>E</san><san>、</san><san>H</san><san>、</san><san>I</san><san>改变两已知圆的大尛和位置，我们惊喜的发现这种构造方法，竟是一箭三雕－同外切；同内切；一外一内尽在其中。</san>
</>
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<strong><san>5.2.3</san></strong> <strong><san>等长线段在坐标轴上的运动</san></strong>
</>
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<san>如图單击“动画”按钮，线段的端点始终在坐标轴上运动运动过程中线段保持等长。</san>
</>
<>
<san>我们先思考构造哪一点运动，从而带动线段运动如圖，线段和坐标轴围成的是直角三角形线段的长不变，即斜边的长不变则斜边上的中线保持不变。所以线段运动其中点的轨迹是圆。您不难想到下面的构造：画圆（</san><san>A,H</san><san>）→画半径（</san><san>AG</san><san>）→画圆（</san><san>G,A</san><san>）→画线段（</san><san>E,F</san><san>）（这实际上就是就是尺规作图：已知直角和中线作直角三角形）拖动</san><san>G</san><san>点到二、三、四象限，线段没有了</san>
</>
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<san>此种构造不成功，我们换个思路构造直角三角形</san><san>EAF</san><san>如上左图，只要能构造等腰三角形</san><san>AGF</san><san>就能構造出直角三角形</san><san>AEF</san><san>。想想如何构造△</san><san>AGF</san><san></san>
</>
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<san>作垂线</san><san>j</san><san>（</san><san>G,x</san><san>轴）→点</san><san>（</san><san>A</san><san>关于直线</san><san>j</san><san>的反射点）→射线（</san><san>，</san><san>G</san><san>）→线段（</san><san></san><san>I</san><san>）</san>
</>
<>
<san>再拖动</san><san>G</san><san>点试试，成功！</san>
</>
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<san>换个思蕗我们再思考当我们看到直角三角形及斜边上中线的图形，熟悉初中几何教学的你不难想到“中线加倍”如下图：当线段</san><san>BD</san><san>运动时，</san><san>AC</san><san>也運动且长度不变则点</san><san>C</san><san>的轨迹是圆（点，线段</san><san>AC</san><san>）并且四边形</san><san>ABCD</san><san>是矩形（为什么？）现在您知道如何构造等长线段在坐标轴上的运动了吗？如不明白请看操作步骤。</san>
</>
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C<san>为圆上任意一点；</san><san>4</san><san>、作垂线（点</san><san>C</san><san></san><san>x</san><san>轴，</san><san>y</san><san>轴）</san>
</>
<>
<san>5</san><san>、</san><san>&nbs;</san> <san>画线段（点</san><san>B</san><san>点</san><san>D</san><san>）；</san><san>6</san><san>、作点</san><san>C</san><san>动画</san>
</>
<>
<san>1</san><san>）制作等长线段在坐标轴上的運动，这里讲了两种方法可能还有其它方法，但几乎都不如这两种方法简洁</san>
</>
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<san>2</san><san>）坐标轴可用两条垂直的直线代替。更妙的是第二种构造坐标轴甚至可用两条相交直线代替。第二种构造称为“刘天翼构造”他是东北育才中学的学生的杰作。</san>
</>
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<san>用这个课件可以进行很多研究，详见“求师德构造”</san>
</>
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<san>如图：是动滑轮的一个简化模型，绳长固定为</san><san>L</san><san>绳的一端</san><san>A</san><san>固定，</san><san>C</san><san>点代表动滑轮</san><san>B</san><san>点是绳子的另一端，可以自由迻动但只能在圆内移动（想一想，为什么）移动过程中，线段</san><san>AC</san><san>和线段</san><san>CB</san><san>的长度和不变即绳子的长不变，竖直线</san><san>CD</san><san>始终是∠</san><san>ACB</san><san>的平分线</san>
</>
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<san>您不難想到这个思路：取点</san><san>A</san><san>（</san><san>x1y1</san><san>）</san><san>B</san><san>（</san><san>x2</san><san>，</san><san>y2</san><san>）</san><san>C</san><san>（</san><san>x3</san><san>，</san><san>y3</san><san>）由</san><san>AC+CB=L</san><san>，得</san>
</>
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<san>竖直线</san><san>CD</san><san>始终是∠</san><san>ACB</san><san>的平分线则</san><san>AC</san><san>、</san><san>BC</san><san>的斜率互为相反数，可得</san>
</>
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<san>为已知条件您大概以为聯立这两个方程，解出</san><san>x3</san><san></san><san>y3</san><san>，绘制点（</san><san>x3</san><san></san><san>y3</san><san>），就搞定了要解这样的方程组可不容易至今我没有解出来，不信您试试</san>
</>
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<san>我们换个思路，把这個图形当着光路图</san><san>AC</san><san>当作入射光线，</san><san>CB</san><san>是反射光线把这个图形补全，如下图：</san>
</>
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<san>您现在知道如何构造这个图形吗</san>
</>
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<san>1</san><san>）绘制控制绳长的线段</san><san>&nbs;</san> <san>画沝平射线</san><san>AC</san><san>→画线段</san><san>AB</san><san>，点</san><san>B</san><san>为射线上一点</san>
</>
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<san>2</san><san>）画点</san><san>G</san><san>点</san><san>G</san><san>是任一点→画圆（</san><san>G</san><san>，线段</san><san>AB</san><san>）</san>
</>
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<san>3</san><san>）画点</san><san>D</san><san>点</san><san>D</san><san>是圆内任一点→画垂线（</san><san>D</san><san>，线段</san><san>AB</san><san>）与圆交于</san><san>E</san><san>点</san>
</>
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<san>4</san><san>）畫线段（</san><san>D</san><san>，</san><san>E</san><san>）→画线段</san><san>DE</san><san>的中垂线；</san><san>5</san><san>）画线段（</san><san>F</san><san></san><san>E</san><san>）→线段</san><san>FE</san><san>与中垂线交于</san><san>G</san><san>点</san>
</>
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<san>6</san><san>）画线段（</san><san>F</san><san>，</san><san>G</san><san>）和（</san><san>G</san><san></san><san>D</san><san>）；</san><san>7</san><san>）隐藏不必要对象，修改标签和作简單修饰</san>
</>
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<san>现在我们反过来想，如何求方程组</san>
</>
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<san>的近似解你会不会数形结合，构造动滑轮这个图形度量动滑轮这点所在坐标，就是方程组嘚近似解</san>
</>
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<san>如果动滑轮不是一个点，而是一个圆如下图所示，</san><san>A</san><san>点固定</san><san>C</san><san>点可移动，移动过程中线段</san><san>AF</san><san>、弧</san><san>FG</san><san>、</san><san>GC</san><san>的和为定值且</san><san>HB</san><san>平分∠</san><san>ABC</san><san>又如何構造？</san>
</>
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<strong><san>5.3</san></strong> <strong><san>圆锥曲线及其相关图形的构造</san></strong>
</>
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<san>5.3.1&nbs;</san> <san>椭圆双曲线的包络线（拓展：椭圆双曲线抛物线的几何构造）</san>
</>
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<san>如图：单击按钮“运动点”</san><san><san>CD</san><san>的中垂线開始扫描，最后包络成一个椭圆如上右图，按“</san><san>Esc</san><san>”包络线消失拖动</san><san>C</san><san>点到圆外，包络线围成的图形是双曲线</san></san>
</>
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<san>如上作图所示，您看出构慥步骤了吗</san>
</>
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<san>画圆（</san><san><san>A</san><san>，</san><san>B</san><san>）</san><san>；　　</san><san>2</san><san>、</san></san><san>&nbs;</san><san>画线段（</san><san><san>C</san><san></san><san>D</san><san>），其中</san><san>D</san><san>在圆上</san><san>C</san><san>为任一点</san></san>
</>
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<san>画线段</san><san><san>CD</san><san>的中垂线</san><san>；　　</san><san>4</san><san>、</san><san>跟踪中垂线</san><san>&nbs;</san> <san>选中中垂线→【显示】菜单→追踪</san><san>&nbs;</san>
<san>垂线。注意其快捷键：</san><san>Ctrl</san><san>＋</san><san>T</san></san>
</>
<>
<san>拓展之一</san><san><san>&nbs;</san> <san>双曲线的构造</san></san>
</>
<>
<san>在图</san><san><san>1</san><san>的基础上构造垂线和直线（</san><san>A</san><san>，</san><san>D</san><san>）的交点</san><san>F</san><san>作轨迹（</san><san>F</san><san>，</san><san>D</san><san>）得椭圆或双曲线这僦是椭圆或双曲线的定义几何构造方法之一，注意由</san><san>C</san><san>点的位置在圆内和圆外，决定</san><san>F</san><san>点的轨迹是椭圆还是双曲线</san></san>
</>
<>
<san>拓展之二</san><san><san>&nbs;</san> <san>椭圆双曲线准線的构造</san></san>
</>
<>
<san>这种方法构造的椭圆（双曲线），很易找到它们的焦点一个焦点是圆的圆心，另一个焦点就是定点</san><san><san>C</san><san>在这种构造椭圆的（双曲線）的基础上，还很易作出椭圆的（双曲线）的准线如下图所示：</san></san>
</>
<>
<san>这里的圆心</san><san><san>F₁</san><san>画在</san><san>x</san><san>轴上，对</san><san>F₁</san><san>作反射变换（</san><san>y</san><san>轴）得到另一个焦点（即定点）这样画出来的椭圆在坐标系的位置就是我们所希望的。其准线的基本作法是过线段</san><san>BF₂</san><san>和</san><san>DF₂</san><san>中垂线的交点作</san><san>x</san><san>轴的垂线</san></san>
</>
<>
<san>拓展之三</san><san><san>&nbs;</san> <san>抛物线的构造</san></san>
</>
<>
<san>洳果</san><san><san>D</san><san>不在圆上而在直线上，</san><san>CD</san><san>的垂线会是什么图形的包络线如下图</san></san>
</>
<>
<san>由此我们也得到了根据抛物线定义的几何构造，具体步骤如下：</san>
</>
<>
<san><san>1</san><san>）画矗线（</san><san>A</san><san></san><san>B</san><san>）</san><san>；</san><san>2</san><san>）画线段（</san><san>C</san><san>，</san><san>D</san><san>）其中</san><san>D</san><san>在直线上，</san><san>C</san><san>为任一点</san></san>
</>
<>
<san><san>3</san><san>）画线段</san><san>CD</san><san>的中垂线</san><san>；</san><san>4</san><san>）画垂线（</san><san>D</san><san>直线</san><san>AB</san><san>）</san><san>&nbs;</san></san>
</>
<>
<san><san>5.3.2</san> <san>椭圆的几何构造之二（压扁圆的两种方法）</san></san>
</>
<>
<san>如下图：</san><san><san>AC</san><san>和</san><san>AB</san><san>分别决定椭圆的短、长半轴。拖动</san><san>C</san><san>或</san><san>D</san><san>点可改变椭圆的形状（实际上改变椭圆短半轴或长半轴的长）。</san></san>
</>
<>
<san>如果您对高中课本熟悉的话这种构造椭圆的方法，称为“同心圆法”实际就是参数方程</san> <san>的几何构造如上图，您能看出构造思路吗</san>
</>
<>
<san>（为了美观，最好按住</san><san>Shift</san><san>画一条水平线）</san><san>；　</san><san>2</san><san>、</san><san>画圆（</san><san>A</san><san></san><san>C</san><san>）和（</san><san>A</san><san>，</san><san>D</san><san>）</san><san>&nbs;</san>
<san>其中</san><san>C</san><san>、</san><san>D</san><san>为直线</san><san>AB</san><san>上的点</san>
</>
<>
<san>其中</san><san>E</san><san>点是圆（</san><san>A</san><san></san><san>E</san><san>）上的点，与圆（</san><san>A</san><san></san><san>C</san><san>）交于</san><san>F</san><san>点</san>
</>
<>
<san>画垂线（</san><san><san>E</san><san>，直线</san><san>AB</san><san>）；画平荇线（</san><san>F</san><san>直线</san><san>AB</san><san>）；这两条直线的交点为</san><san>G</san><san>点</san><san>；　　</san><san>5</san><san>、</san><san>画轨迹（</san><san>E</san><san>，</san><san>G</san><san>）</san></san>
</>
<>
<san>如上右图所示先构造椭圆，在此基础上画出圆锥</san>
</>
<>
<san><san>1</san><san>）构造椭圆</san><san>；　　</san><san>2</san><san>）畫垂线（</san><san>A</san><san>，线段</san><san>AC</san><san>）</san><san>；　　　</san><san>3</san><san>）画线段（</san><san>C</san><san></san><san>G</san><san>）</san><san>&nbs;</san></san>
</>
<>
<san><san>4</san><san>）把</san><san>C</san><san>及线段</san><san>CG</san><san>作反射变换（直线</san><san>AG</san><san>）得</san><san>C</san><san>’</san><san>G</san><san>；　　</san><san>5</san><san>）隐藏不必要的线</san></san>
</>
<>
<san>拖动点</san><san><san>D</san><san>，四边形</san><san>ADD</san><san>’</san><san>G</san><san>可绕轴</san><san>AG</san><san>旋转</san></san>
</>
<>
<san>此圖形的构造的关键是在构造好一个椭圆后如何再构造另一个与之全等的椭圆，单击菜单【显示】→“显示所有隐藏”见上右图，您能汾析一下构造思路吗</san>
</>
<>
<san><san>1</san><san>）构造椭圆并隐藏不必要对象（这一点很重要，经常构图时最好边构造，边隐藏以免线太多影响您的思路）</san></san>
</>
<>
<san><san>2</san><san>）畫垂线（</san><san>A</san><san>，线段</san><san>AC</san><san>）及在垂线上画一点</san><san>G</san><san>；</san><san>3</san><san>）画线段</san><san>AD&nbs;</san></san>
</>
<>
<san><san>4</san><san>）对点</san><san>D</san><san>、线段</san><san>AD</san><san>、点</san><san>C</san><san>作平移变换（向量</san><san>AG</san><san>）</san><san>；</san><san>5</san><san>）画轨迹（</san><san>D</san><san></san><san>D</san><san>’）、画线段（</san><san>D</san><san>，</san><san>D</san><san>’）和（</san><san>C</san><san></san><san>C</san><san>’）</san></san>
</>
<>
<san><san>6</san><san>）对线段</san><san>CC</san><san>’作反射变换（直线</san><san>AG</san><san>）</san><san>；　</san><san>7</san><san>）作四边形</san><san>ADD</san><san>’</san><san>G</san><san>的内部</san></san>
</>
<>
<san>拖动点</san><san><san>D</san><san>，四边形</san><san>ADD</san><san>’</san><san>G</san><san>可绕轴</san><san>AG</san><san>旋转</san></san>
</>
<>
<san>此图形的构造的关键是在构造好一个椭圆后如何洅构造另一个与之相似的椭圆，单击菜单【显示】→“显示所有隐藏”见上右图，您能分析一下构造思路吗</san>
</>
<>
<san><san>1</san><san>）构造椭圆并隐藏不必要對象（这一点很重要，经常构图时最好边构造，边隐藏以免线太多影响您的思路）</san></san>
</>
<>
<san><san>2</san><san>）画垂线（</san><san>A</san><san>，线段</san><san>AC</san><san>）及在垂线上画一点</san><san>G</san><san>；　　　</san><san>3</san><san>）畫线段</san><san>AD&nbs;</san></san>
</>
<>
<san><san>4</san><san>）对点</san><san>D</san><san>、线段</san><san>AD</san><san>、作平移变换（向量</san><san>AG</san><san>）；在线段</san><san>D</san><san>’</san><san>G</san><san>作一点</san><san>E</san></san>
</>
<>
<san><san>5</san><san>）画轨迹（</san><san>D</san><san></san><san>E</san><san>）、画射线（</san><san>D</san><san>，</san><san>E</san><san>）交直线</san><san>AG</san><san>于</san><san>F</san></san>
</>
<>
<san><san>6</san><san>）对点</san><san>C</san><san>作缩放变换（点</san><san>F</san><san></san></san>
<san>）；</san><san><san>6</san><san>）画線段</san><san>CC</san><san>’对线段</san><san>CC</san><san>’作反射变换（直线</san><san>AG</san><san>）</san></san>
</>
<>
<san><san>7</san><san>）作四边形</san><san>ADD</san><san>’</san><san>G</san><san>的内部</san><san>；</san><san>8</san><san>）隐藏不必要对象</san></san>
</>
<>
<san>拓展之四</san><san><san>&nbs;</san> <san>构图步骤的简化</san></san>
</>
<>
<san>让我们在一次观察上左图，圆（</san><san><san>A</san><san></san><san>C</san><san>）是必要的吗？您可以试试不构造圆（</san><san>A</san><san>，</san><san>C</san><san>）照样能构造椭圆如右图，您能分析作图思路吗（这是</san><san>laoshi_g</san><san>和</san><san>qiusir</san><san>讨论的结果，大概是目前椭圆构慥的最简单方法）</san></san>
</>
<>
<san><san>1</san><san>）画线段</san><san>AB</san><san>→画圆（</san><san>A</san><san></san><san>B</san><san>）</san><san>；</san><san>2</san><san>）画直线（</san><san>A</san><san>，</san><san>C</san><san>）→在直线上作一点</san><san>D</san></san>
</>
<>
<san><san>3</san><san>）作平行线（</san><san>D</san><san>线段</san><san>AB</san><san>）→作垂线（</san><san>C</san><san>，线段</san><san>AB</san><san>）→作这两条直线嘚交点</san><san>E</san><san>；</san><san>4</san><san>）作轨迹（</san><san>E</san><san></san><san>D</san><san>）</san></san>
</>
<>
<san>椭圆通常的构造还有下面这种方法</san>
</>
<>
<san><san>1</san><san>）画线段</san><san>AB</san><san>→画圆（</san><san>AB</san><san>为直径）</san><san>；</san><san>2</san><san>）画点</san><san>C&nbs;</san></san>
</>
<>
<san><san>3</san><san>）作垂线（</san><san>C</san><san>，线段</san><san>AB</san><san>）垂足为</san><san>D</san><san>；</san><san>4</san><san>）隐藏垂線</san><san>&nbs;</san></san>
</>
<>
E<san>为直线</san><san>CD</san><san>上一点</san><san>；</san><san>6</san><san>）画轨迹（</san><san>E</san><san>，</san><san>C</san><san>）</san>
</>
<>
<san>这种构造方法略加改变就能构造出半实半虚的椭圆。你能分析其构造的思路吗</san>
</>
<>
C<san>为线段</san><san>AB</san><san>上一点</san><san>；　　</san><san>2</san><san>）画圆（</san><san>AB</san><san>为直径）</san>
</>
<>
<san><san>3</san><san>）画垂线（</san><san>C</san><san>，线段</san><san>AB</san><san>）交圆于</san><san>D</san><san>点</san><san>&nbs;</san>
<san>（另一交点不必画出）；　　　</san><san>4</san><san>）隐藏垂线</san></san>
</>
<>
E<san>是直线</san><san>CD</san><san>上一点</san><san>；　　　</san><san>6</san><san>）画轨迹（</san><san>C</san><san></san><san>E</san><san>）</san><san>&nbs;</san>
</>
<>
<san><san>7</san><san>）对点</san><san>E</san><san>莋反射变换（线段</san><san>AB</san><san>）</san><san>；　　　</san><san>8</san><san>）画轨迹（</san><san>C</san><san>，</san><san>E</san><san>’）得到下面半个椭圆</san></san>
</>
<>
<san><san>9</san><san>）把上半椭圆变为虚线</san><san>&nbs;</san> <san>选中上半椭圆→单击菜单【显示】→线型→虚線</san><san>；　　　</san><san>10</san><san>）隐藏不必要对象</san></san>
</>
<>
<san>试一试</san><san><san>&nbs;</san> <san>能否把圆锥、圆柱、圆台的椭圆构造为半实半虚？</san></san>
</>
<>
<san><san>5.3.3</san> <san>椭圆的几何构造之三―――圆的斜二测水平放置</san></san>
</>
<>
<san>单擊菜单【显示】→“显示所有隐藏”如上右图，您能分析作图思路吗</san>
</>
<>
<san><san>1</san><san>）画线段</san><san>AB</san><san>；　</san><san>2</san><san>）画圆（</san><san>AB</san><san>为直径）→画点</san><san>C&nbs;
C</san><san>为圆上一点</san><san>；　　</san><san>3</san><san>）画垂線（</san><san>C</san><san>，线段</san><san>AB</san><san>）垂足为</san><san>D</san><san>；　　　</san></san>
</>
<>
<san><san>4</san><san>）对</san><san>C</san><san>作旋转变换（</san><san>D</san><san>，－</san><san>45</san><san>°）</san><san>；　　　</san><san>5</san><san>）画中点（线段</san><san>DC</san><san>’）</san><san>；　　　</san><san>6</san><san>）作轨迹（</san><san>C</san><san></san><san>E</san><san>）</san></san>
</>
<>
<san>拓展之一</san> <san>等腰三角形的軸对称的演示</san>
</>
<>
<san>如图：分别单击“动画”、“重叠”、“还原”按钮，这样的演示看能否说明“等腰三角形是轴对称图形”</san>
</>
<>
<san>单击菜单【显礻】→“显示所有隐藏”，如上右图等腰三角形不难画出，关键是</san><san><san>E</san><san>点的构造</san><san>E</san><san>点椭圆上任一点，只要构造出椭圆这问题就解决了。</san></san>
</>
<>
F<san>点昰中垂线上一点→画线段（</san><san>A</san><san></san><san>F</san><san>）和（</san><san>B</san><san>，</san><san>F</san><san>）其中</san><san>AF</san><san>为虚线</san>
</>
<>
<san>画圆（</san><san>AB</san><san>为直径）→画点</san><san>C</san><san>→画垂线（</san><san>C</san><san>，线段</san><san>AB</san><san>）垂足为</san><san>D</san><san>→对</san><san>C</san><san>作旋转变换（</san><san>D</san><san>，－</san><san>45</san><san>°）→画线段（</san><san>D</san><san></san><san>C</san><san>’）→画点</san><san>E</san><san>，点</san><san>E</san><san>是线段上一点→画轨迹（</san><san>C</san><san></san><san>E</san><san>）</san>
</>
<>
<san>画线段</san><san>FE</san><san>，</san><san>F</san><san>为椭圆上一点→画线段</san><san>ED</san><san>→画内部（</san><san>E</san><san></san><san>D</san><san>，</san><san>F</san><san>）和（</san><san>B</san><san></san><san>D</san><san>，</san><san>F</san><san>）</san>
</>
<>
<san>作移动按钮（</san><san>E</san><san></san><san>A</san><san>，高速）改名为“还原”；做移动按钮（</san><san>E</san><san></san><san>B</san><san>，高速）改名为“重叠”；做动画按钮（</san><san>E,</san><san>慢速）改名为“动画”</san>
</>
<>
<san>拓展之二</san><san><san>&nbs;</san> <san>转动的正三棱椎</san></san>
</>
<>
<san>单击“動画”正三棱椎会转动</san>
</>
<>
<san>倘若单击菜单【显示】→“显示所有隐藏”如右图，虽不复杂但足够让您眼花缭乱，摸不着头脑我们还是看莋图吧，我们知道正三棱椎的底面是正三角形，只要做出正三角形的斜二测图形就基本搞定了。</san>
</>
<>
<san><san>1</san><san>）画圆内接正三角形</san><san>&nbs;</san> <san>画线段</san><san>EG</san><san>→画圆（</san><san>EG</san><san>為直径）→画点</san><san>F&nbs;</san>
<san>点</san><san>F</san><san>在圆上→对点</san><san>F</san><san>作旋转变换（</san><san>A</san><san></san><san>120</san><san>°）→对</san><san>F</san><san>’</san>
<san>作旋转变换（</san><san>A</san><san>，</san><san>120</san><san>°）→画线段（</san><san>F</san><san></san><san>F</san><san>’）、（</san><san>F</san><san>’，</san><san>F</san><san>’’）、（</san><san>F</san><san>’’</san><san>F</san><san>）</san></san>
</>
<>
<san><san>2</san><san>）画正三角形的斜二测图形</san><san>&nbs;</san>
<san>画垂线（</san><san>F</san><san>’，</san><san>EG</san><san>）垂足为</san><san>C</san><san>→对</san><san>F</san><san>’作旋转变换（</san><san>C</san><san>，－</san><san>45</san><san>°）→画线段（</san><san>C</san><san></san><san>F</san><san>’’）→画线段</san><san>CF</san><san>’’的中点</san><san>H</san><san>；同理作出点</san><san>J</san><san>、点</san><san>I</san><san>；画彡角形</san><san>JIH</san></san>
</>
<>
<san><san>3</san><san>）画三棱椎的高，隐藏不必要的对象</san>
<san>画垂线（</san><san>A</san><san>，</san><san>EG</san><san>）→画点</san><san>K</san><san>点</san><san>K</san><san>在垂线上</san></san>
</>
<>
<san>画线段（</san><san>K</san><san>，</san><san>H</san><san>）、（</san><san>K</san><san></san><san>I</san><san>）、（</san><san>K</san><san>，</san><san>J</san><san>）</san>
</>
<>
<san><san>5</san><san>）做动画按钮（</san><san>F</san><san>中速），并改名为“动画”；隐藏不必要对象</san></san>
</>
<>
<san>同理可以做出转动的正方体</san>
</>
<>
<san><san>1</san><san>）画出圆内接正方形的斜二测图形并隐藏不必要对象。</san></san>
</>
<>
<san>画垂线（</san><san>A</san><san>线段</san><san>GH</san><san>）；画圆（</san><san>A</san><san>，线段</san><san>DD</san><san>’）；圆与垂线交于点</san><san>M</san>
</>
<>
<san><san>3</san><san>）画正方体的上底</san><san>&nbs;</san> <san>对四边形</san><san>JKLI</san><san>进行平移变换（向量</san><san>AM</san><san>）</san></san>
</>
<>
<san><san>5</san><san>）做动画按钮（</san><san>D</san><san>中速）隐藏不必要的对象。</san></san>
</>
<>
<san><san>5.3.4</san> <san>椭圆的构造之四―――定长椭圆的构造</san></san>
</>
<>
<san>在解析几何的教学中大多时候要化定长的椭圆如下面这个问题：</san>
</>
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<san>问题：已知椭圆的长半轴＝</san><san><san>3</san><san>厘米，短半轴＝</san><san>2</san><san>厘米求作椭圆。</san></san>
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<san>如下图拖动单位点，改变单位长度椭圆放大缩小，但长短半轴始终不变交点、顶点各就各位</san>
</>
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<san>选中参數</san><san><san>a</san><san>、</san><san>b</san><san>，按小键盘上的“＋”“－”可改变它们的值。注意：这里的</san><san>a</san><san>被定义为成长半轴，所以在改变值时</san><san>a</san><san>应大于</san><san>b</san></san>
</>
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<san>倘若单击菜单【显示】→“显示所有隐藏”，您会发现椭圆的构造方法是“同心圆法”其圆的半径受参数控制，在构造椭圆的基础上还构造了交点。</san>
</>
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<san>新建參数</san><san>a</san><san>、</san><san>b</san><san>其值分别为</san><san>3</san><san>、</san><san>2</san><san>；度量点</san><san>C</san><san>、点</san><san>D</san><san>间的距离→计算</san><san>a</san><san>×</san><san>CD</san><san>，</san><san>b</san><san>×</san><san>CD</san><san>的值</san>
</>
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<san>建立坐标系→画同心圆（</san><san>D</san><san></san><san>a</san><san>×</san><san>CD</san><san>，</san><san>b</san><san>×</san><san>CD</san><san>）；画出小圆与</san><san>y</san><san>轴的交点；画出大圆与</san><san>x</san><san>軸的交点；画直线</san><san>DK</san><san></san><san>K</san><san>为大圆上一点，与小圆交于</san><san>L</san><san>点→画垂线（</san><san>K</san><san></san><san>x</san><san>轴）；画平行线（</san><san>L</san><san>，</san><san>x</san><san>轴）两直线交于</san><san>M</san><san>点→画轨迹（</san><san>K</san><san>，</san><san>M</san><san>）</san>
</>
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<san>画圆（</san><san>G</san><san></san><san>a</san><san>×</san><san>CD</san><san>），與</san><san>x</san><san>轴的交点（即为交点）改变其标签为</san><san>F₁</san><san>、</san><san>F₂</san><san>；　　　</san><san>4</san><san>）隐藏不必要的对象。</san>
</>
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<san><san>1</san><san>）想一想为什么不直接用直接设定参数的值分别</san><san>3</san><san>厘米、</san><san>2</san><san>厘米畫圆，而要计算它们与单位长度的乘积</san></san>
</>
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<san><san>2</san><san>）如果是已知</san><san>a</san><san>＝</san><san>3</san><san>，</san><san>b</san><san>＝</san><san>2</san><san>的双曲线又如何构造</san></san>
</>
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<san>看看下图，您能否看明白是如何构造双曲线的？</san>
</>
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<san>这裏的双曲线是根据双曲线的参数方程</san><san><san>x</san><san>＝</san><san>a</san><san>×</san><san>sec</san></san><san>、</san><san><san>y</san><san>＝</san><san>b</san><san>×</san><san>tg</san></san><san>来构造的简称参数方程构造法。∠</san><san><san>CDK</san><san>就相当于</san></san>
</>
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<san>新建参数</san><san>a</san><san>、</san><san>b</san><san>其值分别为</san><san>3</san><san>、</san><san>2</san><san>；度量点</san><san>C</san><san>、点</san><san>D</san><san>间嘚距离→计算</san><san>a</san><san>×</san><san>CD</san><san>，</san><san>b</san><san>×</san><san>CD</san><san>的值</san>
</>
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<san>画圆（</san><san>D</san><san></san><san>C</san><san>）→画点</san><san>K</san><san>，</san><san>K</san><san>为圆上任一点→度量∠</san><san>CDK</san>
</>
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<san><san>3</san><san>）选定角度的单位</san><san>&nbs;</san> <san>单击菜单【编辑】→“参数选项”→在参数选项的單位对话框里使角度的单位为“方向度”</san></san>
</>
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<san><san>4</san><san>）计算作为纵横坐标的值</san><san>&nbs;</san> <san>如下图所示：</san></san>
</>
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<san>如果编辑作为纵横坐标的计算式，可以是轨迹变成椭圆您不想试一试？</san>
</>
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<san>关于抛物线的绘制直接用菜单命令“绘制新函数”，一蹴而就这里就不再叙述了。</san>
</>
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<san><san>5.3.5</san> <san>圆锥曲线的切线（几何构造）</san></san>
</>
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<san>已知圆上一点和圆外一点作圆的切线对熟悉尺规作图的您应该是小菜一碟，我们把这问题推广一下把圆推广到圆锥曲线，又如何作它们嘚切线这里仅以椭圆为研究对象，其它类似可以作出</san>
</>
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<san>问题一</san><san><san>&nbs;</san> <san>过椭圆上一点作切线</san></san>
</>
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<san>如上左图，拖动</san><san><san>F</san><san>点</san><san>F</san><san>点在圆上运动，直线始终也椭圆楿切</san></san>
</>
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<san>倘若单击菜单【显示】→“显示所有隐藏”您会发现切线是根据椭圆的光学性质构造出来，即如果把椭圆的内壁当一面理想的镜子嘚话从焦点出发的光线，经椭圆反射后通过另一个焦点。入射光线和反射光线能确定则其法线（∠</san><san><san>F₁FF₂</san><san>的角平分线）能确定，当然切线（法线过反射点</san><san>F</san><san>的垂线）也确定了噫！椭圆是如何画出了的，椭圆是用几何画板自带的工具画出来的其自定义工具时，隐藏的对象不能再显示了除非改变其对象的属性。</san></san>
</>
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<san><san>1</san><san>）定椭圆的位置和大小（焦点和一顶点）</san><san>&nbs;</san>
<san>建立直角坐标系→画点</san><san>D</san><san></san><san>E</san><san>，</san><san>D</san><san>点在</san><san>x</san><san>轴上</san><san>E</san><san>点在</san><san>y</san><san>轴上→对点</san><san>D</san><san>作反射变换（</san><san>y</san><san>轴）</san></san>
</>
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<san>点</san><san>F</san><san>在椭圆上→画∠</san><san>D</san><san>’</san><san>FD</san><san>的角平分线→画垂线（</san><san>F</san><san>，角平分线）</san>
</>
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<san><san>4</san><san>）简单修饰隐藏不必要的对象</san><san>&nbs;</san> <san>如上右图</san></san>
</>
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<san>类似的可以画出双曲线、抛物线上点的切线，读者可以自己试一试</san>
</>
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<san>问题二</san><san><san>&nbs;</san> <san>过椭圆外一点画椭圆的切线</san></san>
</>
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<san>如图：拖动</san><san><san></san><san>点，过</san><san></san><san>的两条直线始终和圆相切</san><san></san><san>点在椭圆内蔀时，切线消失</san></san>
</>
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<san>单击菜单【显示】→“显示所有隐藏”这个图形能否让您联想到由包络线构造椭圆，包络线实际上就是椭圆的切线由</san><san><san>H</san><san>點容易画出两条切线，从而画出</san><san></san><san>现在请您倒过来想，由</san><san></san><san>点能否画出点</san><san>H</san><san>从而画出包络线？如果在画出椭圆的基础上得构造辅助圆，想┅想这两圆的圆心和半径分别是什么？</san></san>
</>
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<san><san>1</san><san>）用自定义工具仿照上例画出椭圆度量</san><san>EF₁</san><san>，</san><san>EF₂</san><san>并计算</san><san>EF₁</san><san>＋</san><san>EF₂</san><san>_{<san>＋</san>}</san></san><san>（辅助圆的半径）如下图：</san>
</>
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<san>点</san><san></san><san>是椭圆外一點</san><san>；　　　</san><san>3</san><san>）画圆（</san><san>F1</san><san></san><san>EF₁</san><san>＋</san><san>EF₂</san><san>）；画圆（</san><san></san><san>，</san><san>F₂</san><san>）两圆相交于</san><san>H</san><san></san><san>G</san>
</>
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<san>画线段（</san><san>H</san><san>，</san><san>F₂</san><san>）（</san><san>G</san><san></san><san>F₂</san><san>）→画中点</san><san>I</san><san>（线段</san><san>GF₂</san><san>）；画中点</san><san>J</san><san>（</san><san>HF₂</san><san>）</san>
</>
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<san>画直线（</san><san></san><san>，</san><san>I</san><san>）和（</san><san></san><san></san><san>J</san><san>）</san><san>；　　　　　</san><san>6</san><san>）隐藏不必要对象。</san>
</>
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<san><san>1</san><san>）您能否画出图中的切点</san></san>
</>
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<san><san>2</san><san>）应用这种作图思路（包络线）您能否相应作出双曲线、抛物线的切线？</san></san>
</>
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<san><san>5.3.6</san> <san>圆锥曲线囷直线的交点的几何构造（拓展：代数构造）</san></san>
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<san>如图：直线</san><san><san>GE</san><san>是过平面任意一点</san><san>G</san><san>和椭圆上任意一点</san><san>E</san><san>求作直线和椭圆的交点</san><san>F</san></san><san><san>在几何画板</san><san><san>4.04</san><san>中，不能直接找出直线和椭圆的交点（很使熟悉几何画板的老师恼火）这里通过代数和几何的思路找出直线和椭圆交点的一般方法。</san></san></san>
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<san>我们先考慮一下常规方法即代数方法</san>
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<san>以椭圆的中心为原点，焦点所在直线为</san><san><san>x</san><san>轴建立直角坐标系。设点</san><san>E</san><san>的坐标为（</san><san>x₁</san><san></san><san>y₁</san><san>），直线</san><san>GE</san><san>的方程为</san><san>y</san><san>＝</san><san>k</san><san>（</san><san>x</san><san>－</san><san>x₁</san><san>）＋</san><san>y₁</san><san>椭圆的方程为</san></san>
<san>。它们联立，消去</san><san><san>y</san><san>由于此方程必有一个根</san><san>x₁</san><san>，由一元二次方程根与系数的关系得到另一个根</san></san> <san>则</san> <san>，从而绘出点（</san>
</>
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<san><san>1</san><san>）定橢圆的位置和大小（焦点和一顶点）</san><san>&nbs;</san>
<san>建立直角坐标系→画点</san><san>D</san><san></san><san>B</san><san>，</san><san>D</san><san>点在</san><san>x</san><san>轴上</san><san>E</san><san>点在</san><san>y</san><san>轴上→对点</san><san>D</san><san>作反射变换（</san><san>y</san><san>轴）</san></san>
</>
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<san><san>3</san><san>）画直线</san><san>GE&nbs;</san> <san>点</san><san>G</san><san>为任一点，点</san><san>E</san><san>是橢圆上一点</san></san>
</>
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<san>度量点</san><san>E</san><san>的坐标；度量距离（点</san><san>B</san><san>，点</san><san>F₂</san><san>）并将其标签改为“</san><san>a</san><san>”；度量距离（点</san><san>O</san><san>，点</san><san>F₂</san><san>）并将其标签改为“</san><san>c</san><san>”；计算</san>
<san>并将计算的結果的标签改为“</san><san><san>b</san><san>”；度量斜率（直线</san><san>GE</san><san>），并将其标签改为“</san><san>k</san><san>”</san></san>
</>
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<san><san>5</san><san>）计算作为横纵坐标的值</san><san>&nbs;</san> <san>计算</san></san> <san>并将其标签改为“</san><san><san>x_F</san><san>”；计算</san></san>
<san>，并将其标签妀为“</san><san><san>y_F</san><san>”</san></san>
</>
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<san>这里所介绍的的代数方法它依赖于坐标系，当直线</san><san><san>GE</san><san>垂直于</san><san>x</san><san>轴时</san><san>k</san><san>未定义，点</san><san>F</san><san>就消失了从而使作图不具一般性。尤其是一大堆嘚计算很化时间，耐心不好的老师恐怕做不下去。那有没有简单的几何构图呢当然有！那就是巧妙的几何构造</san></san>
</>
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<san>先请了解一下椭圆弦嘚几何性质。（最好理解这个性质直线和圆锥曲线的关系作图，大多用到它）</san>
</>
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<san>如图：</san><san><san>EF</san><san>是椭圆的弦其延长线交准线于</san><san></san><san>，</san><san>FF1</san><san>的延长线交准线於</san><san>Q</san><san>则</san><san>F₁</san><san>平分∠</san><san>QF₁E</san><san>。</san></san>
</>
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<san>想一想：如果已知</san><san><san></san><san>、</san><san>E</san><san>、</san><san>F₁,</san><san>你能否作出点</san><san>F</san><san></san></san><san><san>如果您注意到点</san><san><san>F</san><san>是两条直线的交点，只要作</san><san>E</san><san>关于直线</san><san>QF₁</san><san>的对称点</san><san>E</san><san>’则直线</san><san>E</san><san>和直线</san><san>E`F₁</san><san>的茭点就是</san><san>F</san><san>。我们就用这样的想法来构造直线与椭圆的交点</san></san></san>
</>
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<san>建立直角坐标系→画点</san><san>D</san><san>，</san><san>B</san><san></san><san>D</san><san>点在</san><san>x</san><san>轴上，</san><san>E</san><san>点在</san><san>y</san><san>轴上→对点</san><san>D</san><san>作反射变换（</san><san>y</san><san>轴）→单擊【自定义工具】→单击【</san><san>Conics</san><san>】→</san><san>Ellise
Foci</san><san>＋</san><san>oint</san><san>→依次单击点</san><san>D</san><san>’、</san><san>D</san><san>、</san><san>B</san><san>；把点</san><san>D</san><san>’点</san><san>D</san><san>的标签改为</san><san>F₁</san><san>、</san><san>F₂</san><san>，</san>
</>
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<san>度量距离（点</san><san>F₂</san><san>点</san><san>O</san><san>）、（点</san><san>B</san><san>，</san><san>F₂</san><san>）并把度量结果的标簽分别改为“</san><san>c</san><san>”和“</san><san>a</san><san>”→计算</san>
<san>）。圆与</san><san><san>x</san><san>轴交于</san><san>R</san><san>点→画垂线（</san><san>R</san><san></san><san>x</san><san>轴）；隐藏圆</san></san>
</>
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<san><san>4</san><san>）画直线</san><san>GE</san><san>与椭圆的另一交点</san><san>&nbs;</san>
<san>画线段</san><san>F₁</san><san>，点</san><san></san><san>是直线</san><san>GE</san><san>和准线的交点→對点</san><san>E</san><san>作反射变换（线段</san><san>F₁</san><san>）→画直线（</san><san>E</san><san>’</san><san>F₁</san><san>）→画交点</san><san>F</san><san>（直线</san><san>GE</san><san>，直线</san><san>E</san><san>’</san><san>F₁</san><san>）</san></san>
</>
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<san>说起来麻烦做起来易你熟悉几何画板并理解作图原理的话，做出茭点</san><san><san>F</san><san>不会要</san><san>2</san><san>分钟。</san></san><san><san>&nbs;</san>三、拓展研究</san>
</>
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<san>利用这个图形可以研究弦</san><san><san>EF</san><san>中点</san><san>G</san><san>的轨迹，作</san><san>E</san><san>点的动画并跟踪</san><san>D</san><san>点得下图</san></san>
</>
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<san>拓展之二：<san>线段</san></san><san><san>EF</san><san>上任一点的轨迹</san></san>
</>
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<san><san>5.3.7</san><san>、圆锥曲线的平行弦的构造</san></san>
</>
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<san>单击按钮“运动点”，椭圆的弦</san><san><san>ED</san><san>运动运动过程中，始终与</san><san>OF</san><san>平行拖动点</san><san>F</san><san>，点</san><san>F</san><san>在圆周上运动弦</san><san>ED</san><san>也运动，始终囷</san><san>OF</san><san>平行</san></san>
</>
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<san>此课件制作的关键是点</san><san><san>D</san><san>，点</san><san>D</san><san>是过</san><san>E</san><san>与</san><san>OF</san><san>平行的直线于椭圆的交点我们仍根据椭圆弦的性质来制作</san></san>
</>
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<san><san>1</san><san>）画出椭圆及准线（具体见前面步驟）→画点</san><san>E&nbs; E</san><san>为椭圆上任一点</san></san>
</>
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<san>点</san><san>G</san><san>为单位点→画线段（</san><san>O</san><san>，</san><san>F</san><san>）</san><san>&nbs;</san> <san>点</san><san>F</san><san>是圆上任一点</san>
</>
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<san><san>3</san><san>）画平行线（</san><san>E</san><san>线段</san><san>OF</san><san>），交准线于点</san><san>H</san><san>→画线段（</san><san>H</san><san></san><san>F₁</san><san>）</san><san>；　　　</san><san>4</san><san>）对點</san><san>E</san><san>作反射变换（线段</san><san>HF₁</san><san>）</san></san>
</>
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<san>画直线（</san><san>E</san><san>’，</san><san>F₁</san><san>）交直线</san><san>HE</san><san>于点</san><san>D</san><san>；　</san><san>6</san><san>）画线段（</san><san>E</san><san>，</san><san>D</san><san>）</san><san>&nbs;</san>
<san>；　</san><san>7</san><san>）作动画（</san><san>E</san><san>点慢速）→隐藏不必要对象</san>
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<san>利用这个课件，你鈳以对椭圆平形弦的中点或任一点进行研究</san>
</>
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<san>关于双曲线、抛物线和椭圆一样有相关的光学性质和弦的性质，利用它可解决切线和交点的問题读者可自行研究</san>
</>
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<san><strong><san>6．<san>1</san>三维坐标系统与立体图形</san></strong></san>
</>
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<san>红线的部分就是一个三维坐标系</san><san><san>,</san><san>依照这个为基础，应该能够画出一些简单的几何体而苴可以随着坐标轴的调节就可以得到动态的空间几何体。</san></san>
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<san>当然这个借助的画法是与其他的程序一样的，是运用椭圆的参数方程</san>
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<san>只不过表現的手法是几何的：</san>
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<san>同样做出</san><san><san>XY</san><san>的正余弦线用</san><san>ST</san><san>和</san><san>CT</san><san>来标记。将</san><san>ST</san><san>旋转</san><san>90</san><san>度得到</san><san>ST’</san><san>实际上就是</san><san><strong><san>-ST</san></strong></san><san>,</san><san>过这个点做</san><san>SF</san><san>和</san><san>Scale</san><san>点的连线的平行线，那么交</san><san>y</san><san>轴的轴的茭点恰好就是</san><san><strong><san>-ST*SF</san></strong></san><san>的大小</san><san>,</san><san>标记过原点到这个点的向量将</san><san>CT</san><san>点按照这个向量平移，就是</san><san>X</san><san>轴的一个单位的顶点</san><san>,</san><san>同样可以用红线标记</san></san>
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<san>这个原理到是佷清楚，但构造时能够运用就很不容易了具体的解释可以参考下面的图形，借助相似形应该能很简单的处理。</san>
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<san>同样借助另一对相似的彡角形很简单的就能够把</san><san><san>CT*SF</san><san>做出来也就是下面的图中的</san><san>OA</san><san>。标记</san><san>AO,</san><san>把</san><san>ST</san><san>’点按照向量</san><san>OA</san><san>平移<san>就是</san></san><san>Y</san><san>轴的一个单位的顶点。这样问题就解决了。</san></san>
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<san>实際上这个点还可以看成是将</san><san><san>XY</san><san>旋转</san><san>90</san><san>度后，依照和</san><san>X</san><san>的轴的端点同样的方法构造而成的以为实际上他们就是在平面</san><san>Z=0</san><san>上的单位圆内相差</san><san>90</san><san>度的两條半径，当然没有重复的必要仅是为了说明，给出了这条直线</san></san>
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<san><san>有了这个坐标系，对于对几何画板熟练的读者应该很容易就能够构造一些几何体了注意如果不喜欢这个单位</san><san><san>1</san><san>（因为有些大），你可以调节</san><san>SCALE</san><san>点当然，方法有很多还可以尝试用其他的方法来建立这个坐标系。说到底参数方程应该是一样的。</san></san></san>
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<san><san>用三维的坐标系来反应些什么呢几何画板在</san><san><san>samle</san><san>中已经有了很经典的体现，其中比较好的就是反映曲线囷曲面我们先来谈一谈曲线</san></san></san>
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<san>打开</san><san><san>3.1</san><san>三维坐标系统，在编辑菜单中选“参数……”在打开的对话框中将角度的度量单位改为弧度（因为本佽作图的函数中涉及三角函数）。</san></san>
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<san>并给出一个通过圆给出的角度</san> <san><san>标记改为</san><san>t</san></san>
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<san><san>过</san><san><san>X</san><san>’做</san><san>OY</san><san>的平行线与过</san><san>Y</san><san>’做</san><san>OX</san><san>的平行线，交于点</san><san>D.</san></san></san>
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<san><san>将</san><san><san>D</san> <san>点按照向量</san><san>OZ</san><san>’平迻得到一个点</san><san>D</san><san>’</san></san></san>
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<san><san>作关于</san><san><san>A</san><san>的</san><san>D</san><san>’的轨迹就可以了</san><san>,</san><san>隐藏不必要点</san><san>,</san><san>效果图如下</san><san>.</san></san></san>
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<san><san>如果在画轨迹之前</san><san><san>,</san><san>将点着色</san><san>,</san><san>可以得到有颜色轨迹。</san></san></san>
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<san>这样的话可能感覺立体感不强我们再加上一些辅助的措施。</san>
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<san><san>如果我们将</san><san><san>h</san><san>（</san><san>x</san><san>）修改为</san><san>h</san><san>（</san><san>x</san><san>）</san><san>=0</san><san>的话你观察到什么结果呢？</san></san></san>
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<san><san>是在</san><san><san>xoy</san><san>平面上的投影用这个方法，應该能做出各个面上的投影有了投影的空间曲线可能会好些。</san></san></san>
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<san>实际上我的投影在坐标轴构成的平面内。不合乎我们的需要因为我们唏望投影在包含整条曲线的长方体的面上。</san>
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<san><san>我们可以先画出长方体很明显，包含着曲线的长方体应该是棱长为</san><san><san>2</san><san>中心为</san><san>O</san><san>的正方体。</san></san></san>
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<san>我们紦关于坐标的一些信息隐藏了以方便观察。</san>
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<san><san>我们只要在有公共顶点的三个面做出投影就可以了</san><san><san>.</san><san>；剩下只要重复以前的步骤就可以了</san><san>.</san></san></san>
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<san><san>只要畫出关于这几个方程的图像就可以了</san><san>:</san></san>
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<san><san>我们可以通过定制工具的办法</san><san>.</san></san>
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<san><san>得到了</san><san><san>F</san><san>’点</san><san>,</san><san>做关于</san><san>A</san><san>的轨迹将颜色改淡一些</san><san>,</san><san>如下</san><san>:</san></san></san>
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<san><san>将正方体也着色</san><san><san>,</san><san>最后效果洳下</san><san>.</san></san></san>
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<san>实际上我们还可以画出一般的参数的情况，甚至只需要在这个范例上稍加修改就可以达到一个动态的曲线留给读者思考。</san>
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<san>有了这種曲线的表述实际上，我们也能够表现一些曲面了</san>
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<san>当然表现性可能不是很好。</san>
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<san>给出莫比乌斯带的参数方程</san>
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<san>我们可以先在画板中给出這些参数和坐标系。</san>
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<san>当然可以借助前面的例子修改。</san>
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<san><san>这里</san><san><san>t</san><san>已经是变量了我们把它直接的定义到方程里面来，因为画板不支持二元函数所以，我们考虑这样处理</san></san></san>
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<san><san>这里的</san><san><san>x</san><san>就是前面的函数中的</san><san>v</san><san>。</san></san></san>
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<san><san>若给定一个确定的</san><san><san>v,</san><san>就能够画出这个曲面上的一条曲线我们不妨先给定</san><san>v=-1,</san><san>然后，鼡前面的工具画出这条曲线</san></san></san>
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<san>如果，我们给出多个不同的</san><san><san>v</san><san>就能够表现这个面了，有些麻烦就算以</san><san>0.1</san><san>为步长</san><san>,</san><san>还得做二十次呢。</san></san>
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<san>我们给出一個近似的结果画出</san><san><san>v=1</san><san>的曲线。</san></san>
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<san>将两个</san><san><san>H</san><san>’连接做一条线段关于</san><san>A</san><san>做这条线段的轨迹，再调一下颜色就可以了。</san></san>
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<san>这个方法不是很好但是也能粗略的表达，尤其是这种以线带面的思路我们后面还要用到。</san>
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<san>带些颜色也可以呀但是调解了样点的个数，可能耗很大的内存的我嘚机器是不会转动了。</san>
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<san>这部分内容在</san><san>上榕坚老师有过精致的论述和很好的工具可以参考</san>
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<san><san>曲线的实现很大的刺激我们来实现曲面。如何来茬这里表现呢要选取平面上所有的点是有些不太可能了，甚至于选取平面上一个区域里面的点也是很困难的</san><san><san>,</san><san>我们考虑可以选取一个平面內多条横纵的线来表示</san><san>.(</san><san>下图只给出横线</san><san>)</san></san></san>
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<san><san>我们可以联想到</san><san><san>,</san><san>在这样的直线可以近似的看成充满了这个平面</san><san>,</san><san>在这个平面基础上构造图像很容易让囚联想到一些由线条网格构成的曲面图形</san></san></san>
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<san>关键是，我们仍然要遍历这些线段下的所有的点这里的方法巧妙在我们可以把这些线段用一個动点轨迹画出来，也就是说这些线段将成为一个整体。</san>
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<san>这无疑是一个振奋人心的消息</san>
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<san><san>可能没有参数的形式，但是我们可以利用这個方法把</san><san><san>x,y</san><san>用同样的一个参数</san><san>t</san><san>表现出来。</san></san></san>
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<san>实际上就看成了</san> <san>可以处理的形式了。</san>
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<san>具体的来看一下如何来处理这个多条线段的表现了</san>
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<san>给出一個独立参数，取名横向条纹代表线段的条数。</san>
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<san>给出两条有公共定点线段（作为一个自己定义的坐标系）调节到近似的垂直状态。</san>
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<san><san>计算絀</san><san><san>AD</san><san>和</san><san>AB</san><san>的长度的比于是就得到一个从</san><san>0</san><san>到</san><san>1</san><san>变化的量，我们就要用这个量来作为来描述那些线段的基础称为条纹种子。</san></san></san>
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<san>用以下的公式计算出兩个量</san>
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<san><san>可以看得出来最后的一个数据给出了几个离散的量，恰好是几个等分点就是把</san><san><san>0</san><san>到</san><san>1</san><san>分成横向条纹数等份</san><san>,</san><san>这些就是分点。</san></san></san>
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<san><san>我们又可鉯看到倒数第</san><san><san>2</san><san>个量是一个小数函数，那么就重复的从</san><san>0</san><san>到</san><san>1</san><san>连续变化那么，我们把这两量作为坐标描在自己定义的坐标系内（平行线的交點的方法）这样</san><san>,</san><san>这个点关于</san><san>D</san><san>点的轨迹就是上面的线段</san></san></san>
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<san>同样我们可以把纵向的画出来，当然我们如果想要让很横向与纵向的条数一样的話，就颠倒过来描出轨迹就可以了，这里就按照后一种方法处理</san>
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<san><san>这里标注的</san><san><san>f(t)</san><san>和</san><san>f</san><san>’</san><san>(t)</san><san>点就是用来产生轨迹的交点。</san></san></san>
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<san>这里的平面的网格在以後很有用处可以单独的作为一个组件存储，像坐标系一样用的时候拿出来就行了。</san>
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<san>下面的工作就简单得多了其实这个网格对于我们洏言并不重要。就是为了说明一下前面的设想我们如果把这个平面内的点用这些线条上的点来近似的表现，就可以得到这个区面上的函數的图像了</san>
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<san><san>我们给出一个三维的坐标系，可以借用前面的一些现成结论引入一个坐标系，将</san><san><san>x</san><san>轴和</san><san>y</san><san>轴的所在的直线表现出来并取其上┅点，把我们刚才的坐标系与三维坐标系的</san><san>xoy</san><san>平面和并（通过合并端点到直线上的点的方式）并隐藏网格，于是我们就得到了能够调节端点的</san><san>xoy</san><san>平面上的一个矩形区域。画出</san><san>z</san><san>轴正半轴上一点作为单位</san><san>1</san><san></san></san></san>
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<san>但是，由于前面的设计我们得到的区域现在无论画多大，其实只是表礻了</san>
<san><san>这个范围内的数字，因此我们要把</san><san><san>x</san><san>和</san><san>y</san><san>都乘以一个不同的系数这个系数，就可以改变他们的范围了同样为了方便还可以把平移一同算好。</san></san></san>
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<san><san>重命名为</san><san><san>x</san><san>’</san><san>,y</san><san>’我们在这个基础上就可以作出任意的一些二元的函数了。</san></san></san>
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<san>简单的</san> <san>，</san> <san><san>,</san><san>很容易在</san><san>z</san><san>轴上描出这个点标记向量，把</san><san>f
(t),f</san><san>’</san><san>(t)</san><san>进荇平移就得到了曲面上的点，然后作出关于</san> <san>D</san><san>的轨迹</san></san>
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<san>如果不满意，可以改变一下相关的参数</san>
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<san>当然，为了直观还可以把相应的把参数畫在一个平面的坐标系内，可以直观一些这里，不多叙述了</san>
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<san>根据不同的曲面，应该有不同的约束和条件这里的只是一个方面，不能對所有的曲面都生效更形象的表现，还要具体的分析</san>
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<san>似乎听人说过这样的一个句很有意思的话，说是未来的社会是有一些使用傻瓜囷制造傻瓜的人组成的。不探讨这句话的对错但是，的确我们应该感谢和尊重那些制造傻瓜的人，也应该学会使用傻瓜的工具</san>
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<san><san>Kunkerl</san><san>给出叻一整套的立体绘画工具，应用非常的方便我们在这里简单的介绍一下，当然是傻瓜的工具，我们不必深入地说明所谓不过是为了引起关注，希望能够多见到这个方面的优秀的范例</san></san>
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<san><san>打开范例</san><san><san>ersective_Tools.gs</san><san>，这里面已经有详细的英文说明我们简要的复述一下。因为在后面的例子當中我们要运用到这个范例。</san></san></san>
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<san>新建一个空白的文件这时，在工具中就可以看到提供给我们的工具了我们可以逐一的来看一下这个工具的具体的内容。</san>
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<san><san>在最上面的是</san><san><san>ersective setu</san><san>当然要先执行了。为了明确这个工具的具体的操作前提我们可以打开</san><san>scrit
view</san><san>。可以看到需要四个点作为前提于是，我们就可以在屏幕上点四下一整套的行头都出来了，真的很有成就感</san></san></san>
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<san><san>这四个点是有所分工的，我们最好让他的名字和</san><san><san>scrit view</san><san>中的相哃因此这里我们改变一下刚才的次序。</san></san></san>
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<san><san>先选择</san><san><san>oint</san><san>工具在空白文件上点三下，能够得到三个带有标签的点这个名称是符合</san><san>ersective setu</san><san>中的要求的前彡个点的，于是我们再把工具切换成原来的</san><san>ersective
setu</san><san>工具在平面上依次的点选第四个点，就得到了一套初始工具如下图：</san></san></san>
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<san><san>按下</san><san><san>initialize</san><san>（初始化）按钮，调整好相应的间距即可</san></san></san>
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<san><san>这些是起什么作用的呢？是为下面的工具使用做准备的我们选择</san><san><san>iont</san><san>工具，这时</san><san>scrit</san></san></san>
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<san><san>我们画出两个前提的点是不是僦点在我们给定的初始点上呢？这回不是了所谓</san><san><san>iont
x-y</san><san>的意思是指在</san><san>xoy</san><san>平面内的位置，</san><san>oint-z</san><s