用圆近似抛物线的几何构造
2017-02-01 04:23
本 GeoGebra 范例演示如何利用圆来近似抛物线。构造过程涉及在抛物线上取点，作抛物线的切线与法线，同时构造相关圆的切线与法线，通过交点、向量、正交线、镜像与射线等几何操作，逐步展示圆弧与抛物线之间的逼近关系。该范例直观呈现了曲线局部可用密切圆（曲率圆）近似的微分几何思想，适合用于解析几何与曲线论的教学演示。
所属专题
圆锥曲线
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.307.0" id="2f69950b-fbe9-41c8-86ad-01a90b4ac08f" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="799" height="672" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.5156445556946183" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="481,115,700,550" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3,3" size="173" window="100,100,250,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="412" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="true" location="1" size="367" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="412" height="468"/>
<coordSystem xZero="51.38491463972247" yZero="239.97474518291347" scale="114.56425309864974" yscale="114.56425309864957"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="367" height="468"/>
<coordSystem xZero="214.99999999999994" yZero="314.9999999999999" scale="49.99999999999998" yscale="49.999999999999986"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<spreadsheetView>
<prefCellSize height="36"/>
</spreadsheetView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label ="c" exp="1"/>
<element type="numeric" label="c">
<value val="1.0"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="r" exp="(2 * c)" />
<element type="numeric" label="r">
<value val="2.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="circ" exp="(x - r)^(2) + y^(2) = r^(2)" type="conic" />
<element type="conic" label="circ">
<show object="true" label="false"/>
<condition showObject="bl_c"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="0.0" A3="0.0" A4="-2.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="circ"/>
<output a0="Qc"/>
</command>
<element type="point" label="Qc">
<show object="true" label="false" ev="8"/>
<condition showObject="¬bl_y"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.519956097199612" y="1.3451654343549702" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="PointIn">
<input a0="circ"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="2.4476514248713808" y="1.0631069524664174" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="circ" a1="y = y(P)" a2="1"/>
<output a0="Q'"/>
</command>
<element type="point" label="Q'">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_y"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-47" y="-12"/>
<labelMode val="0"/>
<coords x="0.3059505297608435" y="1.0631069524664174" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="boolean" label="bl_y">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="31" y="123"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="平行光"/>
</element>
<expression label="para" exp="y^(2) = ((4 * c) * x)" type="conic" />
<element type="conic" label="para">
<show object="true" label="false"/>
<condition showObject="bl_p"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="0.0" A1="1.0" A2="0.0" A3="0.0" A4="-2.0" A5="0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="If">
<input a0="bl_y" a1="Q'" a2="Qc"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-29" y="-11"/>
<labelMode val="0"/>
<coords x="0.3059505297608435" y="1.0631069524664174" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="F" exp="(c, 0)" type="point" />
<element type="point" label="F">
<show object="true" label="true"/>
<condition showObject="bl_p"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="O" exp="(r, 0)" type="point" />
<element type="point" label="O">
<show object="true" label="true"/>
<condition showObject="bl_c"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-33" y="30"/>
<labelMode val="0"/>
<coords x="2.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="P" a1="Q"/>
<output a0="il_c"/>
</command>
<element type="vector" label="il_c">
<show object="true" label="false"/>
<condition showObject="bl_c"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="14" y="3"/>
<labelMode val="0"/>
<coords x="-2.1417008951105374" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="P"/>
</element>
<command name="Tangent">
<input a0="Q" a1="circ"/>
<output a0="t_c" a1=""/>
</command>
<element type="line" label="t_c">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_t ∧ bl_c"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-35" y="-1"/>
<labelMode val="0"/>
<coords x="-1.6940494702391564" y="1.0631069524664174" z="-0.611901059521687"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="Q" a1="t_c"/>
<output a0="n_c"/>
</command>
<element type="line" label="n_c">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_t ∧ bl_c"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0631069524664174" y="-1.6940494702391564" z="2.126213904932835"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Mirror">
<input a0="P" a1="n_c"/>
<output a0="P'"/>
</command>
<element type="point" label="P'">
<show object="false" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.2373801122633012" y="-0.8654473394726441" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="Q" a1="P'"/>
<output a0="rQP'"/>
</command>
<element type="ray" label="rQP'">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.9285542919390615" y="0.9314295825024577" z="-1.5802514721825596"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="circ" a1="rQP'" a2="2"/>
<output a0="R"/>
</command>
<element type="point" label="R">
<show object="true" label="true"/>
<condition showObject="bl_c"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.7794408704574876" y="-1.987801215005024" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="Q" a1="R"/>
<output a0="rl_c"/>
</command>
<element type="vector" label="rl_c">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_c"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="9" y="-53"/>
<labelMode val="0"/>
<coords x="1.473490340696644" y="-3.0509081674714413" z="0.0"/>
<lineStyle thickness="6" type="30" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="Q"/>
</element>
<command name="Intersect">
<input a0="para" a1="Line[P, Q]"/>
<output a0="Q'_1" a1="Qp"/>
</command>
<element type="point" label="Q'_1">
<show object="true" label="true"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="Qp">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_p"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-33" y="-20"/>
<labelMode val="0"/>
<coords x="0.2825490980956082" y="1.0631069524664172" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="P" a1="Qp"/>
<output a0="il_p"/>
</command>
<element type="vector" label="il_p">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_p"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.1651023267757727" y="-2.220446049250313E-16" z="0.0"/>
<lineStyle thickness="4" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="P"/>
</element>
<command name="Tangent">
<input a0="Qp" a1="para"/>
<output a0="t_p" a1=""/>
</command>
<element type="line" label="t_p">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_t ∧ bl_p"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.0" y="1.0631069524664172" z="-0.5650981961912164"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="Qp" a1="t_p"/>
<output a0="n_p"/>
</command>
<element type="line" label="n_p">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_t ∧ bl_p"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0631069524664172" y="-2.0" z="2.4265938155313913"/>
<lineStyle thickness="2" type="20" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="para" a1="Line[Qp, Mirror[P, t_p]]" a2="2"/>
<output a0="Rp"/>
</command>
<element type="point" label="Rp">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="215" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.5392078995828955" y="-3.762556524270643" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="Qp" a1="Rp"/>
<output a0="rl_p"/>
</command>
<element type="vector" label="rl_p">
<show object="true" label="false" ev="4"/>
<condition showObject="bl_p"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.2566588014872875" y="-4.82566347673706" z="0.0"/>
<lineStyle thickness="6" type="30" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="Qp"/>
</element>
<element type="boolean" label="bl_t">
<value val="false"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="32" y="164"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="顯示切線法線"/>
</element>
<element type="boolean" label="bl_c">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="165" y="204"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="圓"/>
</element>
<element type="boolean" label="bl_p">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="34" y="201"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
<caption val="拋物線"/>
</element>
<expression label="text1" exp=""移動 P 點的位置觀察 光線在拋物線與圓的反射情況 何時可用圓來近似拋物線？""/>
<element type="text" label="text1">
<show object="true" label="true" ev="43"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<absoluteScreenLocation x="31" y="31"/>
</element>
</construction>
</geogebra>