正方体的任意截面切割与分离演示
2020-04-18 12:00
本GeoGebra范例演示了如何使用任意平面切割正方体，并将其分割为可分离的两部分。通过构造正方体的顶点、棱和面，结合平面与相交命令，动态生成截面多边形。用户可调整平面参数或控制点，观察不同位置和角度的截面形状（如三角形、四边形、五边形或六边形），并直观展示切割后两部分的几何结构。此范例适用于立体几何教学，帮助学生深入理解空间几何体的截面性质，提升空间想象能力。
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<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<expression label="text1" exp=""说明：执行l2,l3得到所有红色曲面""/>
<element type="text" label="text1">
<show object="true" label="false" ev="37"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<absoluteScreenLocation x="59" y="116"/>
</element>
<command name="OrthogonalVector">
<input a0="r"/>
<output a0="u"/>
</command>
<element type="vector3d" label="u">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<coords x="-0.0991070896614108" y="0.5" z="1.0" w="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
</element>
<expression label="o" exp="(I * u)" />
<element type="numeric" label="o">
<value val="0.5"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Sequence">
<input a0=""Surface[" + (l1(i)) + (("/(" * l1(i))) + "*u>=o )+a 2 u,s,0,1,t,0,1]"" a1="i" a2="1" a3="6"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0=""Surface[" + (l1(i)) + (("/(" * l1(i))) + "*u<= o )-a 2 u,s,0,1,t,0,1]"" a1="i" a2="1" a3="6"/>
<output a0="l3"/>
</command>
<element type="list" label="l3">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Surface">
<input a0="(s, 1, t) / (((s, 1, t) * u) ≥ o) + ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="b"/>
</command>
<element type="surfacecartesian3d" label="b">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(1, t, s) / (((1, t, s) * u) ≥ o) + ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="c"/>
</command>
<element type="surfacecartesian3d" label="c">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(s, t, 1) / (((s, t, 1) * u) ≥ o) + ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="d"/>
</command>
<element type="surfacecartesian3d" label="d">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(s, 0, t) / (((s, 0, t) * u) ≥ o) + ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="e"/>
</command>
<element type="surfacecartesian3d" label="e">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(0, t, s) / (((0, t, s) * u) ≥ o) + ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="f"/>
</command>
<element type="surfacecartesian3d" label="f">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(s, t, 0) / (((s, t, 0) * u) ≥ o) + ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="g"/>
</command>
<element type="surfacecartesian3d" label="g">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(s, 1, t) / (((s, 1, t) * u) ≤ o) - ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="h"/>
</command>
<element type="surfacecartesian3d" label="h">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(1, t, s) / (((1, t, s) * u) ≤ o) - ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="i"/>
</command>
<element type="surfacecartesian3d" label="i">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(s, t, 1) / (((s, t, 1) * u) ≤ o) - ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="j"/>
</command>
<element type="surfacecartesian3d" label="j">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(s, 0, t) / (((s, 0, t) * u) ≤ o) - ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="k"/>
</command>
<element type="surfacecartesian3d" label="k">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(0, t, s) / (((0, t, s) * u) ≤ o) - ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="l"/>
</command>
<element type="surfacecartesian3d" label="l">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Surface">
<input a0="(s, t, 0) / (((s, t, 0) * u) ≤ o) - ((a * 2) * u)" a1="s" a2="0" a3="1" a4="t" a5="0" a6="1"/>
<output a0="m"/>
</command>
<element type="surfacecartesian3d" label="m">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="4"/>
<labelMode val="0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<command name="Translate">
<input a0="poly1" a1="Vector[((2 * a) * u)]"/>
<output a0="poly1'"/>
</command>
<element type="polygon3d" label="poly1'">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="255" b="255" alpha="0.6499999761581421"/>
<layer val="4"/>
<labelMode val="0"/>
</element>
<command name="Translate">
<input a0="poly1" a1="Vector[((-2 * a) * u)]"/>
<output a0="poly1'_1"/>
</command>
<element type="polygon3d" label="poly1'_1">
<lineStyle thickness="5" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="255" b="255" alpha="0.6499999761581421"/>
<layer val="4"/>
<labelMode val="0"/>
</element>
</construction>
</geogebra>