Box with Superelliptical Cross-Section
2014-12-29 14:07
This applet demonstrates a 3D box with a superelliptical cross-section. The equation is |x|^a + |y|^a = 1. By varying parameter 'a' from 1 to 2, the cross-section smoothly transitions from a square to a circle. It uses GeoGebra's surface and curve tools to visually explore the geometric construction and cross-sectional changes of the solid.
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<input a0="(-cos(t)^(2 / (z0 + 1)))" a1="sin(t)^(2 / (z0 + 1))" a2="z0 - dz / 2" a3="t" a4="0" a5="1.5708"/>
<output a0="g"/>
</command>
<element type="curvecartesian3d" label="g">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
</element>
<command name="CurveCartesian">
<input a0="(-cos(t)^(2 / (z0 + 1)))" a1="(-sin(t)^(2 / (z0 + 1)))" a2="z0 - dz / 2" a3="t" a4="0" a5="1.5708"/>
<output a0="h"/>
</command>
<element type="curvecartesian3d" label="h">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
</element>
<command name="CurveCartesian">
<input a0="cos(t)^(2 / (z0 + 1))" a1="(-sin(t)^(2 / (z0 + 1)))" a2="z0 - dz / 2" a3="t" a4="0" a5="1.5708"/>
<output a0="i"/>
</command>
<element type="curvecartesian3d" label="i">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
</element>
<command name="Surface">
<input a0="(-cos(t)^(2 / (z0 + 1)))" a1="(s * sin(t)^(2 / (z0 + 1)))" a2="z0 + dz / 2" a3="t" a4="0" a5="1.5707963267948966" a6="s" a7="0" a8="1"/>
<output a0="f12"/>
</command>
<element type="surfacecartesian3d" label="f12">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<levelOfDetail val="5"/>
</element>
<command name="Surface">
<input a0="cos(t)^(2 / (z0 + 1))" a1="((-s) * sin(t)^(2 / (z0 + 1)))" a2="z0 + dz / 2" a3="t" a4="0" a5="1.5707963267948966" a6="s" a7="0" a8="1"/>
<output a0="f14"/>
</command>
<element type="surfacecartesian3d" label="f14">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<levelOfDetail val="5"/>
</element>
<command name="Surface">
<input a0="(s * cos(t)^(2 / (z0 + 1)))" a1="sin(t)^(2 / (z0 + 1))" a2="z0 + dz / 2" a3="t" a4="0" a5="1.5708" a6="s" a7="0" a8="1"/>
<output a0="f11"/>
</command>
<element type="surfacecartesian3d" label="f11">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<levelOfDetail val="5"/>
</element>
<command name="Surface">
<input a0="((-s) * cos(t)^(2 / (z0 + 1)))" a1="(-sin(t)^(2 / (z0 + 1)))" a2="z0 + dz / 2" a3="t" a4="0" a5="1.5708" a6="s" a7="0" a8="1"/>
<output a0="f13"/>
</command>
<element type="surfacecartesian3d" label="f13">
<show object="true" label="false"/>
<objColor r="125" g="125" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<levelOfDetail val="5"/>
</element>
</construction>
</geogebra>