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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Isofläche</span></h1>
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<p><b>Isoflächen</b> sind Flächen, die im Raum benachbarte Punkte gleicher Merkmale oder Werte einer bestimmten Größe wie zum Beispiel <a href="Temperatur" title="Temperatur">Temperatur</a> oder <a href="Dichte" title="Dichte">Dichte</a> miteinander verbinden. Sie sind das dreidimensionale Gegenstück zu <a href="Isolinie" title="Isolinie">Isolinien</a>, die Punkte auf einer Fläche verbinden.
</p><p>Die Bedeutung von <b>Isoflächen</b> liegt in der computergraphischen Visualisierung von <a href="Skalarfeld" title="Skalarfeld">Skalarfeldern</a> bzw. Gittern.
</p><p>In der Medizin verwendet man Isoflächen aus Datensätzen mit <a href="Dichte" title="Dichte">Dichte</a>werten zur Darstellung von Organoberflächen. Die Datensätze entstehen zum Beispiel bei <a href="Computer-Tomographie" class="mw-redirect" title="Computer-Tomographie">Computer-Tomographie</a>-Messungen. Eine andere Anwendung ist die Darstellung von Molekülen, deren Atomlage durch Elektronenmikroskopie bestimmt wurde.
</p><p>Das gängigste Verfahren zur Darstellung von Isoflächen heißt <a href="Marching_Cubes" title="Marching Cubes">Marching Cubes</a> und wurde von Lorensen und Cline 1987 eingeführt.
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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Die <b>Isofläche</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{c}}">
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<mi>S</mi>
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<annotation encoding="application/x-tex">{\displaystyle S_{c}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c9f8b4d1ecb693aefb3372c33479d00103085d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.369ex; height:2.509ex;" alt="{\displaystyle S_{c}}" loading="lazy"></span> zu einem <a href="Skalarfeld" title="Skalarfeld">Skalarfeld</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :\mathbb {R} ^{n}\rightarrow \mathbb {R} }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
<mo>:</mo>
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<mo stretchy="false">→<!-- → --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \varphi :\mathbb {R} ^{n}\rightarrow \mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe9295aaecf847b4730c46222c509ea313c44421.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.646ex; height:2.843ex;" alt="{\displaystyle \varphi :\mathbb {R} ^{n}\rightarrow \mathbb {R} }" loading="lazy"></span> beim <i>Isowert</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathbb {R} }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo>∈<!-- ∈ --></mo>
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<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d47ef490c028656282fd8b18c44c4939bbfff750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {R} }" loading="lazy"></span> ist die Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{c}:=\{\mathbf {v} \in \mathbb {R} ^{n}|\varphi (\mathbf {v} )=c\}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
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<mi>c</mi>
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<mo>:=</mo>
<mo fence="false" stretchy="false">{</mo>
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<mi mathvariant="bold">v</mi>
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<mo>∈<!-- ∈ --></mo>
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<mo stretchy="false">|</mo>
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<mi>φ<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle S_{c}:=\{\mathbf {v} \in \mathbb {R} ^{n}|\varphi (\mathbf {v} )=c\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acf4aedf87fe782d8e40ff8f1d89d735103e0977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.08ex; height:2.843ex;" alt="{\displaystyle S_{c}:=\{\mathbf {v} \in \mathbb {R} ^{n}|\varphi (\mathbf {v} )=c\}}" loading="lazy"></span>.
</p><p>Dreidimensionale <i>Isoflächen</i> werden in der Regel aus einer endlichen Menge von Datenpunkten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\subset \mathbb {R} ^{3}\times \mathbb {R} }">
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<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\subset \mathbb {R} ^{3}\times \mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee7c2b8797e23f620df7dddf33b2debae1dd8f22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.053ex; height:2.676ex;" alt="{\displaystyle {\mathcal {P}}\subset \mathbb {R} ^{3}\times \mathbb {R} }" loading="lazy"></span> (<a href="Gitter_(Geometrie)" title="Gitter (Geometrie)">Gitter</a>) approximiert, beispielsweise durch <a href="Dreiecksnetz" class="mw-redirect" title="Dreiecksnetz">Dreiecksnetze</a>.
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<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Implizite_Fl%C3%A4che" title="Implizite Fläche">implizite Fläche</a></li>
<li><a href="Triangulation_(Fl%C3%A4che)" title="Triangulation (Fläche)">Triangulation (Fläche)</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Hansen, Charles D., Johnson, Chris R.: <i>The Visualization Handbook.</i> Elsevier Academic Press, 2005, ISBN 0-12-387582-X, S. 39ff</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-07-23" href="https://de.wikipedia.org/wiki/?title=Isofl%C3%A4che&oldid=258213653">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
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