Changes

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== References ==

== References ==

== <span id="CompositeSurface"> [http://www.schemacentral.com/sc/niem21/e-gml32_CompositeSurface.html gml:CompositeSurface]</span>==

== <span id="CompositeSurface"> [http://www.schemacentral.com/sc/niem21/e-gml32_CompositeSurface.html gml:CompositeSurface]</span>==

Eine CompositeSurface ist eine Menge <math>C=\lbrace S_1,S_2,...,S_n \rbrace</math> von Polygonen, für die

A set <math>C=\lbrace S_1,S_2,...,S_n \rbrace</math> of polygons is a CompositeSurface for which

# Die Schnittmenge zweier Polygone <math>S_k</math>  und <math>S_l</math>  aus <math>C</math>  ist entweder leer oder besteht nur aus Punkten und/oder Kanten, die auch in den beiden linearen Ringen vorkommen. Bezeichne <math>R_k=(P_0^k,P_1^k,...,P_n^k)</math> den planaren linearen Ring, der das Polygon <math>S</math> definiert. Dann gilt:<br><math>S_i \cap S_k= \begin{cases}\emptyset\\ \lbrace Q_0,Q_1,...,Q_m\rbrace,Q_j=P_k^i\\ \lbrace e_0,e_1,...,e_m\rbrace,e_j=\overline{P_i^kP_{i+1}^k} \end{cases}</math>

# The intersection of two polygons <math>S_k</math>  and <math>S_l</math>  of <math>C</math>  is either empty or contains only points <math>P</math> and/or edges <math>e</math>that are part of both Linear Rings.  

# Jede Kante <math>e_k=\overline{P_i^kP_{i+1}^k}</math> eines linearen Rings <math>R_k=(P_0^k,P_1^k,...,P_n^k)</math> , der ein Polygon <math>S_k \in C</math> definiert, wird höchstens  einmal als Kante <math>e_l=\overline{P_j^lP_{j+1}^l}</math>  in einem linearen Ring <math>R_l=(P_0^l,P_1^l,...,P_m^l)</math>  genutzt, der ein anderes Polygon <math>S_l \in C</math> definiert.<br>Es gilt <math>P_i^k=P_{j+1}^l</math>  und <math>P_{i+1}=P_j^l</math>.

The polygon <math>S</math> is defined by the planar Linear Ring <math>R_k=(P_0^k,P_1^k,...,P_n^k)</math>.

# Die Polygone aus <math>C</math> sind so orientiert, dass die Flächennormale benachbarte Polygone in dieselbe Richtung zeigen.

The intersection of <math>S_i</math> and <math>S_k</math> equals<br>

# Die Vereinigung aller Polygone aus <math>C</math> ohne die Kanten oder Punkte, in denen sich die Polygone berühren, ist isomorph zu einem Polygon.  

<math>S_i \cap S_k= \begin{cases}\emptyset\\ \lbrace Q_0,Q_1,...,Q_m\rbrace,Q_j=P_k^i\\ \lbrace e_0,e_1,...,e_m\rbrace,e_j=\overline{P_i^kP_{i+1}^k} \end{cases}</math>

# Every edge <math>e_k=\overline{P_i^kP_{i+1}^k}</math> of a Linear Ring <math>R_k=(P_0^k,P_1^k,...,P_n^k)</math> , defining a polygon <math>S_k \in C</math> is used exactly once as an edge <math>e_l=\overline{P_j^lP_{j+1}^l}</math>  in a Linear Ring <math>R_l=(P_0^l,P_1^l,...,P_m^l)</math>  defining another polygon <math>S_l \in C</math> with <math>P_i^k=P_{j+1}^l</math>  and  <math>P_{i+1}=P_j^l</math>.

# All polygons in <math>C</math> are oriented such that the normal vectors of neighbouring polygons point to the same direction.

# The union of all polygons in <math>C</math> without the edges or points where the polygons touch, is isomorphic to a polygon.  

Aus (1) und (2) ergibt sich, dass die Oberfläche, die durch  <math>C</math> beschrieben wird, keine sich gegenseitig überlappenden oder durchdringenden Polygone enthalten darf (Polygone berühren sich höchstens in Punkten oder Kanten).

It follows from (i) and (ii) that the surface defined by <math>C</math> , must not contain any overlapping or penetrating polygons, that is, polygons touch at most at points or edges.

== <span id="Solid"> [http://www.schemacentral.com/sc/niem21/e-gml32_Solid.html gml:Solid]</span>==

== <span id="Solid"> [http://www.schemacentral.com/sc/niem21/e-gml32_Solid.html gml:Solid]</span>==

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incorrect: 12 surfaces, no closed (zwei Außenhüllen), correct surface orientation

incorrect: 12 surfaces, no closed (two outer hulls), correct surface orientation

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incorrect: 12 surfaces, not closed (innere und äußere Hülle), correct surface orientation

incorrect: 12 surfaces, not closed (inner and outer hull), correct surface orientation  

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== <span id="CompositeSolid">[http://www.schemacentral.com/sc/niem21/e-gml32_CompositeSolid.html gml:CompositeSolid]</span>==

== <span id="CompositeSolid">[http://www.schemacentral.com/sc/niem21/e-gml32_CompositeSolid.html gml:CompositeSolid]</span>==

Ein CompositeSolid wird durch eine nichtleere Menge  <math>C = \{S_1,..., S_n\}</math> von '''[[#Solid|Solids]]''' (vgl. 12.) beschrieben, für die gilt:

A CompositeSolid is represented by a nonempty set <math>C = \{S_1,..., S_n\}</math> of '''[[#Solid|Solids]]''' (see 10.) with the following properties:

# Der Schnitt der Inneren zweier Solids <math>S_i</math>, <math>S_j</math>, 1 ≤ i ≤ n, 1 ≤ j ≤ n, i ≠ j, ist leer, d.h. entweder sind beide Solids <math>S_i</math>, <math>S_j</math> disjunkt oder beide berühren sich nur in Flächen, Linien oder Punkten

# The intersection of the inner of two solids <math>S_i</math>, <math>S_j</math>, 1 ≤ i ≤ n, 1 ≤ j ≤ n, i ≠ j of <math>C</math> is empty, that is either both solids <math>S_i</math>, <math>S_j</math> are disjoint or both solids are connected only by surfaces, lines or points.

# Sei <math>C'</math> die Vereinigung aller Solids aus <math>C</math>. Dann ist die Begrenzung von <math>C'</math> (die Oberfläche von <math>C'</math> ohne die Flächen oder Punkte, in denen sich die Solids berühren) die Begrenzung eines Solid.  

# Let <math>C'</math> be the union of all solids from <math>C</math>. Then the boundary of <math>C'</math> (the surface of <math>C'</math> without the surfaces or points where the solids touch each other) is the boundary of a solid.  

In CityGML werden bei CompositeSolids ebenso wie bei Solids innere Einschlüsse (Hohlräume) nicht betrachtet.  

In CityGML inner inclusions ( hollow spaces ) are not considered in CompositeSolids, as in solids.  

# Areas of the surface where the data is not sufficiently dense to assure reasonable calculations shall be removed by adding a retention criterion for triangles based on the length of their sides. For any triangle sides exceeding maximum length, the adjacent triangles to that triangle side shall be removed from the surface. # The number of control points is at least three.  

# Areas of the surface where the data is not sufficiently dense to assure reasonable calculations shall be removed by adding a retention criterion for triangles based on the length of their sides. For any triangle sides exceeding maximum length, the adjacent triangles to that triangle side shall be removed from the surface. # The number of control points is at least three.  

# The surface is not affected by the order in which the control points are given. Application schemas may add information based on the ordering of the control points to facilitate the reconstruction of the TIN from the control points.

# The surface is not affected by the order in which the control points are given. Application schemas may add information based on the ordering of the control points to facilitate the reconstruction of the TIN from the control points.