Point clouds are sets of unordered points without connections. They can be generated from 3D scans and used for medical or industrial applications. Point clouds lack properties like textures and normals, so lighting cannot be directly applied. They must be converted into meshes or polygon networks for solid modeling. This can be done through algorithms like triangulation, two-peasant graphs, or marching cubes. Constructive solid geometry uses boolean operations on basic geometric primitives to combine them into complex 3D models. It is commonly used in CAD software for engineering design.

1. 8. Mathematic Modelling of Objects 8.3. Modelling of Point clouds

3. No composition of points (no planes)

4. No predefined orderConclusion

7. 8.3. Modellierung von Punktewolken 8.3.4. Programmierung […] #define M 4 //Größe des Objekts #define ABSTAND 0.1 //Abstand der Einzelpunkte //Zeichne Punkte glBegin(GL_POINT); { intx,y,z; for(int i=0; i<100; i++) { // pro Durchgang: erhöhe den Tiefenwert; Baue Objekt erst Spalten- // dann Zeilenweise auf. z=i%M; x=((int)(i/M))%(M*M); y=(int)(i/(M*M)); // Setze Farbwert glColor4f(0.0f, 0.12f, 0.75f, 0.8f); // Setze Position glVertex3f(ABSTAND*x, ABSTAND*y, ABSTAND*z); } } glEnd(); […]  Vorteil: Bewegung in das Objekt möglich

9. 8.3. Mathematic Modelling of Objects 8.3.6. Conversion point cloud – polygonial net (Meshing) First approach for conversion: Triangulation (triangle construction)  Graham-Algorithmus 2-dimensional: detect point with lowest y-value Afterwards: connection of the other points depending on the angle to seed point, beginning with the lowest interconnect environment points according to angle order

10. 8.3. Mathematic Modelling of Objects 8.3.6. Conversion point cloud – polygonial net (Meshing) Last step: delete concave corner points

11. 8.3. Mathematic Modelling of Objects 8.3.6. Conversion point cloud – polygonial net (Meshing) Second approach: construction of polygon according to human thinking  Two-Peasant – Graphs Detection of points with x-maximum and x-minimum seperation of cloud in upper and lower half order halfs interconnection of sequential points of upper half from min(x) to max(x) Interconnection of sequential points of lower half from max(x) to min(x)

12. 8.3. Modellierung von Punktewolken 8.3.6. Umwandlung Punktewolke - Dreiecksnetz Übergang 2D-3D: Mustererkennung Anwendbar bei Wissen über geometrische Zusammensetzung der Szene (Beispiel: Billard-Tisch) Vergleich einzelner Punktehaufen mit mathematischer Objektbeschreibung Objekt modellieren wenn genügende Anzahl (Schwellwert) an Punkte auf Objektbeschreibung zutrifft Billard: Kugel definiert durch Mittelpunkt und Radius x−x02+y−y02+ z−z02= r2

13. 8.3. Modellierung von Punktewolken 8.3.6. Umwandlung Punktewolke - Dreiecksnetz Suche von Minimum- und Maximum-Punkt für x-, y- und z-Werte Verbindung der Punkte; Halbierung der Strecken; Mittelwerte der Beträge halben Strecken  mgl. Radius P0= xPxMin+xPxMax−xPxMin2zPyMin+yPyMax−yPyMin2zPzMin+zPzMax−zPzMin2 Test nach genannter Formel, ob Oberflächenpunkte zu Kugel zugehörig

14. 8.3. Mathematic Modelling of Objects 8.3.6. Conversion point cloud – polygonial net (Meshing) Last presentedpossiblity: 3D-facettes byMarching Cubes weightingofcornerpoints Constructionofweightedvoxels Check per voxel: cornersexceedingcertainthreshold; indexcontatenationofincludingcorners check Look-Up Table forpolygonconfiguration compositionofvoxel-cuberesultstooneobjects facettenormalscanbesaved inLook-Up Table too

15. 8.3. Modellierung von Punktewolken 8.3.6. Umwandlung Punktewolke - Dreiecksnetz 1 Voxel; Bsp.-Wert: 105 = 1|4|6|7 20+23+25+26=105 105

17. construction: compostion of multiple objects

19. objects with matter

20. combination of geometric primitives

22. cube

23. cylinder

24. cone

25. torus

27. 8.4. Constructive Solid Geometry (CSG) 8.4.3. Application Domain similarapplicationdomainsaspolygonialobjects Computer Aided Design (CAD) forRapid Prototypingand Manufacturing; Product- andJewellery design Spacecraft Simulation; ComponentPrototyping Mechanical Engineering: Modellingofnew Engine Components major CAD systems: Autodesk SolidWorks, Catia, ProEngineer

28. 8.4. Constructive Solid Geometry (CSG) 8.4.3. Application Domain

29. 8.4. Constructive Solid Geometry (CSG) 8.4.4. boolean object combination Union: united set of objects objects are into each other Difference: all points of a basic object, which are not included in the combined object Intersection: Points, included in all combined objects

30. 8.4. Constructive Solid Geometry (CSG) 8.4.4. boolean object combination objectmodelling via treestructure Example…  ⊆ ∩ U Ball Cube Ball Ball …