8. Mathematic Modelling of Objects

8.3. Modelling of Point clouds

8.3.1. Definition

Point clouds are sets of discrete values clustered in a certain area.

In Computer Graphics: unordered set of points in a catesian coordinate system

8.3. Mathematic Modelling of Objects

• uncertain contextual relationship• No composition of points (no planes)• No predefined order

Conclusion

8.3.2. Properties

• Objects don‘t have facettes

Textures can‘t be used• Objects don‘t have face normalsLighting computation not possibleColor of each point must be defined• Each point is rendered (no face culling because of

absence of faces)• Tight-packed points construct point clustersConstruction of solid objects possible

8.3. Mathematic Modelling of Objects

8.3.3. Application Domain

Constuction of solid objects:• medical computer graphics

(Computer Assisted Surgery [CAS])• automotive industry

Point cloud visualization:• voxel graphics (Voxel = Volume

Pixel)• particle systems

8.3. Mathematic Modelling of Objects

8.3.5. professional generation of point cloud objects

• manual generation of point clouds time intensive / not applicable

Technical point recording:• 3D camera records• Laser scans of solid objects• MRI (Magnetic Resonance Imaging)• X-raying

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

8.3. Mathematic Modelling of Objects

• Last step: delete concave corner points

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)

8.3. Mathematic Modelling of Objects

8.3.6. Conversion point cloud – polygonial net (Meshing)

Last presented possiblity: 3D-facettes by Marching Cubes• weighting of corner points• Construction of weighted voxels• Check per voxel: corners exceeding certain threshold;

index contatenation of including corners• check Look-Up Table for polygon configuration• composition of voxel-cube results to one objects• facette normals can be saved in Look-Up Table too

8.3. Mathematic Modelling of Objects

8.3.6. Conversion point cloud – polygonial net (Meshing)

8.3.6. Umwandlung Punktewolke - Dreiecksnetz

8.3. Modellierung von Punktewolken

1 Voxel; Bsp.-Wert: 105 = 1|4|6|7105

8.4. Constructive Solid Geometry (CSG)

• cover: constructed (composed) objects

• construction: compostion of multiple objects

• Application: virtual objects for construction of real objects

• solid: filled; closed• objects with matter

• combination of geometric primitives

• basic blocks

8.4.1. Definition

CSG is a technique for creation of solid objects. complex structures can be build via boolean combination of geometric primitives.

CSG models are convex objects,

build out of co-planar facettes.

8.4. Constructive Solid Geometry (CSG)

• ball• cube• cylinder• cone• torus• facette

8.4.2. Properties

CSG models:• filled and closed• facette normals existent illumination computation possible

convex: all corner facettes co-planar: all corners

are facing in opponent of a facette are in one

directions plane

8.4. Constructive Solid Geometry (CSG)

8.4.3. Application Domain

• similar application domains as polygonial objects• Computer Aided Design (CAD) for Rapid Prototyping and

Manufacturing; Product- and Jewellery design• Spacecraft Simulation; Component Prototyping• Mechanical Engineering: Modelling of new Engine

Components• major CAD systems: Autodesk SolidWorks, Catia,

ProEngineer

8.4. Constructive Solid Geometry (CSG)

8.4.3. Application Domain

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

8.4. Constructive Solid Geometry (CSG)

8.4.4. boolean object combination

• object modelling via tree structure• Example …

U

Ball Ball …

∩

Ball Cube

⊆

8.4. Constructive Solid Geometry (CSG)

8.4.5. Programming

• for generation, storage and rough visualization: CSG Editor• Programming via OpenGL-based API OpenCSG

8.4. Constructive Solid Geometry (CSG)