Overview. Topology as Central Information in Building Models. Basic Set. Topology Topological Space. What is topology?

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Beverly Gordon
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1 Overview What is topology? as Central Inforation in Building Models Frank Gielsdorf Wolfgang uhnt technet Gb Berlin Technical University Berlin Why topology is needed for process odeling? ow can topologic objects associated with geoetric properties? ow can adjustent techniques applied in the reverse engineering? in the construction? Topological Space Syste of subsets defined on a basic set X Basic Set All points of a etric space Exaple: Graph Basic set X of vertices X = { v, v, v, v } X E v v z x Properties:. The set is infinite.. The set is uncountable. Syste of two-valued subsets - edges { v, v} { v, v} { v, v} { v, v } E v E v E Proble: Solution: y It s ipossible to save all points to a data ediu. Mapping of point sets on a finite nuber of topological objects Mapping Mapping p S p n K n S S K K p S p n K n S K = p n { n, n }... S K = { n, n }... p n Proble: The apping topology geoetry is abiguous. Solution: Saving of the apping function. { n, n } S = { x( t) = x + t ( x x ), = 0 x } f : K = t x i A E i A E A... E A

Criterion for a Subset Saving the Mapping Function p S p S n K n point set of etric space apping function point set of etric space p K S K n S K cell of topological space p5 S5 p n5 K5 n The

2 Criterion for a Subset Saving the Mapping Function p S p S n K n point set of etric space apping function point set of etric space p K S K n S K cell of topological space p5 S5 p n5 K5 n The criterion for defining a point set and ap it on a cell is the possibility to describe that set by one function. Exception: Vertex Cell Coplex Partitioned topological space Each point in space belongs to exactly one cell Cells of diension p can be constructed by a fraework of cells of diension (p-) Exaple: Cell coplex of diension D Cell Coplex 0-cell vertex -cell edge -cell esh -cell body vertices (,*) end (,) edges (,*) bound (,*) eshes (,) cover (,*) bodies Topological Space Building Exaple: Civil Engineering at TU Berlin (extract): zones and doors Storey 5 6 Q Q Q.000 Q.0005 Q.0007 Q.00 Q.00 Q.005 Q.007 Coponent Flur0 Q.000 Flur0 Q.0008 Q.005 Q.00 Q.00 Q.008 Flur0 Q.009 Cell Coplex Q Q.00 Q Q.000 Metric Space Set Q Lichthof 59

3 Construction Processes Construction Processes g colun colun ceiling Several activities are necessary to anufacture coponents. Status graphs are suitable to describe the anufacturing process of each coponent. Status graphs Coluns: Ceilings: reinforceent reinforceent concrete concrete stripped concrete required strength for placing stripped support reoved Topological Space Building 0-cell approach Vertex Storey Coponent Edge Line Cell Coplex Mesh -cell approach * * Face Metric Space Set Body Volue and Paraeterization of Planes Mesh Mesh Plane Plane persistent x r d = 0 d x r O : n View n x = d n x = d n x = d SQL-Stateent x r Planes carry (,*) (,*) Meshes

Geoetrical Constraints Exaple: Theoretical Building Coplanarity iplicit 0 floors Parallelis identical noral vectors longitudinal walls transversal walls Orthogonality (in x,y-plane) a = b 0 a b b = a

x, n y, n z, d 0* + * + * = 50 planes noral vectors * coponents + 50 translations d = 59 plane paraeters 0.8%!

4 Geoetrical Constraints Exaple: Theoretical Building Coplanarity iplicit 0 floors Parallelis identical noral vectors longitudinal walls transversal walls Orthogonality (in x,y-plane) a = b 0 a b b = a 0 Nuber of Paraeters 0-Cell Approach: paraeterization by coordinates x, y, z 0 * 0 * + = 0 roos 0 * 8 = 08 points 08 * = 7 coordinates -Cell-Approach: paraeterization by plane paraeters n x-d = 0 n x, n y, n z, d 0* + * + * = 50 planes noral vectors * coponents + 50 translations d = 59 plane paraeters 0.8%! Adjustent Techniques Redundant relative easures can be processed Unique result with accuracy and reliability values observations l, C ll (redundant, uncorrelated) adjustent calculation geoetrical paraeters x, C xx (unique, correlated) Reverse Engineering Construction Exaple: Architectural Survey Classic Absolute Topologic University Bielefeld Departent of Architecture and Civil Engineering, Minden, Relative Adjustent Calculation Absolute ground floor planes 6 distances between -cells 7 distances between 0-cells Process Planning Ipleentation Process Planning Ipleentation

5 Conclusions A topological data odel is the prerequisite for the denotation and the building process planning. A direct paraeterization of surfaces needs considerably fewer paraeters and allows the application of adjustent techniques. Adjustent techniques: Allow a very efficient reverse engineering of geoetry. Provide new ways for the building construction. Thank you for your attention! frank.gielsdorf@technet-gbh.co huhnt@tu-berlin.de 5

Computer Aided Drafting, Design and Manufacturing Volume 26, Number 2, June 2016, Page 13

Computer Aided Drafting, Design and Manufacturing Volume 26, Number 2, June 2016, Page 13 Coputer Aided Drafting, Design and Manufacturing Volue 26, uber 2, June 2016, Page 13 CADDM 3D reconstruction of coplex curved objects fro line drawings Sun Yanling, Dong Lijun Institute of Mechanical