Architectural Indoor Surveying. An Information Model for 3D Data Capture and Adjustment

Transcription

1 Architectural Indoor Surveying. An Information Model for 3D Data Capture and Adjustment Christian Clemen, Frank Gielsdorf Technische Universität Berlin Institute of Geodesy and Geoinformation Science Keywords 3D, data modeling, redundancy, adjustment, BIM Learning Objectives 1. Recognizing a non-standard parameterization of 3D Geometry that suits estimation theory 2. Understanding the need for storing the original geodetic observations in the project database 3. Learning the essential role of topologic information in 3D building models. Abstract Gathering three dimensional geometric data of buildings is becoming increasingly important. Virtual 3D City Models and Building Information Models require reliable information about the inside of buildings. Floor plans are often not available, not updated or not three dimensional. Traditional surveying techniques, such as tacheometry, are too time consuming and too expensive. This technical session shows how to parameterize three dimensional building geometry in a non-standard form that is suitable for fast data acquisition and least squares adjustment. In contrast to existing CAD or GIS data models, the discussed approach does not use any coordinates, instead it utilizes normal vectors to planes. Using this approach, the number of unknowns is significantly reduced, resulting in fewer measurements to be observed. The original measurements and the observation topology are stored in the database enabling statistical testing that allows for robust estimation and stepwise refinement of the level of detail and precision.

2 Architectural Indoor Surveying. An Information Model for 3D Data Capture and Adjustment Christian Clemen, Frank Gielsdorf 1 Introduction From the practical point of view, indoor surveying is designing plots with a CAD tool, where the dimensions are taken from angular and distance measurements. Most often the measurements are taken as a first step and the drawing is done back at the office. But what if measurements are wrong or have been forgotten? What if the outer walls of one floor plan do not aligne with the walls of another floor plan? Probably the surveying staff will then have to return to the building for re-measuring. Figure 1: Common mistakes during indoor surveying Our approach will simplify the surveying process by enabling the engineer, through the usage of a simple software, to sketch the building and attach the observed measurements, allowing checks for consistency and completeness. The resulting building geometry can be viewed on site. This article discusses certain aspects of a conceptual three-dimensional data model that was introduced by [Gründig and Gielsdorf, 2002]. The methodology of this promising model relies on three main ideas. Firstly, topological and geometrical data are specified separately. Explicit topology is compulsory. The explicit specification of a 3D topology facilitates consistency checks during insert, update or delete operation. Secondly, geometry is parameterized with plane surfaces not coordinates. A surface based parameterization reduces the number of unknown parameters in the adjustment, by shifting information, like planarity of a face, from the computational geometry domain to the relational domain. And thirdly, the observations (measurement, relative geometry) are stored in the project database and are used to determine the parameters of the absolute geometry by applying geodetic adjustment techniques. [Thompson and Oosterom, 2008] emphasizes a soaring paradigm: Note that there is a trend in which the original observations and measurements are more often stored in the (cadastral) database, in addition to the resulting interpretations (parcels).

3 3/17 Database normalization is a standard technique to ensure data integrity. Explicitly specified 3D topology also ensures geometric consistency. This article explains a stepwise reduction of redundancy in the 3D data model. Whereas the topological normalization is well discussed in literature the geometric normalization is a subject of recent research. This approach of data normalization is motivated by the need for applying geodetic adjustment techniques to measured (observed) three-dimensional building models. 2 Redundancy If pieces of information are exceeding what is necessary or normal, these pieces are called redundant. Redundancy is characterized by similarity or repetition. The meaning of the word ranges from pure duplication of critical components of an engineering system, to wasted space used to transmit certain data, including tautology in spoken language. (Wikipedia) The GIS/Surveying world uses the term redundancy in context of SDBMS (Spatial Database Management Systems) and in the context of Adjustment Calculation. Database designer try to avoid redundancy. Database Normalization minimizes duplication of information in order to ensure data integrity when data is inserted, updated or deleted. Empirical science like Engineering Surveying, Photogrammetry and Geodesy try to obtain redundant observations in order to increase precision and reliability. The different connotations - negative connotation on the database side and positive connotation on the surveying side - result from different types of information. A database usually represents Deterministic Information (DI). The model of the world does not involve uncertainty and errors. This assumption leads to a contradiction free representation of the real world. Deterministic variables should be stored only once. Here, redundancy means duplication of identical information. Models from Measurements are realized with Stochastic Information (SI). Due to the inherent stochastic properties of observations the model involves information about precision and correlation of all parameters. Although diverse observations are used to describe the same quantity in the real world, every observation is stored, since every measurement is a subject to errors or mistakes. Here redundancy can be considered as statistical degree of freedom [Mikhail, 1976]. Any surveying engineering application benefits from the usage of redundant information by estimating precision and reliability. This article describes how the reduction of deterministic redundancy leads to a data model that supports input, update and deletion of three-dimensional topologic-geometric data and supports redundant measurement. Since the amount of unknown variables is lessened this allows for efficient algorithms concerning stochastic information (adjustment calculation). 3 Three dimensional Topology in Spatial Data models 3.1 Topological models In spatial information models the terms geometry and topology are closely linked. From the mathematical point of view the three-dimensional Euclidean space R 3 is a topological space.

4 4/17 In the context of solid modeling and GIS, geometric information describes position, orientation and shape of objects whereas topology describes relationships between objects. Topologic properties are invariant under certain geometric transformation. This article discusses some aspects of explicit topological boundary representation of threedimensional solids. Out of scope are: Topological predicates from a result of geometric operations and topological networks as used for routing. [Mäntylä, 1988] substantiates the necessity of a topological model with the following properties of the Euclidean space R 3 : The domain of the property set (x, y, z) of the metric space is infinite, uncountable and open. It is impossible to map all points of this space in a unstructured way. The structure of space decomposition (voxel, TEN, B-Rep, Half-Space...) is called topology. Further on [Mäntylä, 1988] distinguishes three separate levels of modeling: Figure 2: A three-level view of modeling [Mäntylä, 1988] Physical objects (terrain, buildings, streets, tunnels, parcels) must be identified and the universe of discourse must be restricted, since the real world cannot be mapped in full detail. Mathematical objects are a suitable idealizations of the real-world physical objects. Diverse mathematical approaches are discussed in literature (regular cell decomposition, TEN, B- Rep, CSG). [Zlatanova, 2004] gives an overview on the history of data models and recent developments. Less effort has been made on the problem of digital representations. Only recently [Thompson, 2007] addresses the finite precision of digitally represented spatial data. Since there is no method to find an explicit representation of shapes in space R 3 an indirect method has to be found. The implicit representation of infinite point-sets can be structured in following classes: Decomposition models. The point set is represented as a collection of simple objects from a fixed collection of primitive object types with a single (!) implicit or explicit gluing operation. Example for decomposition models are: Voxel, Octree, binary space subdivision. Constructive models. The point set is represented as a combination of primitive point sets together with general construction operations. Example for constructive models are: Boolean set operations on half spaces, constructive solid geometry (CSG).

5 5/17 Boundary models. These vector based models decompose the 3d solid with the relation type boundary. Complex n-dimensional Shapes are described indirectly with their (n 1)- dimensional boundary. Topology then is the structure that specifies how objects of lower dimension are combined. Figure 3: Boundary Representation [Bradley, 2006] Only the boundary model will be discussed in this article. 3.2 Topological primitives Topological primitives are topological objects that describe single, non-decomposed elements of a topological complex. Topological primitives are open and connected point sets. Node. A node is a 0-dimensional topological object. A node has no boundary. Edge. An edge is a 1-dimensional topological object. An edge is bounded by two nodes. The interior of an edge is an open and connected point set. Face. A Face is a 2-dimensional topological object. A Face is bounded by a finite set of edges. The boundary is closed and has no self intersection. The interior of a face is an open and connected point set. Faces are 2-manifolds. If a face is topologically the same like a disk -which means it has no holes - it is called a 2-cell or CW-complex. Topological Solid. A topological solid is a 3-dimensional topological object. A solid is bounded by faces. The surface (boundary) is closed, orientable, connected and free of intersection. The interior of a topological solid is an open and connected point set. 3.3 Topologic and geometric consistency As mentioned above, the terms geometry and topology are closely linked. An explicit specified topology helps to ensure geometric consistency. The ISO says that the most important productive use of topology is to accelerate computational geometry. Beyond that, topology allows to ensure consistency during inserting, updating and deleting operations. A simple example for the usage of topology for consistency checks is the Euler-Poincaré formula. The number of topologic primitives in valid solids is invariant. It is used by the (extended) Euler-Poincaré formula. v e + f = 2(s h) + r [Mäntylä, 1988]

6 6/17 In addition to the validation of nodes, edges, faces and solids this simple formula controls the number of rings (hole in a face) and shells. The referential integrity of topologic primitives ensures that the outer boundary and the interior of a topological primitive exists in the model instance. For instance: It can be checked, whether an edge that is supposed to bound a specific face is really instantiated. Cardinality between topologic primitives provides another validation rule. An edge is bounded by exactly two nodes. A mesh is bounded by three edges or more. In linear models a solid is at least bounded by four faces. Topologic Integrity of a ring is ensured if the ring is closed (first vertex equals last vertex) and if distinct vertices are used. Geometric Integrity validates whether all vertices are in the same the plane and whether edges are non-intersecting. If faces have additional interior inner boundaries (holes) these rings have to lie in the same plane as the exterior ring. The interior rings are not allowed to intersect and must follow the orientation rules of the data model. Furthermore interior rings must not overlap nor intersect nor touch the exterior ring. In general one can state, that the algorithms for validating geometric integrity are more expensive than those that are ensured by the data model or the DBMS. Topological encoding transfers certain algorithms from computational geometry to combinatorial algorithms which are well supported by relational databases. 4 Topological Normalization In terms of relational databases normalization means the reduction of multiple stored pieces of information. This chapter does not provide a new 3D topological data model. The aim is to describe the stepwise reduction of redundancy. One technique though is to replace composite relationship (black rhomb) with a shared aggregate (white rhomb). In so-called topology-free, polygon soup or spaghetti data models each face is represented as a collection of coordinate triples (x, y, z) (point composite). This simple structure is suitable to visualization purposes, since there is no need for searching during rendering. It is obvious, that the storage of the point coordinates is redundant. However, more levels of redundancy can be specified Fig4: - Node-redundancy (multiple storage of point geometry) - Edge-redundancy (multiple storage of node-connectivity (edge)) - Face-redundancy (multiple storage of edge-connectivity (face) ) - Solid-redundancy (multiple storage of faces)

7 7/17 Figure 4: Dimension of redundancy. Multiple storage of nodes, edges, loops and faces. In order to avoid node-redundancy, points are uniquely stored in an independent list and are referred by faces. (shared point aggregation). Here, every vertex-coordinate is only stored once. As a next step of normalization the entity type edge is introduced in order to avoid edge-redundancy. Since the normalization of higher-dimensions redundancies must take into account the orientation of the topological primitives, the 2:N relationship is specified in more detail with the attributes start and end. Now a face consists of a list of oriented edges and is stored twice. By switching from edgecomposition to edge-aggregation a real-world edge is only stored once, but is referred by two faces (if the model is a 2-manifold). Since these two faces refer the same edge, but might be of different orientation, the edge is decomposed into two half edges. A half-edge is an entity with a binary orientation flag and a reference to an edge. A face comprises of half edges. If a face has no holes, one (1!) well-ordered set of half-edges (polygon, closed ring, loop) bounds a face. If a face has holes it is bound by one exterior and one or many interior rings. Again, by switching from composition to aggregation a loop is only stored once. In this type of data model each solid stores its own faces. As each face separates two solids the reference direction could be changed. Now, the solids are oriented by the face attributes

8 8/17 behind and front. Orientation of solids allows numeric distinguishing between the inside and outside. Until now all topological dimensions have been checked for redundant information. The topological data structure has been normalized. However, there are diverse ways to structure topological primitives [Ellul, 2005]. It is the objective of this article to demonstrate general methods to reduce redundancy in a rigorous way. 5 Geometric Normalization Until now geometric representation has not been covered. The prevalent method for specifying location is making use of point representation: Every node is attached with three coordinate values (x, y, z) in order to specify position. Additionally implicit (in the mathematic model) or explicit (stored) assumptions are made on planarity of faces, parallelism of faces and perpendicularity of faces. The main idea of the discussed approach is to reduce geometric redundancy by replacing point representation with surface representation.[fig. 5] Figure 5: a) point representation b) surface representation c) plane intersection

9 9/17 A flat surface is represented by its normal vector n = (n x, n y, n z ) T and the orthogonal (shortest) distance d to the origin.this parameterization is known as the Hessian normal form. Since topology is an integral part of the data model, the (relational) database is able to determine which planes intersect in each node. The point coordinates can be calculated on demand, using Cramer s Rule. Figure 6: Planar faces and parallel planes How does this shift from point representation to surface representation affect the objective to reduce geometric redundancy? Planarity of a single face. Planarity of polygons is traditionally seen as a geometric integrity constrained and is checked with run time algorithms up to a certain tolerance. Planarity must be ensured for computational reasons (computational geometry) or visualization purposes. Especially man made objects are designed with planar surfaces. Using surface representation (here: plane based), the discussed model ensures the planarity of a single polygon by means of referential integrity. Planarity of diverse faces. The Face-Plane relationship is of cardinality 1:N. Each face associates exactly one plane. But many faces can associate the same plane. Diverse faces representing the façade could link the same geometrical entity plane. Planarity of different faces is ensured by referential integrity. Parallelism of planes. Fig. 6 illustrates a modern office building. Often planes, which are carrying diverse faces, are parallel to each other. Plane #1, #2 and #3 are parallel. What does this mean in terms of redundancy? Planes are parallel if they have equal rotation; therefore they have the same normal vector. The idea of the discussed data model is to normalize this redundancy by introducing an entity type Normal Vector. Normal vectors are referred by diverse planes (1:N). The translational component d remains with the entity type Plane. Parallelism of planes is ensured by referential integrity.

10 10/17 Figure 7: Parallelism and perpendicularity Perpendicularity of normal vectors. If, for example, two vertical walls are perpendicular to each other, the normal vectors components are switched in the non-zero elements and the sign of the n y component changes. Therefore the data model introduces the entity type Parameter. The Parameter entity is referred by the normal vector. Since the sign might change, the Normal Vector - Parameter association is qualified by a Boolean variable that indicates a change in algebraic sign. This model is called plane-based and geometrically normalized parameterization. 6 Numerical Examples In this chapter the degree of redundancy is calculated. Here, the model instance is seen as a set of atomic piece of information, where each piece has the size 1. One piece of information is either positive or negative. In terms of deterministic information a positive piece needs to be stored whereas a negative piece needs to be checked. The total sum of information equals zero when all the data has been checked. In terms of stochastic information (i.e. Least-Squares-Adjustment) a positive piece can be seen as one unknown in the equation system, with the negative piece being a condition or an observation. The total sum of information equals zero when the so called minimal configuration is present. 6.1 Cuboid Point aggregate. A cuboid needs 9 pieces of information (3 Translation, 3 Rotation, 3 Dimensions). With point representation 8 points have to be stored. Since each point has 3 components this equals 24 pieces of information. Thus the overhead of information (redundancy) can be calculated with 24-9=15. This means, that 15 consistency checks have to be made. These are:

11 11/17 - Planarity of faces (-6) - Parallelism of opposite faces. Normal vector dot product equals -1 (-3) - Perpendicularity of pairs of parallel faces (-3) - Orientation of faces (inside or outside) (-3) Normalized to Parallelism of Planes. 3 normal vectors need to be stored if the cardinality plane-normal vector is 1:N. 6 translational components d must be attached to each plane. So the number of stored values is 3*3 + 6 = 15. There still is a redundancy of 6. So 6 negative pieces of information need to be specified: - Absolute value of the 3 normal vectors equals 1 (-3) - Perpendicularity of all three normal vectors (-3) Normalized to Perpendicularity of normal vectors. In man made objects like buildings most walls are vertical whereas floor and ceiling are horizontal. With the cuboid example this means that 2 rotational components are fixed. Under the assumption of being parallel to the z-axis a cube needs 7 pieces of information (3 Translation, 1 Rotation, 3 Dimensions). Instead of storing three normal vectors 6 values are stored (α 1 = 0; α 2 = 0; α 3 = 1; α 4 = a; α 5 = b; α 6 = 0) whereas the normal vectors refer to theses values α 1 0 α 4 a α 5 b n 1 = α 2 = 0 ; n 2 = α 5 = b ; n 3 = α 4 = a α 3 1 α 6 0 α 6 0 α 1 0 α 4 a α 5 b n 4 = α 2 = 0 ; n 5 = α 5 = b ; n 6 = +α 4 = a α 3 1 α 6 0 α translational components d must be attached to each plane. So the number of stored values is = 12. There still is a redundancy of 5. So 5 negative pieces of information need to be specified: - α 1, α 2, α 3, α 6, are fixed (-4) - a, b are rotation parameters so a 2 + b 2 = 1 ( 1) Even with this very simple example of a cube one can find many ways of parameterization. The decision, which type of parameterization is applied, depends on the application. 6.2 Building Fig.8 illustrates the application of a plane-based parameterization to a building. The model consists of 412 nodes. With a point aggregate model 1236 coordinate values are stored. In the discussed plane based parameterization 104 translational components d and 14 α are stored. Under the assumption of coplanarity, parallelism and perpendicularity of vertical walls, the amount of stored geometric information is reduced about 90%.

12 12/17 Figure 8: Plane based parameterization with a residential house [Clemen and Gründig, 2007] 7 Models from Measurement 7.1 General Method Surveying engineers solve inverse problems. An inverse problem is estimating the model parameters from a set of measured data. In the case of architectural indoor surveying the model parameters to be estimated are those that describe the building geometry. In the discussed parameterization these are n x, n y, n z and d for each plane. The data given are measurements taken with surveying instruments. Since surveying instruments provide precise measurements but no information about the topological configuration of the survey, the engineer (manually) specifies the observation topology. From a methodical point of view the observation topology must not be confounded with the object topology described in section 3 and 4. Figure 9: Observation Topology and Object Topology [Clemen et al., 2005] Due to the statistic properties of measurements and the desired probabilistic redundancy an adjustment algorithm is applied. A successful adjustment delivers a contradiction free (adjusted) observations, estimated model parameters (unknowns), and stochastic information about precision and reliability.

13 13/ Database Design: The Surveying Engineer point of view The adjustment provides a solution for the unknowns. But the estimated model parameters are stochastically correlated. Therefore it would NOT be a good idea to store the estimated parameters in the project database and neglect the original observations. A stochastically correct representation would have to store the Variance-Covariance matrix of the unknown parameters, which would need much storage. Figure 10: What are the primary data? The discussed model takes into account, that the surveyed observations (directions, distances, height differences) are stochastically uncorrelated and thus can be stored with a single attribute, the variance, only. From the surveying engineer point of view, the parameters of the absolute geometry are a database view on the primary data (original measurements). What is this approach good for in everyday engineering life? A stochastically correct parameterization allows applying statistical testing during the process of geometric and topologic generalization. Storing the original measurements is robust, since gross errors and poor functional models (EDM offset, scale) are identifiable. 7.3 Adjustment (Functional Model) The functional model describes the deterministic properties of the physical situation under consideration. For parametric adjustment the observations l and the residuals v are expressed as a (nonlinear) function of the unknowns x. The unknowns are of type (normalized) normal vector n and orthogonal distance d to the origin of the local coordinate system. n = n, n = n 2 x + n 2 y + n 2 z = 1 (1) This article describes three types of equations: Distances between topological primitives, angular constraints of topological primitives, spherical coordinates on topological primitives.

14 14/ Distances between topological primitives Distances are measured with ruler, measuring tape, laser distance meter or simply by pacing. In the Euclidean space the distance is calculated between two points. The discussed parameterization is not based on point coordinates but on surface parameters. The functional model of the distance observation depends on the dimension of the topological primitives. Since the topological primitives edge and face represent point sets, additional constraints to this point sets must be attached to the model. Distance Face-Face. If a distance l between the parallel faces i and j is measured the observation equation distance face-face is described as: l + v = d j d i (2) The geometric solution must ensure parallelism of the two planes that carry the faces. This is realized by the constraint, that the dot product of the normal vectors equals 1. n i, n j = 1 (3) Distance Face-Edge. If a distance l between the faces i and edge j k, i.e. is the intersection of the faces j and k, is measured the observation equation distance face-edge is described as: l + v = n i, p jk + d i... [ p jk = ( n j, n k, n j ] n k ) T 1 (dj, d k, 0) T (4) The functional model needs to ensure that face i and edge j k are parallel n i, n j n k = 0 (5) Distance Face-Node. If a distance l between the faces i and node j k l, i.e. is the intersection of the faces j, k and l, is measured the observation equation distance face-node is described as: l + v = n i, pjkl + d i... [ pjkl = ( n j, n k, ] n l ) T 1 (dj, d k, d l ) T (6) Distance Edge-Edge. If a distance l between the two edges i j and k l is measured the observation equation distance edge-edge is described as: l + v = p ij p kl... [ p ij = ( n i, n j, n i ] n j) T 1 (di, d j, 0) T ; [ p kl = ( n k, n l, n k ] n l ) T 1 (dk, d l, 0) T (7) n i n j, n k n l = 0 (8) Distance Edge-Node. If a distance l between the edges i j and node k l m is measured the observation equation distance edge-node is described as: l + v = p ij p klm... [ p ij = ( n i, n j, n i ] n j) T 1 (di, d j, 0) T ; [ p klm = ( n k, n l, ] n m) T 1 (dk, d l, d m) T (9)

15 15/17 Distance Node-Node. If a distance l between the nodes i j k and node l m n is measured the observation equation distance node-node is described as: l + v = p ijk p lmn... [ p ijk = ( n i, n j, ] n k ) T 1 (di, d j, d k ) T ; [ p lmn = ( n l, n m, ] n n) T 1 (dl, d m, d n) T (10) Angular Constraints between topological primitives Angular constraints are included to the model for ensuring the observational integrity (like parallelism of faces that are connected by a distance measurement) and to find a mathematical description for obvious situations like wall perpendicular to floor or floor parallel to ceiling. In order to facilitate statistical testing, these conditions are introduced as so called pseudo observation and therefore are a subject to random error. Angular Constraints Face-Face. If parallelism or perpendicularity of two faces is not specified implicitly by references, the pseudo-observation l is parameterized as follows: } 0 + v = n 1 + v i, { perpendicular n j (11) parallel Angular Constraints Face-Edge. If parallelism or perpendicularity of faces i and edge j k is not specified implicitly by references, the pseudo-observation is parameterized as follows: } 0 + v = n 1 + v i, n j { parallel n k perpendicular Angular Constraints Edge-Edge. If parallelism or perpendicularity of the two edges i j and k l is not specified implicitly by references, the pseudo-observation is parameterized as follows: } 0 + v = n 1 + v i n j, n k { parallel n l perpendicular Spherical coordinates on topological primitives Local spherical coordinates (r, θ, φ) T are taken with the total station at position (x T, y T, z T ) T. Since the instrument axis of total station is adjusted to the vertical, only one rotational degree of freedom remains per instrumental set up. The three components of (r, θ, φ) T are considered to stochastically independent. The observation of type spherical point i on face j is described as a conditional equation: f(x, l +v) = p i, n j d j = (r i + v r ) sin (θ i + v θ ) cos (φ i + v φ + ω T ) (r i + v r ) sin (θ i + v θ ) sin (φ i + v φ + ω T ) (r i + v r ) cos (θ i + v θ ) + x T y T z T, n j d j = 0 (12) The observation of type spherical point i on edge j k is modeled as a system of two equations f(x, l + v) = p i, n j d j = 0 f(x, l + v) = p i, n k d k = 0 (13)

16 16/17 The observation of type spherical point i on node j k l is modeled as a system of three equations f(x, l + v) = p i, n j d j = 0 f(x, l + v) = p i, n k d k = 0 (14) f(x, l + v) = p i, n l d l = Post Adjustment Statistics Post Adjustment Statistics are applied in order to - test the general goodness of fit and detect whether mistakes occur in observations or the functional/statistical model. ( χ 2 -Test) - test if two normal vectors are stochastically equal. If true, the normal vectors would be unified by interchanging the references in the database. (Student s t-test) - test if two orthogonal distances d of parallel planes are stochastically equal. If true the two planes will be unified (Student s t-test). - Test for outliers due to bad face-plane assignment. (z-test on residuals v i ) 8 Outlook This article shows in detail, how topological and geometrical normalization reduces the amount of redundantly stored information. [Clemen and Gründig, 2007] showed how practical applications, i.e. data capture tools, can benefit from this data model by applying adjustment techniques. [Huhnt and Gielsdorf, 2006] showed that the conceptual model does not only fit to geo-information tasks but also to diverse aspects of civil engineering design. Further research will show which degree of normalization is suitable to which application or problem domain. In addition, curved surfaces will be introduced to the data model. This extended data model and its impact on topological and geometrical validation routines are subject to further work. REFERENCES [Bradley, 2006] Bradley, bradley/arch/archkomp/. [Clemen and Gründig, 2007] Clemen, C. and Gründig, L., The imap principle in building information systems. Proceedings of XXX FIG Working Week 2007, Hong Kong, China. [Clemen et al., 2005] Clemen, C., Gründig, L. and Gielsdorf, F., Reverse engineering for generation of 3d-building-information-models applying random varibles in computer aided design. Proceedings of XXIII FIG Working Week 2005,Cairo, Egypt. [Ellul, 2005] Ellul, C., e. a., A generic topological data structure for 3d data. Topology and Spatial Databases Workshop, Glasgow, UK.

17 17/17 [Gründig and Gielsdorf, 2002] Gründig, L. and Gielsdorf, F., Geometrical modeling for facility managment systems applying surface parameter. XXII FIG Federation Internationale des Geometres Congress, Washington D.C., USA. [Huhnt and Gielsdorf, 2006] Huhnt, W. and Gielsdorf, F., Topological information as leading information in building product models. Proceedings of 17th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany. [Mäntylä, 1988] USA. Mäntylä, M., Solid Modeling. Computer Science Press, Rockville, [Mikhail, 1976] Mikhail, E., Observation and Least Squares. University Press of America, Lanham-New York-London. [Thompson, 2007] Thompson, R., Towards a Rigorous Logic for Spatial Data Representation. PhD thesis, TU Delft, Netherlands. [Thompson and Oosterom, 2008] Thompson, R. and Oosterom, P. v., Mathematically provable correct implementation of integrated 2D and 3D representations. Springer. Berlin. [Zlatanova, 2004] Zlatanova, S., e. a., Topological models and frameworks for 3d spatial objects. Journal of Computers & Geosciences. Biographical Notes Christian Clemen, born Graduated in 2004 as Dipl.-Ing. in Surveying from Technical University of Berlin. Since 2004 Assistant at the Department of Geodesy and Geoinformation, Technical University of Berlin. Dr.-Ing. habil. Frank Gielsdorf, born Graduated in 1987 as Dipl.-Ing. in Surveying from Technical University of Dresden. Obtaining doctoral degree in 1997 from Technical University of Berlin. Since 1995 Assistant Professor at the Department of Geodesy and Geoinformation, Technical University of Berlin. Contact Dipl.-Ing. Christian Clemen Technische Universität Berlin Institute for Geodesy and Geoinformation Science Secretary H20 Strasse des 17. Juni 135 D Berlin GERMANY clemen@fga.tu-berlin.de website: