Modelling and validation of 3D cadastral objects

Modelling and validation of 3D cadastral objects R.M. Thompson Queensland Government, Australia/ Delft University of Technology, GIS Technology Section, Delft, The Netherlands

P.J.M van Oosterom Delft University of Technology, GIS Technology Section, Delft, The Netherlands

ABSTRACT: As the value of land in the urban regions of the world increases, there is a trend towards the subdivision of property rights in 3D. That is to say, rights to land may be replaced by rights to the space above and below the land. This increases the complexity of the descriptions that are needed to define the regions of space to which the property rights apply. The simple plans of subdivision that are used in defining property rights on the surface of the earth are being replaced by far more complex 3D spatial definitions. An important issue in the framing of these definitions is that they must be correct and unambiguous, because an error or ambiguity in the definition of the extent of a property can lead to expensive legal disputes. This paper explores an axiomatic approach to the problem of defining what is a valid definition of a 3D Cadastral objects (parcels). This paper addresses three problems: 1. Modelling and validation of a single cadastral parcel; 2. Modelling and validation of a set of cadastral parcels; 3. Modelling and validation of a complete spatial partition. 1 INTRODUCTION There is strong current trend in urban areas for property rights to be defined in 3D. That is to say, rights to land may be replaced by rights to the space above and below it, increasing the complexity of the definitions of the regions of space. This paper explores an axiomatic approach to the problem of defining what is a valid definition of a 3D Cadastral objects (parcels). The paper attempts to be neutral in terms of the storage method used, and the details of any interchange formats involved. It does, however, assume the node, edge, face, volume paradigm (ISO-TC211 2003). It does not consider other possible representations, such as for example the Regular Polytope (Thompson 2007; Thompson and Oosterom 2010), or a set of constraints (Rigaux, Scholl, Segoufin and Grumbach 2003). It also focuses specifically on issues that arise because of the use of finite precision arithmetic such as floating point. Many of the issues addressed here do not apply if infinite precision rational numbers are used (Lema and Güting 2002), but the aim here is to achieve a similar level of rigour using the more widely and available, faster, and less storage-intensive floating point representation. Floating point arithmetic leads to accumulation of errors that must be taken into consideration if reliable and repeatable results are to be achieved. Intuitively, the object being defined here – the parcel, is a unit of space which is the minimal unit for the definition of property rights (such as ownership). It is a generalisation of the familiar 2D land (property) parcel, or “lot”. The FIG joint commission 3 and 7 working group on 3DCadastres (http://www.gdmc.nl/3DCadastres/) defines a 3D parcel as the spatial unit against which (one or more) unique and homogeneous rights (e.g. ownership right or land use right), responsibilities or restrictions are associated to the whole entity, as included in a Land Administration system (based on ISO 19152). Homogenous means that the same combination of rights equally apply within the whole 3D spatial unit. Unique means that this is the largest spatial unit for which this is true. Making the unit any larger would result in the combination of rights not

being homogenous. Making the unit smaller would result in at least 2 neighbour 3D parcels with the same combinations of rights. The individual land parcel should comprise a continuous unit (although some grouping of parcels into a multi-parcel will be necessary in some cases). The 3D parcel should be defined in such a way that the classical 2D parcel can be seen as a specialization (variant). In the emerging standard ISO19152, this is done by recognising that a “2D parcel” is in reality a column of space with no top or bottom defined. This reflects the legal situation in many jurisdictions – where there is no legally defined limit to the vertical extent of a 2D parcel. Individual parcels may be in contact with one another, and often share common faces, but base parcels (primary interests) are not permitted to overlap. Most jurisdictions allow “secondary interests” in land/space to be defined which may overlap primary interests. A further desirable attribute of a set of cadastral parcels is completeness –all the space within the area of interest must be accounted for – with no gaps of undefined space between parcels. This paper is restricted to parcels defined by planar surfaces, but further research is planned to remove this restriction. Section 2 reviews current research in the field and positions this paper in relation to it. The rationale for validation is discussed, the precision of calculation is discussed, and cadastral issues are presented. Section 3 formulates the axioms, discussing the rationale and consequences of adopting them. Section 4 discusses practicalities and implementation issues related to validation and automatic correction of parcels. Section 5 introduces some of the issues raised by introducing partially unbounded parcels. Section 6 addresses specific algorithmic issues in more detail. An important fact when considering cadastral information is that there can be significant variation in the detailed requirements, especially in the validation rules. This is treated in Section 7. Section 8 suggests future research. The appendix gives an outline of the point in parcel test. 2 BACKGROUND The representation of 3D objects using CAD (Computer Aided Design/Drafting) is not new, and significant work has been done on ensuring that the computer-based model is valid and is a good representation of the real-world object as it exists, or as it is to be constructed. The problem with the cadastral parcel is slightly different. Cadastral parcels are not “real-world” objects, although they may be associated with them. A cadastral parcel is a theoretical definition of space. One result of this is that the validation rules of cadastral parcels may have differences from the rules in CAD for physical objects. The approach taken by Oracle Spatial is described in (Kazar, Kothuri, van Oosterom and Ravada 2008). This provides a clear description of the rules for validating 3D geometries that are to be imported into the Oracle database, but does not address the issue of the finite arithmetic in a computational representation. Oracle does not allow faces with inner rings; see Figure 1c.

A

B

C

Note – there is no shared face between the two parts of this object

Figure 1. Cases A and B are disallowed by the axioms of Gröger and Plümer (which requires the surfaces to be 2-manifold). Case C is disallowed by the Oracle Spatial rules (which disallow inner rings on a face).

(Gröger and Plümer 2005) Gives a set of axioms that define a “2.8D” coverage – which has many of the required attributes of the problem domain, but with restrictions. One of the restrictions – the inability to model bridges or tunnels has been removed in a later paper (Gröger and

Plümmer 2011a), and extended to solid objects and a space partition in (Gröger and Plümmer 2011b). A remaining restriction is that the surfaces are required to be 2-manifold; see Figure 1a and 1b. Brugman, Tijssen and van Oosterom (2011) discuss the related problem of creating a 3D partition of space in a general context, but do not consider the issue of finite precision of the arithmetic apart from briefly discussing the difficulty of assuring planarity of faces. 2.1 Reasons to validate spatial data It is important to keep in mind why validation is necessary. The action of validating data, particularly in 3D can be time-consuming and therefore costly. Unless a benefit results, it must be questioned. There are 4 aims that can be identified to justify validation rules: Aim 1. To ensure that the definition is unambiguous. For example, in the Cadastral domain, to avoid argument in a court of law, rather the intent should be rigorously defined. As an example, if a volume is defined by surface patches which do not close, then it may be debatable whether a point in space is within or outside the volume; see Figure 2.

Figure 2. Is the point within the region? (Which has two faces missing).

Aim 2. To allow “Programming by Contract”. If data in a database is known to satisfy a set of axioms, the number of special cases that must be accommodated in an application program is dramatically reduced. E.g. given axioms “Different nodes have different coordinates” and “each edge has two distinct nodes as end points” it is not necessary to test for zero length edges. Aim 3. To ensure that data can be transferred without loss of integrity and without being rejected by the recipient. This can be difficult, as the definitions of validity checks applied by commercial software are not always clearly stated. Even worse, tight validation is seen as a selling point in commercial software, and is presented as “finding errors missed by other systems”. Aim 3 can only be achieved where the validation rules are agreed, or at least well documented. Aim 4. To prevent errors in the database. This is the least valid reason. It is quite easy to have incorrect data that pass validation. It is also possible to have a correct representation which is invalid because it contains topological failures that are smaller than the accuracy of the data (although the validity checks fail, the representation still correctly represents the real world feature); see Figure 3.

p

Figure 3. Invalid data (in 2D) that is correct. The hole in the polygon crosses the outer boundary – failing validation. On the other hand, the all point positions are correct within the measurement limits.

2.2 Finite precision In the 2D world of GIS, the problem of finite computation precision has been handled by the concept of normalisation (Milenkovic 1988), or some variant. This paper extends the concept into 3D. It is critical to the approach that there exists a tolerance value ε with the characteristic that arithmetic operations can be assured to give a result that is correct to an order of magnitude smaller that ε (Milenkovic uses 1/10). This gives rise to a question of point identity (A ≡ B). There is a distinction to be made between points which are close together (distance D(A, B) < d); at zero distance apart (D(A, B) = 0); close but the calculated distance rounds to zero (D(A, B) is calculated as zero); and points which are identified as the same point logically (e.g. the meeting of several faces). Here it is assumed that points are uniquely identified, and so the statement that points must be a minimum of distance of ε apart excludes the possibility that two points are at the same 3D location. An alternate strategy uses equality of the coordinates as a test for identity – as, for example A = (xa,ya,za), B = (xb,yb,zb), A ≡ B ⇔ xa = xb, ya = yb, za = zb. This may have problems if floating point coordinate values are used unless consistency of arithmetic can be guaranteed. The universal use of IEEE floating could make this acceptable (IEEE 1985; Goldberg 1991), but, for example, Java does not use IEEE floating point. A further alternative, defines a second tolerance ι