# Extension of RCC*-9 to Complex and Three-Dimensional Features and Its Reasoning System

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## Abstract

**:**

## 1. Introduction

## 2. Definition of Geometric Features

#### 2.1. Features in 2D Space

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Definition 6.**

**Definition 7.**

**Definition 8.**

**Definition 9.**

**Definition 10.**

**Definition 11.**

**Definition 12.**

**Definition 13.**

#### 2.2. Features in 3D Space

**Definition 14.**

**Definition 15.**

**Definition 16.**

**Definition 17.**

**Definition 18.**

**Definition 19.**

**Definition 20.**

## 3. Definition of RCC*-9

**Definition 21.**

**Definition 22.**

**Definition 23.**

**Definition 24.**

**Definition 25.**

**Definition 26.**

**Definition 27.**

**Definition 28.**

**Definition 29.**

**Definition 30.**

**Definition 31.**

**Definition 32.**

**Definition 33.**

**Definition 34.**

**Definition 35.**

## 4. Demonstration That RCC*-9 Is JEPD

#### 4.1. The Case of Lines and Regions in 2D

**Experiment 1.**

**Experiment 2.**

**Experiment 3.**

#### 4.2. The Case of Points in 2D

**Experiment 4.**

#### 4.3. The Case of 3D Features

**Equivalence 1.**

**Proof.**

**Equivalence 2.**

**Equivalence 3.**

**Proof.**

**Equivalence 4.**

**Equivalence 5.**

**Equivalence 6.**

**Equivalence 7.**

**Equivalence 8.**

**Proof.**

**Equivalence 9.**

**Equivalence 10.**

**Equivalence 11.**

**Equivalence 12.**

**Proof.**

**Equivalence 13.**

**Proof.**

**Equivalence 14.**

**Equivalence 15.**

**Experiment 5.**

**Experiment 6.**

## 5. Spatial Reasoning

**Experiment 7.**

**.**We considered a dataset of 100 random simple polygons and calculated the 10,000 relations holding among them. We repeated the random generation 10 times. Therefore, the composition table was overall filled using 10,000,000 cases of composition. The obtained composition table is the same as in Cui et al. [5]. This thus provides evidence that the new definitions of RCC*-9 with respect to RCC-8 do not modify the nature of the relations in the case of simple regions.

**Experiment 8.**

**.**We considered a dataset of mixed regions (100 complex regions with holes and disconnected components, 100 simple regions, and 100 complex regions made up of two disconnected small squares, plus about 10 rare configurations) to maximize the probability of finding all combinations of relations. The random generation of polygons was repeated 10 times for a total of 297,910,000 compositions. As a result, we obtained the composition table in Table 1. In the table, the symbol U represents the universal relation, that is, the disjunction of all nine base relations of RCC*-9. This table corresponds to the composition table that appeared in [35] with the exception of the cases $\mathsf{TPP}\oplus \mathsf{CR}\phantom{\rule{4pt}{0ex}}=\mathsf{TPP}\phantom{\rule{4pt}{0ex}}$ and $\mathsf{CR}\oplus \mathsf{TPPi}\phantom{\rule{4pt}{0ex}}=\mathsf{TPPi}$, which we did not find in this experiment for complex regions. (Indeed, from the analyses of subsequent experiments, it can be seen that these two cases of composition are never possible, and, therefore, they were erroneously included in ([35]).)

**Experiment 9.**

**.**We considered a random set of simple lines (100 were polylines with 7 vertices, 100 horizontal segments, and 100 vertical segments). The number of calculated compositions is similar to the previous experiment. The composition table (Table 2) reveals that there are more composition cases that were not included in Clementini and Cohn [35]. These additional cases are $\mathsf{EC}\oplus \mathsf{EC}=\{\mathsf{NTPPi},\mathsf{NTPP}\}$, $\mathsf{EC}\oplus \mathsf{NTPP}=\mathsf{EC}$, $\mathsf{EC}\oplus \mathsf{NTPPi}=\mathsf{EC}$, $\mathsf{NTPP}\oplus \mathsf{EC}=\mathsf{EC}$, and $\mathsf{NTPPi}\oplus \mathsf{EC}=\mathsf{EC}$. From the composition table, we can also see that there are cases that are not possible for simple lines but that were possible for complex regions: $\mathsf{TPP}\oplus \mathsf{TPPi}=\mathsf{CR}$, $\mathsf{TPP}\oplus \mathsf{NTPPi}=\mathsf{CR}$, $\mathsf{NTPP}\oplus \mathsf{TPPi}=\mathsf{CR}$, $\mathsf{NTPP}\oplus \mathsf{NTPPi}=\mathsf{CR}$, $\mathsf{TPPi}\oplus \mathsf{CR}=\{\mathsf{TPPi},\mathsf{NTPPi}\}$, $\mathsf{NTPPi}\oplus \mathsf{CR}=\{\mathsf{TPPi},\mathsf{NTPPi}\}$, $\mathsf{CR}\oplus \mathsf{TPP}=\{\mathsf{NTPP},\mathsf{TPP}\}$, and $\mathsf{CR}\oplus \mathsf{NTPP}=\{\mathsf{NTPP},\mathsf{TPP}\}$.

**Experiment 10.**

**.**The experiment for complex lines was similar to the previous experiment, adding 200 complex lines made up of self-intersections and disconnected components to the previous set of simple lines for each random generation. No variations in the composition table were discovered. Hence, the composition tables for simple lines and complex lines are the same.

**Experiment 11.**

**.**In this experiment, we include composition cases that can be obtained from relations between simple regions, relations between simple lines, and relations between regions and lines. With a random set of 330 simple lines and 200 simple polygons and repeating the random generation 10 times, the composition table was filled with 1,488,770,000 cases of composition, obtaining the result of Table 3. We discovered the following cases that are realizable with compositions involving both lines and regions that were not included in previous experiments: $\mathsf{EC}\oplus \mathsf{PO}=\mathsf{TPPi}$, $\mathsf{EC}\oplus \mathsf{TPPi}=\{\mathsf{CR},\mathsf{NTPPi},\mathsf{TPPi},\mathsf{PO}\}$, $\mathsf{EC}\oplus \mathsf{NTPPi}=\mathsf{TPPi}$, $\mathsf{EC}\oplus \mathsf{CR}=\mathsf{TPPi}$, $\mathsf{PO}\oplus \mathsf{EC}=\mathsf{TPP}$, $\mathsf{PO}\oplus \mathsf{TPP}=\mathsf{EC}$, $\mathsf{TPP}\oplus \mathsf{EC}=\{\mathsf{CR},\mathsf{NTPP},\mathsf{TPP},\mathsf{PO}\}$, $\mathsf{TPP}\oplus \mathsf{TPPi}=\{\mathsf{NTPPi},\mathsf{NTPP}\}$, $\mathsf{NTPP}\oplus \mathsf{EC}=\mathsf{TPP}$, $\mathsf{NTPP}\oplus \mathsf{TPP}=\mathsf{TPP}$, $\mathsf{TPPi}\oplus \mathsf{PO}=\mathsf{EC}$, $\mathsf{TPPi}\oplus \mathsf{TPP}=\mathsf{EC}$, $\mathsf{TPPi}\oplus \mathsf{NTPP}=\{\mathsf{EC},\mathsf{TPPi}\}$, $\mathsf{TPPi}\oplus \mathsf{NTPPi}=\mathsf{TPPi}$, $\mathsf{TPPi}\oplus \mathsf{CR}=\mathsf{EC}$, $\mathsf{NTPPi}\oplus \mathsf{TPP}=\{\mathsf{EC},\mathsf{TPP}\}$, $\mathsf{CR}\oplus \mathsf{EC}=\mathsf{TPP}$, and $\mathsf{CR}\oplus \mathsf{TPP}=\mathsf{EC}$.

**Experiment 12.**

**.**In this experiment, we generated a scenario made up of a mixing of about 600 previously considered features, simple and complex regions and lines, including multipoints as well. We did not discover any changes from the previous experiment. Hence, the composition table for complex features is the same as Table 3 for simple features.

**Experiment 13.**

**.**In this final experiment, we used the 3D scenarios from Experiments 5 and 6. As we already discussed, the number of 3D features that we could consider was limited due to the high computation time. We used a random distribution of 30 simple polyhedrons and then a random distribution of 24 simple polyhedrons, 8 convex polygons, and 8 segments. For simple polyhedrons, the resulting composition table is very similar to the composition table for simple regions in 2D space, while for mixed 3D simple features the composition table is close to the composition table of simple features in 2D. Unfortunately, the small number of involved relations was not sufficient to fill the composition table with all possible results. The result from this experiment is partial: the composition tables for simple polyhedrons and simple 3D features are a subset of the respective composition tables in 2D, but we did not discover evidence of all the entries.

## 6. Implementation of Experiments

`Relate`function to be found in the OGC Simple Features Specification [6]. For the

`Relate`function, we used the implementation provided by the “Shapely” library in Python (https://pypi.org/project/Shapely/, accessed on 10 November 2023). The function returns a string that represents a set of nine values for the DE+9IM matrix introduced in [8]. The string expresses the matrix by rows, where an “

`F`” stands for an empty intersection and values 0, 1, and 2 express the dimension of the intersection if the intersection is not empty. DE+9IM relations can be transformed in the corresponding 9IM string (each character 0, 1, or 2 is transformed to a “

`T`”, expressing a non-empty intersection). The equivalent RCC*-9 relation can be found by applying the correspondence in Table 4. The symbol “

`*`” in the pattern indicates that both values “

`T`” and “

`F`” are possible. (This latter table also appeared in [35], but it has been updated in this paper following the modified definitions of RCC*-9.)

#### 6.1. Assessment of the JEPD Properties

#### 6.2. Finding Composition Tables

#### 6.3. Implementation of 3D Experiments

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The subsumption hierarchy of RCC*-9 relations. A link between two relations represents an implication from the lower one to the upper one. The shaded area includes the relations that have the same syntactical definitions as in RCC-8.

**Figure 2.**Illustrations of the NTPP definition: (

**a**) two simple regions; (

**b**) two simple lines; (

**c**) a simple line and a simple region; (

**d**) two simple bodies.

**Figure 3.**Illustrations of the TPP definition: (

**a**) two simple regions; (

**b**) two simple lines; (

**c**) a simple line and a simple region; (

**d**) two simple bodies.

**Figure 4.**Illustrations of the $\mathsf{PO}$ relation: (

**a**) two simple regions; (

**b**) a simple line and a simple region; (

**c**) two simple lines; (

**d**) a simple region and a simple body.

**Figure 5.**Illustrations of the $\mathsf{EC}$ relation: (

**a**) two simple regions; (

**b**) a simple line and a simple region; (

**c**) two simple lines; (

**d**) a simple region and a simple body.

**Figure 6.**Illustrations of the $\mathsf{CR}$ (

**a**) a simple region and a simple line; (

**b**) two simple lines; (

**c**) two complex regions with separations; (

**d**) a simple region and a simple body.

**Figure 7.**Experiment 1: (

**a**) random generation of simple regions and lines; (

**b**) percentages of obtained RCC*-9 relations.

**Figure 8.**Experiment 2: (

**a**) random generation of complex regions; (

**b**) percentages of obtained RCC*-9 relations.

**Figure 9.**Experiment 3: (

**a**) random generation of complex features; (

**b**) percentages of obtained RCC*-9 relations.

**Figure 10.**Experiment 4: percentages of obtained RCC*-9 relations in the case of random multipoints and complex features.

**Figure 11.**Samples of 3D RCC*-9 relations: (

**a**) an EC relation between two polyhedrons; (

**b**) a PO relation between a polygon and a polyhedron; (

**c**) a CR between a segment and a polyhedron.

**Figure 12.**Experiment 5: (

**a**) random generation of polyhedrons; (

**b**) percentages of obtained RCC*-9 relations.

**Figure 13.**Experiment 6: (

**a**) random generation of 3D features; (

**b**) percentages of obtained RCC*-9 relations.

r_{2} | DC | EC | PO | TPP | NTPP | TPPi | NTPPi | EQ | CR | |

r_{1} | ||||||||||

DC | U | DC, EC, PO, TPP, NTPPCR | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | DC | DC | DC | DC, EC, PO, TPP, NTPP, CR | |

EC | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, PO, TPP, TPPi, EQ, CR | DC, EC, PO, TPP, NTPP, CR | EC, PO, TPP, NTPP, CR | PO, TPP, NTPP, CR | DC, EC | DC | EC | DC, EC, PO, TPP, NTPP, CR | |

PO | DC, EC, PO, TPPi, NTPPiCR | DC, EC, PO, TPPi, NTPPi, CR | U | PO, TPP, NTPP, CR | PO, TPP, NTPP, CR | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | PO | U∖EQ | |

TPP | DC | DC, EC | DC, EC, PO, TPP, NTPP, CR | TPP, NTPP | NTPP | DC, EC, PO, TPP, TPPi, EQ, CR | DC, EC, PO, TPPi, NTPPi, CR | TPP | DC, EC, PO, NTPP, CR | |

NTPP | DC | DC | DC, EC, PO, TPP, NTPP, CR | NTPP | NTPP | DC, EC, PO, TPP, NTPP, CR | U | NTPP | DC, EC, PO, TPP, NTPP, CR | |

TPPi | DC, EC, PO, TPPi, NTPPi, CR | EC, PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | PO, TPP, TPPi, EQ | PO, TPP, NTPP, CR | TPPi, NTPPi | NTPPi | TPPi | PO, TPPi, NTPPi, CR | |

NTPPi | DC, EC, PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | PO, TPP, NTPP, TPPi, NTPPi, EQ, CR | NTPPi | NTPPi | NTPPi | PO, TPPi, NTPPi, CR | |

EQ | DC | EC | PO | TPP | NTPP | TPPi | NTPPi | EQ | CR | |

CR | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | U∖EQ | PO, TPP, NTPP, CR | PO, TPP, NTPP, CR | DC, EC, PO, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | CR | U |

r_{2} | DC | EC | PO | TPP | NTPP | TPPi | NTPPi | EQ | CR | |

r_{1} | ||||||||||

DC | U | DC, EC, PO, TPP, NTPPCR | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | DC | DC | DC | DC, EC, PO, TPP, NTPP, CR | |

EC | DC, EC, PO, TPPi, NTPPi, CR | U | DC, EC, PO, TPP, NTPP, CR | EC, PO, TPP, NTPP, CR | EC, PO, TPP, NTPP, CR | DC, EC | DC, EC | EC | DC, EC, PO, TPP, NTPP, CR | |

PO | DC, EC, PO, TPPi, NTPPiCR | DC, EC, PO, TPPi, NTPPi, CR | U | PO, TPP, NTPP, CR | PO, TPP, NTPP, CR | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | PO | U∖EQ | |

TPP | DC | DC, EC | DC, EC, PO, TPP, NTPP, CR | TPP, NTPP | NTPP | DC, EC, PO, TPP, TPPi, EQ | DC, EC, PO, TPPi, NTPPi | TPP | DC, EC, PO, NTPP, CR | |

NTPP | DC | DC, EC | DC, EC, PO, TPP, NTPP, CR | NTPP | NTPP | DC, EC, PO, TPP, NTPP | U∖CR | NTPP | DC, EC, PO, TPP, NTPP, CR | |

TPPi | DC, EC, PO, TPPi, NTPPi, CR | EC, PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | PO, TPP, TPPi, EQ | PO, TPP, NTPP, CR | TPPi, NTPPi | NTPPi | TPPi | PO, CR | |

NTPPi | DC, EC, PO, TPPi, NTPPi, CR | EC, PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | PO, TPP, NTPP, TPPi, NTPPi, EQ, CR | NTPPi | NTPPi | NTPPi | PO, CR | |

EQ | DC | EC | PO | TPP | NTPP | TPPi | NTPPi | EQ | CR | |

CR | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | U∖EQ | PO, CR | PO, CR | DC, EC, PO, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | CR | U |

r_{2} | DC | EC | PO | TPP | NTPP | TPPi | NTPPi | EQ | CR | |

r_{1} | ||||||||||

DC | U | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | DC | DC | DC | DC, EC, PO, TPP, NTPP, CR | |

EC | DC, EC, PO, TPPi, NTPPi, CR | U | DC, EC, PO, TPP, NTPP, TPPi, CR | EC, PO, TPP, NTPP, CR | EC, PO, TPP, NTPP, CR | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, TPPi | EC | DC, EC, PO, TPP, NTPP, TPPi, CR | |

PO | DC, EC, PO, TPPi, NTPPiCR | DC, EC, PO, TPP, TPPi, NTPPi, CR | U | EC, PO, TPP, NTPP, CR | PO, TPP, NTPP, CR | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | PO | U∖EQ | |

TPP | DC | DC, EC, PO, TPP, NTPP, CR | DC, EC, PO, TPP, NTPP, CR | TPP, NTPP | NTPP | U | DC, EC, PO, TPPi, NTPPi, CR | TPP | DC, EC, PO, NTPP, CR | |

NTPP | DC | DC, EC, TPP | DC, EC, PO, TPP, NTPP, CR | TPP, NTPP | NTPP | DC, EC, PO, TPP, NTPP, CR | U | NTPP | DC, EC, PO, TPP, NTPP, CR | |

TPPi | DC, EC, PO, TPPi, NTPPi, CR | EC, PO, TPPi, NTPPi, CR | EC, PO, TPPi, NTPPi, CR | EC, PO, TPP, TPPi, EQ | EC, PO, TPP, NTPP, TPPi, CR | TPPi, NTPPi | TPPi, NTPPi | TPPi | EC, PO, TPPi, NTPPi, CR | |

NTPPi | DC, EC, PO, TPPi, NTPPi, CR | EC, PO, TPPi, NTPPi, CR | PO, TPPi, NTPPi, CR | EC, PO, TPP, TPPi, NTPPi, CR | PO, TPP, NTPP, TPPi, NTPPi, EQ, CR | NTPPi | NTPPi | NTPPi | PO, TPPi, NTPPi, CR | |

EQ | DC | EC | PO | TPP | NTPP | TPPi | NTPPi | EQ | CR | |

CR | DC, EC, PO, TPPi, NTPPi, CR | DC, EC, PO, TPP, TPPi, NTPPi, CR | U∖EQ | EC, PO, TPP, NTPP, CR | PO, TPP, NTPP, CR | DC, EC, PO, NTPPi, CR | DC, EC, PO, TPPi, NTPPi, CR | CR | U |

RCC*-9 | 9IM |
---|---|

$\mathsf{DC}(x,y)$ | Relate(x,y,"FF*FF****") |

$\mathsf{EC}(x,y)$ | Relate(x,y,"F*TT**T**") ∨ |

Relate(x,y,"FTT***T**") ∨ | |

Relate(x,y,"F*T*T*T**") | |

$\mathsf{NTPP}(x,y)$ | Relate(x,y,"*FF*FFT**") |

$\mathsf{TPP}(x,y)$ | Relate(x,y,"*TF**F***") ∨ |

Relate(x,y,"**F*TF***") ∧ | |

¬ Relate(x,y,"TFFFTFFFT") | |

$\mathsf{CR}(x,y)$ | Relate(x,y,"T*TFFTT**") ∨ |

Relate(x,y,"TFT*F*TT*") ∨ | |

Relate(x,y,"TFTFFFTFT") ∨ | |

Relate(x,y,"TTTFFFTTT") ∨ | |

Relate(x,y,"TFTTFTTFT") | |

$\mathsf{PO}(x,y)$ | Relate(x,y,"TTTT**T**") ∨ |

Relate(x,y,"T*T*T*T**") | |

$\mathsf{NTPPi}(x,y)$ | Relate(x,y,"**TFF*FF*") |

$\mathsf{TPPi}(x,y)$ | Relate(x,y,"***T**FF*") ∨ |

Relate(x,y,"****T*FF*") ∧ | |

¬ Relate(x,y,"TFFFTFFFT") | |

$\mathsf{EQ}(x,y)$ | Relate(x,y,"TFFF*FFFT") |

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**MDPI and ACS Style**

Clementini, E.; Cohn, A.G.
Extension of RCC*-9 to Complex and Three-Dimensional Features and Its Reasoning System. *ISPRS Int. J. Geo-Inf.* **2024**, *13*, 25.
https://doi.org/10.3390/ijgi13010025

**AMA Style**

Clementini E, Cohn AG.
Extension of RCC*-9 to Complex and Three-Dimensional Features and Its Reasoning System. *ISPRS International Journal of Geo-Information*. 2024; 13(1):25.
https://doi.org/10.3390/ijgi13010025

**Chicago/Turabian Style**

Clementini, Eliseo, and Anthony G. Cohn.
2024. "Extension of RCC*-9 to Complex and Three-Dimensional Features and Its Reasoning System" *ISPRS International Journal of Geo-Information* 13, no. 1: 25.
https://doi.org/10.3390/ijgi13010025