# A 27-Intersection Model for Representing Detailed Topological Relations between Spatial Objects in Two-Dimensional Space

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{+}-intersection model [18], the uncertain intersection and difference model [19] and the double straight line four-intersection models [20] have been proposed. All of the above models consider the content invariant (i.e., the emptiness or non-emptiness) of the intersections. The 4IM and 9IM were compared on spatial relations, and the result was that 9IM with the content invariant provides more details than does the 4IM [9]. A complete classification of spatial relations was demonstrated using the Voronoi-based nine-intersection model [21]. Topological relations for spatial scene [22,23] and compound spatial objects [24] were also presented. We can conclude that the 4IM and 9IM, as well as extended models based on these have been widely studied.

## 2. Methods

#### 2.1. Spatial Data Model

**Definition**

**1:**

**Definition**

**2:**

**Definition**

**3**

#### 2.2. Different Topological Invariants for Topological Relations

^{9}) binary topological relations can be distinguished in theory. However, not all 512 binary topological relations can be implemented. The 9IM can distinguish 33 relations between two simple lines, 19 relations between a simple line and a region and 8 relations between two simple regions [8]. The value of the intersection in Equation (1) can be drawn from {0, 1}, where a 0 means that the intersection is null and a 1 represents that the intersection is not null,

^{9}) topological relations can be distinguished in theory. However, not all 262,144 topological relations can be implemented, and some topological relations have no practical significance.

#### 2.3. A Comprehensive Model for Topological Relations

**Proposition**

**1:**

**Proof:**

**Proposition**

**2:**

**Condition**

**1:**

**Proof:**

**Condition**

**2:**

**Proof:**

**Proposition**

**3:**

**Proof:**

**Proposition**

**4:**

**Proof:**

## 3. Results

## 4. Discussion

#### 4.1. The Refinement of Topological Relations

#### 4.2. The Interoperability of the 27IM

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Spatial objects including interior, boundary and exterior. (

**a**) A point with its interior, boundary and exterior. (

**b**) A line with its interior, boundary and exterior. (

**c**) A region with its interior, boundary and exterior.

**Figure 3.**The formalism of the 27-intersection model (27IM) and the corresponding geometric interpretations between two points.

**Figure 4.**The formalism of the 27IM and the corresponding geometric interpretations between a point and a line.

**Figure 5.**The formalism of the 27IM and the corresponding geometric interpretations between a point and a region.

**Figure 7.**The formalism of the 27IM and the corresponding geometric interpretations between a line and a region.

**Figure 8.**The formalism of the 27IM and the corresponding geometric interpretations between two regions.

**Figure 9.**Examples of the topological relations between two spatial objects. (

**a**–

**c**) Topological relations between Line A and Line B. (

**d**–

**f**) Topological relations between line A and region B. (

**g**–

**i**) topological relations between Region A and Region B.

Point/Point | Point/Line | Point/Region | Line/Line | Line/Region | Region/Region | |
---|---|---|---|---|---|---|

${\chi}_{0}({A}^{o}\cap {B}^{o})$ | √ | √ | √ | √ | × | × |

${\chi}_{1}({A}^{o}\cap {B}^{o})$ | × | × | × | √ | √ | × |

${\chi}_{2}({A}^{o}\cap {B}^{o})$ | × | × | × | × | × | √ |

${\chi}_{0}({A}^{o}\cap \partial B)$ | × | √ | √ | √ | √ | × |

${\chi}_{1}({A}^{o}\cap \partial B)$ | × | × | × | × | √ | √ |

${\chi}_{2}({A}^{o}\cap \partial B)$ | × | × | × | × | × | × |

${\chi}_{0}({A}^{o}\cap {B}^{-})$ | √ | √ | √ | × | × | × |

${\chi}_{1}({A}^{o}\cap {B}^{-})$ | × | × | × | √ | √ | × |

${\chi}_{2}({A}^{o}\cap {B}^{-})$ | × | × | × | × | × | √ |

${\chi}_{0}(\partial A\cap {B}^{o})$ | × | × | × | √ | √ | × |

${\chi}_{1}(\partial A\cap {B}^{o})$ | × | × | × | × | × | √ |

${\chi}_{2}(\partial A\cap {B}^{o})$ | × | × | × | × | × | × |

${\chi}_{0}(\partial A\cap \partial B)$ | × | × | × | √ | √ | √ |

${\chi}_{1}(\partial A\cap \partial B)$ | × | × | × | × | × | √ |

${\chi}_{2}(\partial A\cap \partial B)$ | × | × | × | × | × | × |

${\chi}_{0}(\partial A\cap {B}^{-})$ | × | × | × | √ | √ | × |

${\chi}_{1}(\partial A\cap {B}^{-})$ | × | × | × | × | × | √ |

${\chi}_{2}(\partial A\cap {B}^{-})$ | × | × | × | × | × | × |

${\chi}_{0}({A}^{-}\cap {B}^{o})$ | √ | × | × | × | × | × |

${\chi}_{1}({A}^{-}\cap {B}^{o})$ | × | √ | × | √ | × | × |

${\chi}_{2}({A}^{-}\cap {B}^{o})$ | × | × | √ | × | √ | √ |

${\chi}_{0}({A}^{-}\cap \partial B)$ | × | √ | × | √ | × | × |

${\chi}_{1}({A}^{-}\cap \partial B)$ | × | × | √ | × | √ | √ |

${\chi}_{2}({A}^{-}\cap \partial B)$ | × | × | × | × | × | × |

${\chi}_{0}({A}^{-}\cap {B}^{-})$ | × | × | × | × | × | × |

${\chi}_{1}({A}^{-}\cap {B}^{-})$ | × | × | × | × | × | × |

${\chi}_{2}({A}^{-}\cap {B}^{-})$ | √ | √ | √ | √ | √ | √ |

**Table 2.**Topological relation as described by the 9IM, dimensionally-extended nine-intersection matrix (DE-9IM), separation number extended nine-intersection matrix (SNE-9IM) and 27IM.

9IM | DE-9IM | SNE-9IM | 27IM | |
---|---|---|---|---|

Figure 9a | $\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}0& -1& 1\\ -1& -1& 0\\ 1& 0& 2\end{array}\right]$ | $\left[\begin{array}{ccc}2& 0& 3\\ 0& 0& 2\\ 3& 2& 2\end{array}\right]$ | $\left[\begin{array}{ccccccccc}2& 0& 0& 0& 0& 0& 0& 3& 0\\ 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 3& 0& 2& 0& 0& 0& 0& 2\end{array}\right]$ |

Figure 9b | $\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}1& -1& 1\\ -1& -1& 0\\ 1& 0& 2\end{array}\right]$ | $\left[\begin{array}{ccc}2& 0& 3\\ 0& 0& 2\\ 3& 2& 2\end{array}\right]$ | $\left[\begin{array}{ccccccccc}0& 2& 0& 0& 0& 0& 0& 3& 0\\ 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 3& 0& 2& 0& 0& 0& 0& 2\end{array}\right]$ |

Figure 9c | $\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}1& -1& 1\\ -1& -1& 0\\ 1& 0& 2\end{array}\right]$ | $\left[\begin{array}{ccc}3& 0& 4\\ 0& 0& 2\\ 4& 2& 3\end{array}\right]$ | $\left[\begin{array}{ccccccccc}1& 2& 0& 0& 0& 0& 0& 4& 0\\ 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 4& 0& 2& 0& 0& 0& 0& 3\end{array}\right]$ |

Figure 9d | $\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}1& 0& 1\\ -1& -1& 0\\ 2& 1& 2\end{array}\right]$ | $\left[\begin{array}{ccc}1& 2& 2\\ 0& 0& 2\\ 2& 2& 1\end{array}\right]$ | $\left[\begin{array}{ccccccccc}0& 1& 0& 2& 0& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 0& 2& 0& 2& 0& 0& 0& 1\end{array}\right]$ |

Figure 9e | $\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}1& 1& 1\\ -1& -1& 0\\ 2& 1& 2\end{array}\right]$ | $\left[\begin{array}{ccc}1& 2& 2\\ 0& 0& 2\\ 2& 2& 1\end{array}\right]$ | $\left[\begin{array}{ccccccccc}0& 1& 0& 0& 2& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 0& 2& 0& 2& 0& 0& 0& 1\end{array}\right]$ |

Figure 9f | $\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}1& 1& 1\\ -1& -1& 0\\ 2& 1& 2\end{array}\right]$ | $\left[\begin{array}{ccc}2& 3& 2\\ 0& 0& 2\\ 3& 3& 1\end{array}\right]$ | $\left[\begin{array}{ccccccccc}0& 2& 0& 1& 2& 0& 0& 2& 0\\ 0& 0& 0& 0& 0& 0& 2& 0& 0\\ 0& 0& 3& 0& 3& 0& 0& 0& 1\end{array}\right]$ |

Figure 9g | $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}2& 1& 2\\ 1& 0& 1\\ 2& 1& 2\end{array}\right]$ | $\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccccccccc}0& 0& 1& 0& 1& 0& 0& 0& 1\\ 0& 1& 0& 2& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 1& 0& 0& 0& 1\end{array}\right]$ |

Figure 9h | $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}2& 1& 2\\ 1& 1& 1\\ 2& 1& 2\end{array}\right]$ | $\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccccccccc}0& 0& 1& 0& 1& 0& 0& 0& 1\\ 0& 1& 0& 0& 2& 0& 0& 1& 0\\ 0& 0& 1& 0& 1& 0& 0& 0& 1\end{array}\right]$ |

Figure 9i | $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ | $\left[\begin{array}{ccc}2& 1& 2\\ 1& 1& 1\\ 2& 1& 2\end{array}\right]$ | $\left[\begin{array}{ccc}1& 1& 1\\ 2& 3& 1\\ 2& 2& 1\end{array}\right]$ | $\left[\begin{array}{ccccccccc}0& 0& 1& 0& 1& 0& 0& 0& 1\\ 0& 2& 0& 1& 2& 0& 0& 1& 0\\ 0& 0& 2& 0& 2& 0& 0& 0& 1\end{array}\right]$ |

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**MDPI and ACS Style**

Shen, J.; Zhou, T.; Chen, M. A 27-Intersection Model for Representing Detailed Topological Relations between Spatial Objects in Two-Dimensional Space. *ISPRS Int. J. Geo-Inf.* **2017**, *6*, 37.
https://doi.org/10.3390/ijgi6020037

**AMA Style**

Shen J, Zhou T, Chen M. A 27-Intersection Model for Representing Detailed Topological Relations between Spatial Objects in Two-Dimensional Space. *ISPRS International Journal of Geo-Information*. 2017; 6(2):37.
https://doi.org/10.3390/ijgi6020037

**Chicago/Turabian Style**

Shen, Jingwei, Tinggang Zhou, and Min Chen. 2017. "A 27-Intersection Model for Representing Detailed Topological Relations between Spatial Objects in Two-Dimensional Space" *ISPRS International Journal of Geo-Information* 6, no. 2: 37.
https://doi.org/10.3390/ijgi6020037