# Linking Off-Road Points to Routing Networks

## Abstract

**:**

## 1. Introduction

**Veracity**Every dataset that provides spatial data is incomplete and inaccurate in the open-world assumption, for instance, the dynamic reshaping of infrastructure.

- First, the network’s graph might not be planar. When projecting a non-planar graph to a plane, the image has intersecting lines that represent two edges. By merely looking at the image, we cannot differentiate whether it is a crossing or whether one street tunnels under the other. A geometrical query, unaware of this fact, might deliver results that do not match real-world expectation. Let us, for instance, consider a long bridge crossing a valley. A query for the nearest access point to the routing network of the valley might suggest this bridge as a result, although it is not directly accessible from the valley.
- A second problem arises when very distant points are taken into consideration. For large routing networks, it is not possible to project them on a compact plane that respects both the angles and distances of the Earth. That is because the Earth is not isomorphic to any compact flat model. For instance, let us take a look at the Mercator projection [3]. It gained popularity due to its accuracy of angles and is still used for course information in marine navigation systems [5]. Furthermore, the projection provides a good approximation for nearby spatial reasoning. Unfortunately, the projected shapes of the objects get distorted with respect to angle and length. More precisely, the distortion correlates with the objects’ distances to the equator. Hence, the projection exaggerates the distances and sizes of spatial objects near pole regions. This makes spatial objects with large differences in latitude incomparable, and is a main complaint of the Mercator projection. In order to cope with large-distance reasoning, we have to leave the concept of a single chart and move towards a concept that represents a “truer” form of the Earth. In other words, distances between distant objects are retrieved by a geodesic line instead of some projection.

**Research Hypothesis.**Given a (global) routing network that is based on a geometric representation, we want to enhance the network in such a way that queries based on geometric information are answered by matching the geometric position with the network. This should be conducted in such a way that respects the nature of the geometric representation.

**Remark**

**1**

**.**The representation of the Earth as a spheroid can be formalized by the local parametrizations

#### 1.1. Related Work

- Extracted from digital imagery;
- Collected by sensors (e.g., GPS-enabled tracking devices);
- Created manually.

#### 1.2. Structure of the Paper

## 2. Preliminaries

#### 2.1. Graphs

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**2.**

**Definition**

**3.**

**Lemma**

**1.**

**Proof.**

- From $\ell \left(P\right)\ge 0\phantom{\rule{3.33333pt}{0ex}}\forall P$, we obtain the non-negativity of ℓ.
- As $\ell \left(P\right)>0$ for all walks from u to $v\in V,u\ne v$, we obtain by definition $\ell \left(u,v\right)>0$ for all $u,v\in V,u\ne v$. As $c(v,v)=0$ for each $v\in V$, we can conclude that $\ell \left(\right(v,v\left)\right)=0$ for each $v\in V$. Thus, we yield the positive definiteness of ℓ.
- If we define the concatenation of walks by $({v}_{1},\dots ,{v}_{n})\circ ({w}_{1},\dots ,{w}_{n}):=$$$\left\{\begin{array}{cc}({v}_{1},\dots ,{v}_{n},{w}_{2},\dots ,{w}_{n})\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{v}_{n}={w}_{1},\hfill \\ \phantom{\rule{4.pt}{0ex}}\mathrm{undefined}\phantom{\rule{4.pt}{0ex}}\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

**Remark**

**3.**

#### 2.2. Manifolds

**Definition**

**4.**

- ${\bigcup}_{j\in J}{V}_{j}=M$.
- $\forall j,k\in J:{V}_{j}\cap {V}_{k}\ne \varnothing \Rightarrow {\phi}_{j}\circ {\phi}_{k}^{-1}:{\phi}_{k}({V}_{k}\cap {V}_{j})\to {\phi}_{j}({V}_{k}\cap {V}_{j})$is a homomorphism, called the transition map.

#### 2.3. Routing Networks

**Definition**

**5.**

- $i{\mid}_{V}:V\hookrightarrow N\subset {\mathbb{R}}^{3}$with${\bigcup}_{e\in E}i\left(e\right)=N;$
- For each$n\in N$, there exists an$m\in M$and an$h\in \mathbb{R}$such that$m+h\nu \left(m\right)=n$(embedding property). This property is based on the fact that street segments may not be on the surface. Informally, the parameter h counts for the altitude difference between the ground and street; e.g., bridges have positive heights and tunnels negative heights. This separation between spherical data and altitude is also common when dealing with the WGS84 format that encodes points on M by latitude and longitude [22].
- Let${\pi}_{M}\circ i:V\to M$be the (canonical) projection of the graph onto M. Then, we assume that for all$e\in E$there exists some$j\in J$with${\pi}_{M}\circ i\left(e\right)\subset {V}_{j}$, i.e., we stipulate that the complete image of each edge is contained in at least one chart’s image (containment property).
- For each edge$e=(a,b)\in E$, there is a line${\gamma}_{e}:[0,1]\to N$with${\gamma}_{e}\left(0\right)=i\left(a\right),{\gamma}_{e}\left(1\right)=i\left(b\right),$and$i\left(e\right)=Im{\gamma}_{e}$, i.e., ${\gamma}_{e}$is a linear continuous, parameterized, geodesic line with a and b as endpoints (line-string property).
- Two of these lines ${\gamma}_{1}$ and ${\gamma}_{2}$ are called equivalent when ${\gamma}_{1}={\gamma}_{2}$ or ${\gamma}_{1}(1-\xb7)={\gamma}_{2}$. If we denote equivalence with ∼ and define the quotient set $\mathsf{\Gamma}:=\left\{{\gamma}_{e}:e\in E\right\}/\sim $, then $\left\{\gamma (0,1):\gamma \in \mathsf{\Gamma}\right\}$ shall be a disjoint partition of $N\backslash i\left(V\right)$(disjoint property).

**Remark**

**4.**

#### 2.4. Lower-Bounding Metrics

**Definition**

**6.**

**Example**

**1.**

**Example**

**2.**

- A common local lower-bounding metric is the beeline when the network is mapped to a local plane. It would be tedious to adhere multiple charts by the transition map in order to calculate the beeline between two distant points. Fortunately, it is easy to calculate the geodesic on M by approximation [23]. For example, Vincenty’s algorithm [24] is a good approximation for calculating geodesics between two different points on the surface of the Earth.
- The Euclidean metric in ${\mathbb{R}}^{3}$ is a lower-bounding metric. In particular, this metric is also a lower bound of the local beeline because it is allowed to neglect the curvature of the manifold M, cf. Figure 1.

**Definition**

**7.**

## 3. Linkage to Network

- The resulting routing network stays valid. In particular, the graph shall remain connected, i.e., there is an edge with ${v}^{\prime}$ as an end.
- The modified geometric representation ${i}^{\prime}$ maps ${i}^{\prime}\left({v}^{\prime}\right)$ to M.
- $i\left({v}^{\prime}\right)\in M$. Informally, this means that a pedestrian can access ${v}^{\prime}$ from the ground.
- d is still a valid lower-bounding metric.

**Definition**

**8.**

- ${i}^{\prime}:{E}^{\prime}\cup {V}^{\prime}\to {N}^{\prime}$ with ${i}^{\prime}{\mid}_{V}=i{\mid}_{V}$ and ${i}^{\prime}{\mid}_{(E\cap {E}^{\prime})}=i{\mid}_{(E\cap {E}^{\prime})}$.
- ${V}^{\prime}=V\cup \left\{{v}^{\prime}\right\}$ with ${i}^{\prime}\left({v}^{\prime}\right)=a$ for some ${v}^{\prime}$ (that is not necessarily part of V).
- ${i}^{\prime}({V}^{\prime}\cup {E}^{\prime})\subset {N}^{\prime}$; hence, $a\in {N}^{\prime}\cap M$.
- There exist some $x,y\in V$ with $({v}^{\prime},x)\in {E}^{\prime}$ and $(y,{v}^{\prime})\in {E}^{\prime}$.
- ${c}^{\prime}:{E}^{\prime}\to \mathbb{R}$ with ${c}^{\prime}{\mid}_{E\cap {E}^{\prime}}=c$.
- ${G}^{\prime}$ respects the metric d.

**Example**

**3.**

- 1.
- If there exists some$v\in V$such that$i\left(v\right)=a$, then${G}^{\prime}:=G$.
- 2.
- If there exists an edge$e\in E$such that$a\in i\left(e\right)\backslash i\left(V\right)$, then there is some$\lambda \in (0,1)$such that${\gamma}_{e}\left(\lambda \right)=a$, where${\gamma}_{e}$is the line induced by$i\left(e\right)$due to the line-string property. Let$x,y\in V$be the vertices connected by e such that$e=(x,y)$. We add a new vertex${v}^{\prime}\notin V$with$i\left({v}^{\prime}\right)=a$. Let us set${c}^{\prime}(x,{v}^{\prime}):=\lambda c(x,y)\phantom{\rule{4.pt}{0ex}}and\phantom{\rule{4.pt}{0ex}}$${c}^{\prime}({v}^{\prime},y):=(1-\lambda )c(x,y)$. If $(y,x)\in E$, we further set, due to$Im{\gamma}_{(x,y)}=Im{\gamma}_{(y,x)}$, ${c}^{\prime}({v}^{\prime},x):=\overleftarrow{\lambda}c(y,x)\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}{c}^{\prime}(y,{v}^{\prime}):=(1-\overleftarrow{\lambda})c(y,x)$with$\overleftarrow{\lambda}\in (0,1)$such that${\gamma}_{(y,x)}\left(\overleftarrow{\lambda}\right)=a$. In the end, we define${E}^{\prime}$as the set E without$\left\{(x,y),(y,x)\right\}$, but with the new edges used above -$\left\{(x,{v}^{\prime}),({v}^{\prime},y)\right\}$, and $\{({v}^{\prime},x),(y,{v}^{\prime})\}$ if $(y,x)\in E$.Let ℓ and ${\ell}^{\prime}$ be the quasimetrics that are induced by c and ${c}^{\prime}$, respectively. Then, ${\ell}^{\prime}{\mid}_{V\times V}=\ell $ holds due to $\ell \left(x,y\right)=c(x,y)={c}^{\prime}(x,{v}^{\prime})+{c}^{\prime}({v}^{\prime},y)={\ell}^{\prime}\left(x,y\right)$. Thus, we yield the beeline property with ${i}^{\prime}(x,{v}^{\prime})={\gamma}_{e}\left([0,\lambda ]\right),{i}^{\prime}({v}^{\prime},y)={\gamma}_{e}\left([\lambda ,1]\right)$. With the same arguments, we are able to preserve the properties of a routing network for $(y,x)$, if $(y,x)\in E$. Because we have not changed the image of i, we can just set ${N}^{\prime}:=N$ and hence we are finished.
- 3.
- Otherwise, we are certain that $a\notin N$. We use a not yet defined method to create a new network ${G}^{\prime}=({V}^{\prime},{E}^{\prime},{c}^{\prime},{i}^{\prime},{N}^{\prime})$ with some ${E}^{\prime}$, ${c}^{\prime}$ and ${N}^{\prime}$.

**Counter-Example**

**1.**

- $d\left(i\right(e),a)<d\left(i\right(u),a)$and hence$d(i\left(e\right),a)\le min\left\{d\left(i\right(x),a),d\left(i\right(y),a)\right\}$;
- There exist $\alpha <0,w\in {\dot{\gamma}}_{e}^{\perp}\left(0\right)$ and $\lambda ,\mu \in [0,1]$ such that$${\gamma}_{e}\left(\lambda \right)+w=i\left(v\right)\mathit{and}{\gamma}_{e}\left(\mu \right)+\alpha w=a,$$

**Example**

**4**

**.**We define the edge linkage ${G}^{\prime}:=\sigma (a,G)$ of $G:=(V,E,c,i,N)\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}\mathcal{G}$ for a point $a\in M$ that is based on linear interpolation and implement the last procedure as follows: First, search for an

- 1.
- $\lambda =0$ or $\lambda =1$, i.e., $d\left(i\right(e),a)=d\left(i\right(x),a)$ or $d\left(i\right(e),a)=d\left(i\right(y),a)$. Without loss of generality, let the equation $d\left(i\right(e),a)=d(x,a)$ hold. Now we can apply a vertex linkage on G by Counter-Example 1 (setting $u\leftarrow x$). This linkage holds the disjoint property.
- 2.
- Otherwise, $\lambda \in (0,1)$. Hence, there exists some $b\in N$ with $b\in i\left(e\right)$ and $d\left(i\right(e),a)=d(b,a)$. We apply linear interpolation on G with b and yield a new network $\sigma (b,G)=:\tilde{G}:=(\tilde{V},\tilde{E},\tilde{c},\tilde{i},N)$ with $\tilde{i}\left(\tilde{v}\right)=b$ for some $\tilde{v}\in \tilde{V}$. Note that the rule applied by linear interpolation will not modify N. If we exchange G with $\tilde{G}$, the first case holds, since there exists an edge $\tilde{e}:={argmin}_{\tilde{v}\in \tilde{E}}d(\tilde{i}\left(\tilde{v}\right),a)$ with ${min}_{\tilde{\lambda}\in [0,1]}d({\gamma}_{\tilde{e}}\left(\tilde{\lambda}\right),a)\in \left\{0,1\right\}$.

**Proof.**

**Example**

**5**

**.**If there exists some $v\in V$ with ${min}_{v\in V}d(i\left(v\right),a)<r$, then add a to $i\left(v\right)$, i.e., ${i}^{\prime}{\mid}_{V\backslash \left\{v\right\}}=i$ and ${i}^{\prime}\left(v\right)=i\left(v\right)\cup \left\{a\right\}$. We define the r-snap${\sigma}_{r}:M\times \mathcal{G}\to \mathcal{G}$ with a threshold $0\le r<\infty $ as a modification of our linear interpolation approach (Example 3), which makes the above rule its highest priority. Choosing the right r is situation-dependent, e.g., a small r would be more preferable in network-dense areas. Note that this approach resembles “snapping” in computer graphics.

## 4. Implementation

- d was given by
`ST_Distance`. - ℓ was computed by a shortest path algorithm on pgRouting’s network. Common algorithms such as Dijkstra and A* were available.
`ST_ShortestLine`represented the geodesic function g.- ${argmin}_{a\in A}d(a,v)$ with $A\subseteq E$ or $A\subseteq V$: We used
`ST_DWithin`as a pre-filter for distant edges/vertices before the exact distances to all remaining edges/vertices were calculated. - The split of an edge $(u,v)$ into $(u,{v}^{\prime})$ and $({v}^{\prime},u)$ with $i\left({v}^{\prime}\right)\in i(u,v)$ was performed by calling
`ST_Line_Locate_Point`and then generating both new edges with two calls of`ST_Line_Substring`.

#### 4.1. Evaluation

#### 4.2. Expectations on Larger Datasets

#### 4.3. Outlook

## 5. Conclusions

## Funding

`JP21K17701`and

`JP21H05847`.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Curvature on an ellipsoid embedded in ${\mathbb{R}}^{3}$. The ellipsoid’s surface is visualized in dashed green color. We take two points u and v on this ellipsoid. If we cut the ellipsoid along these two points, we obtain a circle on which both points lie. The arc (blue color) that connects both points is a geodesic $g(u,v)$ of the surface [25]. A straight line (red color) would be shorter than the geodesic and hence the ${l}^{2}$-norm ${\u2225\xb7\u2225}_{2}^{{\mathbb{R}}^{3}}$ is a lower bound.

**Figure 4.**Vertex linkage creating a new edge intersecting with e. After this operation, the graph is no longer planar, a setting we want to avoid.

**Figure 5.**Edge Linkage. We link a point a to a newly created node b splitting the former edge e of the routing network.

**Figure 6.**Architecture design chart. Our Java middleware parses a query from a user, fetches POIs from a spatial database, amends temporarily the routing network and finally runs a routing query on the pgRouting database.

**Figure 7.**Linking point a to a routing network with the beeline crossing an obstacle O (dark blue). We consider the vertices on the convex hull of O and start creating a path (green arrows) from a to along the vertices on the convex hull of O to the routing network, where we find u and w to be the two closest nodes to a with respect to the Euclidean distance.

**Table 1.**Running time for linkage and routing on the setting described in Section 4.1. The ratio of execution time between linkage and routing is almost constant with variance in the number of nodes $\left|V\right|$.

$\left|\mathit{V}\right|$ | Linkage | Routing | Ratio |
---|---|---|---|

100 | 0.40 s | 8.06 s | 4.96% |

200 | 0.74 s | 15.21 s | 4.87% |

400 | 1.33 s | 32.26 s | 4.12% |

800 | 2.47 s | 59.69 s | 4.14% |

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Köppl, D.
Linking Off-Road Points to Routing Networks. *Algorithms* **2022**, *15*, 163.
https://doi.org/10.3390/a15050163

**AMA Style**

Köppl D.
Linking Off-Road Points to Routing Networks. *Algorithms*. 2022; 15(5):163.
https://doi.org/10.3390/a15050163

**Chicago/Turabian Style**

Köppl, Dominik.
2022. "Linking Off-Road Points to Routing Networks" *Algorithms* 15, no. 5: 163.
https://doi.org/10.3390/a15050163