> For the complete documentation index, see [llms.txt](https://academy.carto.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://academy.carto.com/advanced-spatial-analytics/spatial-analytics-for-bigquery/step-by-step-tutorials/a-nyc-subway-connection-graph-using-delaunay-triangulation.md).

# A NYC subway connection graph using Delaunay triangulation

<div align="left"><figure><img src="/files/YUiOye9yS8uvdGFnFppD" alt="Advanced difficulty banner" width="175"><figcaption></figcaption></figure></div>

Providing a good network connection between subway stations is critical to ensure an efficient mobility system in big areas. Let’s imagine we need to design a well-distributed subway network to connect the stations of a brand-new subway system. A simple and effective solution to this problem is to build a Delaunay triangulation of the predefined stations, which ensures a good connection distribution.

For this particular example we are choosing the New York city subway stations to build the cited triangulation. The following query will construct the triangulation using the `ST_DELAUNAYLINES` function from the processing module of the Analytics Toolbox.

{% tabs %}
{% tab title="carto-un" %}

```sql
WITH data AS (
    SELECT ARRAY(
        SELECT geom FROM `cartobq.docs.nyc_subway_stations`
    ) AS array_points
),
delaunay_array AS (
    SELECT `carto-un`.carto.ST_DELAUNAYLINES(array_points) AS nested
    FROM data
),
delaunay_triangles AS (
    SELECT geom
    FROM delaunay_array, UNNEST(nested) AS geom
)
select * from delaunay_triangles
```

{% endtab %}

{% tab title="carto-un-eu" %}

```sql
WITH data AS (
    SELECT ARRAY(
        SELECT geom FROM `cartobq.docs.nyc_subway_stations`
    ) AS array_points
),
delaunay_array AS (
    SELECT `carto-un-eu`.carto.ST_DELAUNAYLINES(array_points) AS nested
    FROM data
),
delaunay_triangles AS (
    SELECT geom
    FROM delaunay_array, UNNEST(nested) AS geom
)
select * from delaunay_triangles
```

{% endtab %}

{% tab title="manual" %}

```sql
WITH data AS (
    SELECT ARRAY(
        SELECT geom FROM `cartobq.docs.nyc_subway_stations`
    ) AS array_points
),
delaunay_array AS (
    SELECT carto.ST_DELAUNAYLINES(array_points) AS nested
    FROM data
),
delaunay_triangles AS (
    SELECT geom
    FROM delaunay_array, UNNEST(nested) AS geom
)
select * from delaunay_triangles
```

{% endtab %}
{% endtabs %}

In the visualization below, each subway station is represented by a point where the color and size represent the number of received connections.

{% embed url="<https://clausa.app.carto.com/map/631a843f-d1e9-49c9-9e75-45ad02df1abf>" %}

This map has been created by grouping the delauney connections by station ID, using the below SQL:

```sql
//WITH data AS (
```

As a curiosity, the average number of connections between the existing New Yorks subway stations is close to six, which indicates a balanced distribution of the stations according to the [Delaunay triangulation properties](https://en.wikipedia.org/wiki/Delaunay_triangulation#Properties).

<img src="/files/4m1BK9j4Wq34gat4HHd2" alt="EU flag" data-size="line"> This project has received funding from the [European Union’s Horizon 2020](https://ec.europa.eu/programmes/horizon2020/en) research and innovation programme under grant agreement No 960401.


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