# Spatial relationships and operations¶

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%matplotlib inline

import pandas as pd
import geopandas

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countries = geopandas.read_file("zip://./data/ne_110m_admin_0_countries.zip")


## Spatial relationships¶

An important aspect of geospatial data is that we can look at spatial relationships: how two spatial objects relate to each other (whether they overlap, intersect, contain, .. one another).

The topological, set-theoretic relationships in GIS are typically based on the DE-9IM model. See https://en.wikipedia.org/wiki/Spatial_relation for more information.

(Image by Krauss, CC BY-SA 3.0)

### Relationships between individual objects¶

Let's first create some small toy spatial objects:

A polygon (note: we use .squeeze() here to to extract the scalar geometry object from the GeoSeries of length 1):

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belgium = countries.loc[countries['name'] == 'Belgium', 'geometry'].squeeze()


Two points:

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paris = cities.loc[cities['name'] == 'Paris', 'geometry'].squeeze()
brussels = cities.loc[cities['name'] == 'Brussels', 'geometry'].squeeze()


And a linestring:

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from shapely.geometry import LineString
line = LineString([paris, brussels])


Let's visualize those 4 geometry objects together (I only put them in a GeoSeries to easily display them together with the geopandas .plot() method):

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geopandas.GeoSeries([belgium, paris, brussels, line]).plot(cmap='tab10')


You can recognize the abstract shape of Belgium.

Brussels, the capital of Belgium, is thus located within Belgium. This is a spatial relationship, and we can test this using the individual shapely geometry objects as follow:

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brussels.within(belgium)


And using the reverse, Belgium contains Brussels:

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belgium.contains(brussels)


On the other hand, Paris is not located in Belgium:

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belgium.contains(paris)

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paris.within(belgium)


The straight line we draw from Paris to Brussels is not fully located within Belgium, but it does intersect with it:

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belgium.contains(line)

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line.intersects(belgium)


### Spatial relationships with GeoDataFrames¶

The same methods that are available on individual shapely geometries as we have seen above, are also available as methods on GeoSeries / GeoDataFrame objects.

For example, if we call the contains method on the world dataset with the paris point, it will do this spatial check for each country in the world dataframe:

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countries.contains(paris)


Because the above gives us a boolean result, we can use that to filter the dataframe:

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countries[countries.contains(paris)]


And indeed, France is the only country in the world in which Paris is located.

Another example, extracting the linestring of the Amazon river in South America, we can query through which countries the river flows:

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amazon = rivers[rivers['name'] == 'Amazonas'].geometry.squeeze()

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countries[countries.crosses(amazon)]  # or .intersects

**REFERENCE**: Overview of the different functions to check spatial relationships (*spatial predicate functions*): * equals * contains * crosses * disjoint * intersects * overlaps * touches * within * covers See https://shapely.readthedocs.io/en/stable/manual.html#predicates-and-relationships for an overview of those methods. See https://en.wikipedia.org/wiki/DE-9IM for all details on the semantics of those operations.

## Let's practice!¶

We will again use the Paris datasets to do some exercises. Let's start importing them again:

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districts = geopandas.read_file("data/paris_districts.geojson").to_crs(epsg=2154)

**EXERCISE: The Eiffel Tower** The Eiffel Tower is an iron lattice tower built in the 19th century, and is probably the most iconic view of Paris. The location of the Eiffel Tower is: x of 648237.3 and y of 6862271.9. * Create a Shapely point object with the coordinates of the Eiffel Tower and assign it to a variable called eiffel_tower. Print the result. * Check if the Eiffel Tower is located within the Montparnasse district (provided). * Check if the Montparnasse district contains the bike station location. * Calculate the distance between the Eiffel Tower and the bike station (note: in this case, the distance is returned in meters).
Hints * The Point class is available in the shapely.geometry submodule * Creating a point can be done by passing the x and y coordinates to the Point() constructor. * The within() method checks if the object is located within the passed geometry (used as geometry1.within(geometry2)). * The contains() method checks if the object contains the passed geometry (used as geometry1.contains(geometry2)). * To calculate the distance between two geometries, the distance() method of one of the geometries can be used.
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# Import the Point geometry
from shapely.geometry import Point

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# %load _solved/solutions/03-spatial-relationships-operations1.py

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# %load _solved/solutions/03-spatial-relationships-operations2.py

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# Accessing the Montparnasse geometry (Polygon)
district_montparnasse = districts.loc[52, 'geometry']
bike_station = stations.loc[293, 'geometry']

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# %load _solved/solutions/03-spatial-relationships-operations3.py

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# %load _solved/solutions/03-spatial-relationships-operations4.py

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# %load _solved/solutions/03-spatial-relationships-operations5.py

**EXERCISE: In which district in the Eiffel Tower located?** In previous exercise, we constructed a Point geometry for its location, and we checked that it was not located in the Montparnasse district. Let's now determine in which of the districts of Paris it *is* located. * Create a boolean mask (or filter) indicating whether each district contains the Eiffel Tower or not. Call the result mask. * Filter the districts dataframe with the boolean mask and print the result.
Hints * To check for each polygon in the districts dataset if it contains a single point, we can use the contains() method of the districts GeoDataFrame. * Filtering the rows of a DataFrame based on a condition can be done by passing the boolean mask into df[..].
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# Construct a point object for the Eiffel Tower
eiffel_tower = Point(648237.3, 6862271.9)

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# %load _solved/solutions/03-spatial-relationships-operations6.py

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# %load _solved/solutions/03-spatial-relationships-operations7.py

**EXERCISE: How far is the closest bike station?** Now, we might be interested in the bike stations nearby the Eiffel Tower. To explore them, let's visualize the Eiffel Tower itself as well as the bikes stations within 1km. To do this, we can calculate the distance to the Eiffel Tower for each of the stations. Based on this result, we can then create a mask that takes True if the station is within 1km, and False otherwise, and use it to filter the stations GeoDataFrame. Finally, we make a visualization of this subset. * Calculate the distance to the Eiffel Tower for each station, and call the result dist_eiffel. * Print the distance to the closest station (which is the minimum of dist_eiffel). * Select the rows the stations GeoDataFrame where the distance to the Eiffel Tower is less than 1 km (note that the distance is in meters). Call the result stations_eiffel.
Hints * The distance() method of a GeoDataFrame works element-wise: it calculates the distance between each geometry in the GeoDataFrame and the geometry passed to the method. * A Series has a min() method to calculate the minimum value. * To create a boolean mask based on a condition, we can do e.g. s < 100.
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# %load _solved/solutions/03-spatial-relationships-operations8.py

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# %load _solved/solutions/03-spatial-relationships-operations9.py

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# %load _solved/solutions/03-spatial-relationships-operations10.py

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# Make a plot of the close-by restaurants
ax = stations_eiffel.to_crs(epsg=3857).plot()
geopandas.GeoSeries([eiffel_tower], crs='EPSG:2154').to_crs(epsg=3857).plot(ax=ax, color='red')
import contextily
ax.set_axis_off()


## Spatial operations¶

Next to the spatial predicates that return boolean values, Shapely and GeoPandas also provide operations that return new geometric objects.

Binary operations:

Buffer:

For example, using the toy data from above, let's construct a buffer around Brussels (which returns a Polygon):

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geopandas.GeoSeries([belgium, brussels.buffer(1)]).plot(alpha=0.5, cmap='tab10')


and now take the intersection, union or difference of those two polygons:

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brussels.buffer(1).intersection(belgium)

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brussels.buffer(1).union(belgium)

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brussels.buffer(1).difference(belgium)


Another useful method is the unary_union attribute, which converts the set of geometry objects in a GeoDataFrame into a single geometry object by taking the union of all those geometries.

For example, we can construct a single object for the Africa continent:

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africa_countries = countries[countries['continent'] == 'Africa']

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africa = africa_countries.unary_union

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africa

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print(str(africa)[:1000])

**REMEMBER**: GeoPandas (and Shapely for the individual objects) provides a whole lot of basic methods to analyse the geospatial data (distance, length, centroid, boundary, convex_hull, simplify, transform, ....), much more than the few that we can touch in this tutorial. * An overview of all methods provided by GeoPandas can be found here: http://geopandas.readthedocs.io/en/latest/reference.html

## Let's practice!¶

EXERCISE: What are the districts close to the Seine?

Below, the coordinates for the Seine river in the neighbourhood of Paris are provided as a GeoJSON-like feature dictionary (created at http://geojson.io).

Based on this seine object, we want to know which districts are located close (maximum 150 m) to the Seine.

• Create a buffer of 150 m around the Seine.
• Check which districts intersect with this buffered object.
• Make a visualization of the districts indicating which districts are located close to the Seine.

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districts = geopandas.read_file("data/paris_districts.geojson").to_crs(epsg=2154)

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# created a line with http://geojson.io
s_seine = geopandas.GeoDataFrame.from_features({"type":"FeatureCollection","features":[{"type":"Feature","properties":{},"geometry":{"type":"LineString","coordinates":[[2.408924102783203,48.805619828930226],[2.4092674255371094,48.81703747481909],[2.3927879333496094,48.82325391133874],[2.360687255859375,48.84912860497674],[2.338714599609375,48.85827758964043],[2.318115234375,48.8641501307046],[2.298717498779297,48.863246707697],[2.2913360595703125,48.859519915404825],[2.2594070434570312,48.8311646245967],[2.2436141967773438,48.82325391133874],[2.236919403076172,48.82347994904826],[2.227306365966797,48.828339513221444],[2.2224998474121094,48.83862215329593],[2.2254180908203125,48.84856379804802],[2.2240447998046875,48.85409863123821],[2.230224609375,48.867989496547864],[2.260265350341797,48.89192242750887],[2.300262451171875,48.910203080780285]]}}]},
crs={'init': 'epsg:4326'})

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# convert to local UTM zone
s_seine_utm = s_seine.to_crs(epsg=2154)

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import matplotlib.pyplot as plt
fig, ax = plt.subplots(figsize=(20, 10))
districts.plot(ax=ax, color='grey', alpha=0.4, edgecolor='k')
s_seine_utm.plot(ax=ax)

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# access the single geometry object
seine = s_seine_utm.geometry.squeeze()

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# %load _solved/solutions/03-spatial-relationships-operations11.py

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# %load _solved/solutions/03-spatial-relationships-operations12.py

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# %load _solved/solutions/03-spatial-relationships-operations13.py

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# %load _solved/solutions/03-spatial-relationships-operations14.py

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