Simulating Geographical Points Around the Eiffel Tower

The coordinates of the Eiffel Tower are (Longitude: 48.8584° N, Latitude: 2.2945° E). I am interested in randomly generating 100 points that are located within a 12 KM radius of the Eiffel Tower. In other words, I would like to randomly generate 100 pairs of (Longitude, Latitude) that are located within a 12 KM radius of the Eiffel Tower.

According to this question here (Simple calculations for working with lat/lon and km distance?), the following formulas can be used to convert Longitude and Latitude to KM:

• Latitude: 1 deg = 110.574 km
• Longitude: 1 deg = 111.320*cos(latitude) km

Thus, if I want to find out a 12 KM radius, the corresponding maximum ranges should be:

• Max Latitude Range: 12 * (1/110.574) = 0.1085246
• Max Longitude Range: 111.320*cos(0.1085246) = 110.6651 -> 1/110.6651 = 0.009036273

Using this information, I tried to simulate points and plot them:

# for some reason, you have to call "long" as "lat" and vice versa - otherwise the points appear in the wrong locations all together 
id = 1:100
long = 2.2945 + rnorm( 100, 0.1085246 , 1)
lat = 48.8584 + rnorm( 100, 0.009036273 , 1)

my_data = data.frame(id, lat, long)

library(leaflet)
my_data %>% 
  leaflet() %>% 
  addTiles() %>% 
  addMarkers(clusterOption=markerClusterOptions())

But these points do not appear near the Eiffel Tower - some of them are even in Belgium! :

I reduced the variance and now the points appear closer:

# reduce variance
id = 1:100
long = 2.2945 + rnorm( 100, 0.1085246 , 0.01)
lat = 48.8584 + rnorm( 100, 0.009036273 , 0.01)

my_data = data.frame(id, lat, long)

library(leaflet)
my_data %>% 
  leaflet() %>% 
  addTiles() %>% 
  addMarkers(clusterOption=markerClusterOptions())

But this of course required some guess work and playing around - ideally, I would like a more "mathematical approach".

• Is there some standard formula I can use to make sure that no matter what initial coordinate I choose (e.g. Eiffel Tower, Statue of Liberty, etc.), the randomly generated points will always fall in a certain radius?

One option is to use the sf package. The function st_buffer will allow you to create a 12 km circle around your starting point, and st_sample will allow you to take 100 random points within that circle.

Create the data

library(sf)
library(dplyr)

pt_sf <- data.frame(long = 2.2945, lat = 48.8584) %>%
  st_as_sf(coords = c("long", "lat"), crs = 4326)


buff <- pt_sf %>%
  st_buffer(dist = units::as_units(12, 'km'))

buff_sample <- st_sample(buff, 100)

Plot it

library(leaflet)

leaflet(pt_sf) %>% 
  addTiles() %>% 
  addCircleMarkers(color = 'red') %>%
  addPolygons(data = buff) %>%
  addMarkers(data = buff_sample, clusterOption=markerClusterOptions())