Points within a distance from a point

I have been trying to git this right, in fact it is working but the distance is wrong:

1- I have a collection of points
2- I have a point within which I need to find all points that are 200m far
3- Normally, It should return 6 points, however it returns only 4

library(sf)
library(geosphere)


#My points
x<-c(2.3009132,2.2999853,2.2995872,2.2985374,2.2991502,2.2984043,2.3054471,2.3009132,2.3048155,2.3014964)
y<-c(48.8511847,48.8505062,48.8502346,48.8495305,48.8499405,48.8494376,48.8542721,48.8511847,48.853842,48.8515819)
y<-c(48.8511847,48.8505062,48.8502346,48.8495305,48.8499405,48.8494376,48.8542721,48.8511847,48.853842,48.8515819)
df<-data.frame(x=x,y=y)
#Transforming to SF object 
sdf<-st_transform(st_as_sf(df, coords = c("x", "y"), 
                           crs = 4326, agr = "constant"),3857)
#My point to which I need to calculte
pnt<- st_transform( 
    st_sfc(st_point(x = c(2.3009132, 48.8511847)), crs = 4326), 3857)
#A buffer of 200m arround my point
buffer <- st_buffer(pnt,200)
#getting points within the buffer
intr <- st_intersects(sdf, buffer, sparse=F)
#transforming back to lon/lat
sdf <- st_transform(sdf,4326)

#getting the selected points
sdf<-sdf[which(unlist(intr)),]

#Only 4 points were found
> sdf
Simple feature collection with 4 features and 0 fields
geometry type:  POINT
dimension:      XY
bbox:           xmin: 256033.2 ymin: 6249533 xmax: 256201.4 ymax: 6249715
epsg (SRID):    3857
proj4string:    +proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +wktext +no_defs
                  geometry
1 POINT (256136.5 6249648)
2 POINT (256033.2 6249533)
3 POINT (256136.5 6249648)
4 POINT (256201.4 6249715)

#To verify I have calculated the distance to my point
t_sdf<-df%>% mutate(d = by(df, 1:nrow(df), function(row) {
+     distHaversine(c(row$x, row$y), c(2.3009132, 48.8511847), r = 6378137)
+ }))
#6 points are less than 200m to my point
> t_sdf %>% arrange(d)
          x        y         d
1  2.300913 48.85118   0.00000
2  2.300913 48.85118   0.00000
3  2.301496 48.85158  61.48172
4  2.299985 48.85051 101.61024
5  2.299587 48.85023 143.59844
6  2.299150 48.84994 189.36954
.....

Solution

Straight line (Euclidean) distances between a certain set of points vary across different projections.

As discussed here, 3857 is not a great choice for distance calculations. If we use a projection built for Northern France, where the points are located (perhaps EPSG:27561 - NTF (Paris) / Lambert Nord France), we get the expected results.

sdf <- st_as_sf(df, coords = c("x", "y"), 
                crs = 4326, agr = "constant")
sdf_proj <- st_transform(sdf, 3857)
sdf_France_proj <- st_transform(sdf, 27561)

Great circle distances

> st_distance(pnt, sdf, by_element = T) %>% sort()
Units: [m]
 [1]   0.00000   0.00000  61.50611 101.64007 143.64481 189.43485 253.46142 267.67954 411.50933 478.11306

Mercator (3857) Euclidean distances

> st_distance(pnt_proj, sdf_proj, by_element = T) %>% sort()
Units: [m]
 [1]   0.00000   0.00000  93.43522 154.41781 218.22699 287.78463 385.04306 406.64205 625.14635 726.33083

France projection (27561) Euclidean distances

> st_distance(pnt_France_proj, sdf_France_proj, by_element = T) %>% sort()
Units: [m]
 [1]   0.00000   0.00000  61.50289 101.63477 143.63731 189.42497 253.44822 267.66559 411.48772 478.08794

And here are the plots with buffers:

library(gridExtra)

buffer_proj <- st_buffer(pnt_proj, 200)
buffer_France_proj <- st_buffer(pnt_France_proj, 200)

proj <- ggplot() + g
eom_sf(data = buffer_proj, fill = "blue", alpha = 0.5) + 
  geom_sf(data = sdf_proj, color = "red") + 
  ggtitle("Mercator (3857)")

France_proj <- ggplot() + 
  geom_sf(data = buffer_France_proj, fill = "blue", alpha = 0.5) + 
  geom_sf(data = sdf_proj, color = "red") + 
  ggtitle("France proj (27561)")

grid.arrange(proj, France_proj, ncol = 1)