Shared segment of parallel lines

I need your help!!

I am working on a problem where I identify approximately parallel lines. From the two lines that I have deemed parallel, I want to calculate the length of the segment that has a shared domain or range, or both domain and range.

In these examples I am using truly parallel lines for sake of simplicity.

There are four scenarios that we have to solve for: positive slope, negative slope, no slope, and undefined slopes.

Helper functions:

# calculate range of x values 
x_range <- function(x) {
  bbox <- sf::st_bbox(x)
  return(c(bbox$xmin, bbox$xmax))
}

# calculate range of y values 
y_range <- function(x) {
  bbox <- sf::st_bbox(x)
  return(c(bbox$ymin, bbox$ymax))
}

# calculate overlapping range between two ranges
overlap_range <- function(r1, r2) {
  if (r1[2] < r2[1] || r2[2] < r1[1]) {
    return(NA)
  } else {
    return(c(max(r1[1], r2[1]), min(r1[2], r2[2])))
  }
}

find_overlaps <- function(a, b) {
  list(
    x_overlap = overlap_range(x_range(a), x_range(b)),
    y_overlap = overlap_range(y_range(a), y_range(b))
  )
}

overlap_rect <- function(x) {
  bbox <- sf::st_bbox(
    c(
      xmin = x$x_overlap[1], 
      xmax = x$x_overlap[2],
      ymin = x$y_overlap[1],
      ymax = x$y_overlap[2]
    )
  )
  sf::st_as_sfc(bbox)
}

Positive Slope

The first scenario is the shared positive slope.

Question:

How do I find the coordinates of the contained line segment to calculate the length? The solution should be able to handle the scenario where x and y are flipped as well.

# Positive Slope Scenario
x <- wk::wkt(
  c(
    "LINESTRING(0.0 0.0, 2.0 2.0)", # target line
    "LINESTRING(0.5 0.75, 2.5 2.75)" # one we've deemed parallel
  )
) 
plot(x)

We can see that these two lines are parallel. We find their overlapping range:

overlap <- find_overlaps(x[1], x[2])
overlap

$x_overlap
[1] 0.5 2.0

$y_overlap
[1] 0.75 2.00

What we want to calculate is the length of the red line segment contained by the bounding box.

plot(x, col = c("red", "black"))
plot(overlap_rect(overlap), add = TRUE)

Negative Slope Scenario

We have a very similar scenario. But this time with a negative slope. The solution should be able to handle if I want to find each line segment if x and y are swapped.

x <- wk::wkt(
  c(
    "LINESTRING(0 2, 2 0)", # target line
    "LINESTRING(1 0.5, 3 -1.5)" # deemed parallel
  )
)

(overlap <- find_overlaps(x[1], x[2]))

$x_overlap
[1] 1 2

$y_overlap
[1] 0.0 0.5

plot(x)
plot(overlap_rect(overlap), border = "red", add = TRUE)

Undefined Slope Scenario

Here, our overlap is only in one dimension as opposed to two. It may be more simple?

I think the answer here is is y_max - y_min.

x <- wk::wkt(
  c(
    "LINESTRING(2 1.5, 2 2.5)", # the target line 
    "LINESTRING(1 1, 1 2)", # deemed parallel
    "LINESTRING(2 1.5, 2 2)" # overlap range (1D)
  )
)

(overlap <- find_overlaps(x[1], x[2]))

$x_overlap
[1] NA

$y_overlap
[1] 1.5 2.0

plot(x[1:2])
plot(x[3], lwd = 3, lty = 1, col = "red", add = TRUE)

Segment Length:

Here is how we can calculate the overlap in the y-dimension:

overlap <- find_overlaps(x[1], x[2])
y_over <- overlap$y_overlap
y_over[2] - y_over[1]

[1] 0.5

No slope scenario

Similar to the undefined slope. We have a one dimensional overlap. I think the answer here is x_max - x_min.

x <- wk::wkt(
  c(
    "LINESTRING(0 1, 2 1)", # target feature
    "LINESTRING(1 2, 3 2)", # deemed parallel
    "LINESTRING(1 1, 2 1)" # overlap range (1D)
  )
)


plot(x[1:2])
plot(x[3], lwd = 3, lty = 1, col = "red", add = TRUE)

Segment Length:

Here is how we can calculate the overlap in the x-dimension:

overlap <- find_overlaps(x[1], x[2])
x_over <- overlap$x_overlap
x_over[2] - x_over[1]

[1] 1