Is there way in ggplot2 to place text on a curved path?

Question

Is there a way to put text along a density line, or for that matter, any path, in ggplot2? By that, I mean either once as a label, in this style of xkcd: 1835, 1950 (middle panel), 1392, or 2234 (middle panel). Alternatively, is there a way to have the line be repeating text, such as this xkcd #930 ? My apologies for all the xkcd, I'm not sure what these styles are called, and it's the only place I can think of that I've seen this before to differentiate areas in this way.

Note: I'm not talking about the hand-drawn xkcd style, nor putting flat labels at the top

I know I can place a straight/flat piece of text, such as via annotate or geom_text, but I'm curious about bending such text so it appears to be along the curve of the data.

I'm also curious if there is a name for this style of text-along-line?

Example ggplot2 graph using annotate(...):

Above example graph modified with curved text in Inkscape:

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Edit: Here's the data for the first two trial runs in March and April, as requested:

df <- data.frame(
  monthly_run = c('March', 'March', 'March', 'March', 'March', 'March', 'March', 
                  'March', 'March', 'March', 'March', 'March', 'March', 'March', 
                  'April', 'April', 'April', 'April', 'April', 'April', 'April', 
                  'April', 'April', 'April', 'April', 'April', 'April', 'April'),
  duration    = c(36, 44, 45, 48, 50, 50, 51, 54, 55, 57, 60, 60, 60, 60, 30,
                  40, 44, 47, 47, 47, 53, 53, 54, 55, 56, 57, 69, 77)
  )

ggplot(df, aes(x = duration, group = monthly_run, color = monthly_run)) + 
  geom_density() + 
  theme_minimal()`

Answer 1

In the end, this question prompted Teun van den Brand (@teunbrand) and I to develop the geomtextpath package, which is now on CRAN.

So now the question could be answered much more directly and simply:

library(geomtextpath)

ggplot(df, aes(x = duration, color = monthly_run)) + 
  geom_textdensity(aes(label = monthly_run, hjust = monthly_run,
                       vjust = monthly_run), size = 6) +
  scale_hjust_manual(values = c(0.4, 0.55)) +
  scale_vjust_manual(values = c(1.1, -0.2)) +
  scale_y_continuous(limits = c(0, 0.06)) +
  theme_minimal() +
  theme(legend.position = "none")