Point labels perpendicular to a curve in ggplot2

I would like to label points on a sine function so that the labels are always legible. In a sine wave plot in which θ ranges from 0 to 2π, sin(θ) ranges from −1 to +1. Thus, the plot’s xy ratio is

\text{plot ratio}= \frac{2\pi-0}{1- (-1)}=\pi

The first derivative of the sine function is the cosine function. In my plot, the slope of the tangent line at each point is

\text{tangent slope} = \text{plot ratio}\times \cos(\theta) The angle of the tangent line’s slope is the arctan of the slope. I would like to place the label perpendicular to the tangent line so I will add 90 degrees (i.e., π/2 radians).

\text{text angle}=\tan^{-1}(\text{tangent slope})+\pi/2

Now I need a pair of functions that will convert this angle into the right values for ggplot2’s hjust and vjust arguments. I have added the angle2vjust and angle2hjust functions to the WJSmisc package, but I have defined them here as well:

angle2vjust <- function(theta, multiplier = 1, as_degrees = FALSE) {
  if (as_degrees) theta <- theta * pi / 180
  (((sin(theta + pi) + 1) / 2) - 0.5) * multiplier + 0.5
}

angle2hjust <- function(theta, multiplier = 1, as_degrees = FALSE) {
  if (as_degrees) theta <- theta * pi / 180
  (((cos(theta + pi) + 1) / 2) - 0.5) * multiplier + 0.5
}

Now we plot the sine function with labels. I have used geom_richtext from the ggtext package because it allows me to set a white background along with padding and margins.

library(tidyverse)
library(ggtext)

plot_ratio <- pi

tibble(theta = seq(0, 2 * pi, length.out = 13),
            y =  sin(theta),
            tangent_slope = cos(theta) * plot_ratio,
            text_angle = atan(tangent_slope) + pi / 2) %>%
  ggplot(aes(theta, y)) +
  geom_richtext(aes(label = formatC(y, digits = 2, format = "f"),
                    vjust = angle2vjust(text_angle),
                    hjust = angle2hjust(text_angle)),
                label.color = NA,
                label.padding = unit(1, "pt"),
                label.margin = unit(5, "pt"),
                size = 4) +
  geom_point() +
  stat_function(fun = sin) +
  scale_x_continuous("&theta;",
                     breaks = seq(0, 2 * pi, 
                                  length.out = 13),
                     minor_breaks = NULL,
                     labels = function(x) round(x * 180 / pi)) +
  scale_y_continuous("sin(&theta;)") +
  coord_fixed(ratio = plot_ratio, clip = "off") +
  theme_minimal(base_size = 16) +
  theme(axis.title.x = element_markdown(),
        axis.title.y = element_markdown())

If you do not mind turning your head to one side or the other, a somewhat easier method is to set the label’s vjust to a negative value and rotate the labels by the angle of the tangent line:

tibble(theta = seq(0, 2 * pi, length.out = 13),
            y = sin(theta),
            tangent_slope = cos(theta) * plot_ratio,
            text_angle = atan(tangent_slope)) %>%
  ggplot(aes(theta, y)) +
  geom_richtext(aes(label = round(y, 2),
                    angle = text_angle * 180 / pi),
                vjust = 0,
                label.color = NA,
                label.padding = unit(1, "pt"),
                label.margin = unit(2, "pt"),
                size = 4) +
  geom_point() +
  stat_function(fun = sin) +
  scale_x_continuous("θ",
                     breaks = seq(0, 2 * pi, 
                                  length.out = 13),
                     minor_breaks = NULL,
                     labels = function(x) round(x * 180 / pi)) +
  scale_y_continuous("sin(θ)") +
  coord_fixed(ratio = plot_ratio, clip = "off") +
  theme_minimal(base_size = 16) +
  theme(axis.title.x = element_markdown(),
        axis.title.y = element_markdown())

What if you do not know the function’s first derivative? You can approximate the slope of the tangent line by comparing a function’s output of each point with the output of a slightly deviated point. Here is a plot of the normal cumulative distribution function.

# Small change in x
dx <- .00000001
plot_ratio <- 6

tibble(x = seq(-3,3,.5),
       y = pnorm(x),
       tangent_slope = plot_ratio * (pnorm(x + dx) - y) / dx,
       text_angle = atan(tangent_slope) + pi / 2,
       degrees = text_angle * 180 / pi) %>% 
  ggplot(aes(x,y)) + 
  geom_point() +
  geom_richtext(aes(label = formatC(y, 2, format = "f"),
                    vjust = angle2vjust(text_angle),
                    hjust = angle2hjust(text_angle)),
                label.color = NA,
                label.padding = unit(1, "pt"),
                label.margin = unit(5, "pt"),
                size = 4) +
  stat_function(fun = pnorm) + 
  theme_minimal(base_size = 16) + 
  theme(axis.title.x = element_markdown(),
        axis.title.y = element_markdown()) +
  coord_fixed(ratio = plot_ratio) + 
  scale_x_continuous("*x*", breaks = -3:3) +
  scale_y_continuous("&Phi;(*x*)")