How to draw a shape (ellipse or oval) following so

I am trying to plot rings of trees and calculate their areas. However, I have noticed that in reality not all rings have symmetric radii like a circle. I have data measurements of 4 radii, and I would like to plot rings (or any similar shape) following each point of every radio like this example (this figure was done manually with vectors in PowerPoint):

the problem is that in R I found only the possibility to plot these rings with the circles option from the symbols() function, and I got this graph:

using this R script:

data <- data.frame(
a = c(1,4,5,8, 10),
b = c(1, 3,7,9, 10),
c = c(2, 6, 8, 9 ,10),
d = c(1, 3, 4, 7, 9) )

data$y <- (data$a - data$b)/2 # y position
data$x <- (data$d - data$c)/2 # x position
data$z <- rowMeans(data[,1:4]) # radio length

symbols(x = data$x, y = data$y, circles=data$z, 
        xlim = c(-10, 10)*1.5, ylim = c(-10, 10)*1.5, inches = F, fg = "orange", lwd = 2)

I have checked some packages with functions to draw ellipses (elliplot, ellipse, ellipseplot, car, etc), but I don't like their functions. I am not interested in use these packages, on the contrary I would like to write an own code.

My idea is to plot a shape which best meets the real figure of a ring with my data values of the four radii, it can be an ellipse, oval, etc.

With a circle I am using only data of one radio (in my example, the mean of all radii). With a ellipse would be better, because I can use at least two values, the major-axis (A+B), and the minor-axis (C+D). But would be great to draw a shape that use the values of four radii (A, B, C, D) or even more radii.

Here a guy drew a very nice superellipse using a R script, and another one drew some ellipses likes rings also in R.

However, I don't know how to use their methods to my specific problem.

If somebody have idea how to start drawing at least an ellipse in R would be nice. But would be great to know how to draw a shape (oval, ellipse, etc.) using the values of four radii and finally calculate their area.

I would appreciate very much your help or any direction to do that.

UPDATE:

Thanks @cuttlefish44 for your excellent answer, that was very useful to explain tree growth to my students. However, most tropical trees have very irregular shapes and now I am wondering to know if can I draw this other shape with an additional radio "E" and the radii axes at different positions like this scheme:

any direction would be very useful for me.

If A & B are on y-axis and C & D are on x-axis, it isn't difficult to calculate the parameters of ellipses. I used optim() to get params (Note: this approach has tiny error, such as 2.439826e-12).

data manipulation

 # change all data into xy coordinates and make ring-factor
library(reshape2); library(dplyr)

data <- data.frame(
  a = c(1, 4, 5, 8, 10),
  b = c(1, 3, 7, 9, 10) * -1,
  c = c(2, 6, 8, 9, 10) * -1,
  d = c(1, 3, 4, 7, 9) )

data <- t(data)
colnames(data) <- LETTERS[1:ncol(data)]   # ring-factor
df <- melt(data, value.name = "x")        # change into long-form

df$y <- df$x                              # make xy coordinates
df[df$Var1=="a"|df$Var1=="b", "x"] <- 0
df[df$Var1=="c"|df$Var1=="d", "y"] <- 0

calculation of center coordinates, ox & oy

center <- df %>% group_by(Var2) %>% summarize(sum(x)/2, sum(y)/2) %>% as.data.frame()

calculation of parameters of ellipse; semi-major and -minor axis, ra & rb

opt.f <- function(par, subset, center) {     # target function
  ox <- center[[1]]                          # par[1] and par[2] are ra and rb
  oy <- center[[2]]
  x <- subset$x
  y <- subset$y
  sum(abs((x - ox)^2/par[1]^2 + (y - oy)^2/par[2]^2 - 1))   # from ellipse equation
}

lev <- levels(df$Var2)

## search parameters
res <- sapply(1:length(lev), function(a) 
  optim(c(1,1), opt.f, subset = subset(df, Var2 == lev[a]), 
        center = center[a, 2:3], control = list(reltol = 1.0e-12)))

res  # result. you can get detail by res[,1etc]. values are not 0 but much nearly 0

function to plot (Probably some packages have similar one)

radian <- function(degree) degree/180*pi
plot.ellipse <- function(ox, oy, ra, rb, phi=0, start=0, end=360, length=100, func=lines, ...) {
  theta <- c(seq(radian(start), radian(end), length=length), radian(end))
  if (phi == 0) {
    func(ra*cos(theta)+ox, rb*sin(theta)+oy, ...)
  } else {
    x <- ra*cos(theta)
    y <- rb*sin(theta)
    phi <- radian(phi)
    cosine <- cos(phi)
    sine <- sin(phi)
    func(cosine*x-sine*y+ox, sine*x+cosine*y+oy, ...)
  }
}

draw

plot(0, type="n", xlim=c(-10, 10), ylim =c(-10, 10), asp=1, xlab="x", ylab="y", axes = F)
axis(1, pos=0);axis(2, pos=0, las=2)
points(df$x, df$y)
for(a in 1:length(lev)) plot.ellipse(ox = center[a, 2], oy = center[a, 3], 
                                     ra = res[,a]$par[1], rb = res[,a]$par[2], length=300)

area <- sapply(res[1,], function(a) pi * a[1] * a[2])