---
title: "eulerr under the hood"
author: "Johan Larsson"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{eulerr under the hood}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: eulerr.bib
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  echo = FALSE,
  warning = FALSE,
  message = FALSE,
  fig.height = 3,
  fig.width = 3,
  collapse = TRUE,
  comment = "#>"
)

library(grid)
library(RConics)
library(eulerr)
library(lattice)

pal <- c("#999999", "#E69F00", "#56B4E9", "#009E73", "#F0E442",
         "#0072B2", "#D55E00", "#CC79A7")

ellipse_arc <- function(saxes = c(1, 1),
                        loc = c(0, 0),
                        theta = 0,
                        n = 200,
                        rng = c(0, 2*pi)) {
  b <- min(saxes[1], saxes[2])
  a <- max(saxes[1], saxes[2])
  d2 <- (a - b) * (a + b)
  if (length(rng) == 1)
    phi <- rng - theta
  else
    phi <- seq(rng[1], rng[2], len = n) - theta
  sp <- sin(phi)
  cp <- cos(phi)
  r <- a * b/sqrt((saxes[2] * cp)^2 + (saxes[1] * sp)^2)
  P <- matrix(nrow = n, ncol = 2)
  P[, 1] <- r * cp
  P[, 2] <- r * sp

  if (theta != 0) {
    P <- P %*% matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)),
                      byrow = TRUE, nrow = 2, ncol = 2)
  }
  P <- P + matrix(loc[1:2], nrow = nrow(P), ncol = 2, byrow = TRUE)
  P
}

set.seed(1)
```

## Introduction

**eulerr** relies on an extensive machinery to turn user input into
a pretty Euler diagram. Little of this requires any
tinkering from the user. To make that happen, however, **eulerr**
needs to make several well-formed decisions about the design of the
diagram on behalf of the user, which is not a trivial task.

This document outlines the implementation of **eulerr** from input
to output. It is designed to be an evolving documentation on the innards of the
program.

## Input

Euler diagrams present relationships between sets, wherefore the data must
describe these relationships, either directly or indirectly.
**eulerr** allows several alternatives for this data, namely,

* intersections and relative complements 
  $$
  A \setminus B = 3 \quad B \setminus A = 2 \quad A \cap B=1,
  $$
* unions and identities
  $$
  A=4 \quad B=3 \quad A \cap B=1,
  $$
* a matrix of binary (or boolean) indices,
  $$
  \begin{bmatrix}
    \mathbf{A} & \mathbf{B}\\
    1          & 0     \\
    1          & 0     \\
    1          & 0     \\
    1          & 1     \\
    0          & 1     \\
    0          & 1,
  \end{bmatrix}
  $$
* a list of sample spaces
  $$
  \begin{array}{l}
      A = \{a,\,b,\,c,\,d\}\\
      B = \{a,\,e,\,f,\}
    \end{array},
  $$
* or a two- or three-way table

                    $A$      $A^\mathsf{c}$
  ----------------  ---     ----------------
          $B$        1             2
   $A^\mathsf{c}$    3             0

As an additional feature for the matrix form, the user may supply a factor
variable with which to split the data set before fitting the diagram,
which sometimes improves diagrams where the set relationships vary
across categories.

Whichever type of input is provided, **eulerr** will translate it to the first
and second types, *intersections and relative complements* and
*unions and identities*, which will be used in the steps to come.

The Euler diagram is then fit in two
steps: first, an initial layout is formed with circles using only the
sets' pairwise relationships. Second, this layout is fine-tuned taking
all $2^N-1$ intersections into consideration.

## Initial layout

For our initial layout, we adopt a constrained version of
multi-dimensional scaling (MDS) that has been adapted from
**venn.js** [@Frederickson_2016], which in turn is a modification of
an algorithm used in **venneuler** [@wilkinson_exact_2012].
In it, we consider only the pairwise intersections between sets, attempting to
position their respective shapes so as to minimize the difference between the
separation between their centers
required to obtain an optimal overlap
and the actual overlap of the shapes in the diagram.

This problem is unfortunately intractable for ellipses, being that there is an
infinite number of ways by which we can position two ellipses to obtain a given
overlap. Thus, we restrict ourselves to circles in our initial layout, for 
which we can use the
circle--circle overlap formula to numerically find the
required distance, $d$, for each pairwise relationship.

$$
\begin{aligned}
O_{ij} = & r_i^2\arccos\left(\frac{d_{ij}^2 + r_i^2 - r_j^2}{2d_{ij}r_i}\right) +
r_j^2\arccos\left(\frac{d_{ij}^2 + r_j^2 - r_i^2}{2d_{ij}r_j}\right) -\\
&\quad \frac{1}{2}\sqrt{(-d_{ij} + r_i + r_j)(d_{ij} + r_i - r_j)(d_{ij} - r_i + r_j)(d_{ij} + r_i + r_j)},
\end{aligned}
$$

where $r_i$ and $r_j$ are the radii of the circles representing the $i$^th^ and
$j$^th^ sets respectively, $O_{ij}$ their overlap, and $d_{ij}$ their separation.

```{r, fig.cap = "The circle--circle overlap is computed as a function of the discs' separation ($d_{ij}$), radii ($r_i,r_j$), and area of overlap ($O_{ij}$)."}
c0 <- ellipseToConicMatrix(c(1, 1), c(0, 0), 0)
c1 <- ellipseToConicMatrix(c(0.7, 0.7), c(1.2, 0), 0)
pp <- intersectConicConic(c0, c1)

theta0 <- atan2(pp[2, 1], pp[1, 1])
theta1 <- atan2(pp[2, 2], pp[1, 2])

phi0 <- atan2(pp[2, 1], pp[1, 1] - 1.2)
phi1 <- atan2(pp[2, 2], pp[1, 2] - 1.2)

seg <- rbind(ellipse_arc(c(0.7, 0.7), c(1.2, 0), 0, n = 50, c(-pi, phi0)),
             ellipse_arc(c(1, 1), c(0, 0), 0, n = 100, c(theta0, theta1)),
             ellipse_arc(c(0.7, 0.7), c(1.2, 0), 0, n = 50, c(phi1, pi)))

ospot <- c(mean(seg[, 1]), mean(seg[, 2]))

xyplot(1~1, xlim = c(-1.1, 2), ylim = c(-1.5, 1.1), asp = "iso",
       xlab = NULL, ylab = NULL,
       scales = list(draw = FALSE),
       par.settings = list(axis.line = list(col = "transparent")),
       panel = function() {
         panel.polygon(seg, col = "steelblue1", alpha = 0.5,
                       border = "transparent")
         grid::grid.circle(0, 0, r = 1, default.units = "native",
                           gp = gpar(fill = "transparent"))
         grid::grid.circle(1.2, 0, r = 0.7, default.units = "native",
                           gp = gpar(fill = "transparent"))
         pBrackets::grid.brackets(1.2, -1.2, 0, -1.2, h = 0.05)
         pBrackets::grid.brackets(-1, 0, 0, 0, h = 0.05)
         pBrackets::grid.brackets(1.2, 0, 1.9, 0, h = 0.05)
         panel.lines(c(0, 0), c(0, -1.2), lty = 2, col = "grey65")
         panel.lines(c(1.2, 1.2), c(0, -1.2), lty = 2, col = "grey65")
         panel.text(0.6, unit(-1.3, "native"),
                    labels = expression(italic(d[ij])), pos = 1)
         panel.text(ospot[1], ospot[2], labels = expression(italic(O[ij])))
         panel.text(-0.5, 0.1, labels = expression(italic(r[i])), pos = 3,
                    default.units = "native")
         panel.text(1.55, 0.1, labels = expression(italic(r[j])), pos = 3,
                    default.units = "native")
         panel.points(c(0, 1.2), c(0, 0), pch = 21, col = 1, cex = 1,
                      fill = "white")
       })
```

Setting $r_i = \sqrt{F_i/\pi}$, where $F_i$ is
the size of the $i$^th^ set, we are able to obtain $d$ numerically using
the squared difference between $O$ and the desired overlap
as loss function,

$$
\mathcal{L}(d_{ij}) = \left(O_{ij} - (F_i \cap F_j)  \right)^2, \quad \text{for } i <
j \leq n,
$$

which we optimize using `optimize()`^[According to the documentation, `optimize()` consists of a "combination of golden section search and successive parabolic interpolation." from **stats**.]

For a two-set combination, this is all we need to plot an exact diagram, given
that we now have the two circles' radii and separation and may place the circles
arbitrarily as long
as their separation, $d$, remains the same. This is not, however, the case
with more than two sets.

With three or more sets, the circles need to be arranged so that
they interfere minimally with one another. In some cases, the set
configuration allows this to be accomplished flawlessly,
but often, compromises must me made. As is often the case
in this context, this turns out to be another optimization problem.
It can be
tackled in many ways; **eulerr**'s approach is based on a method
developed by @Frederickson_2015, which the author describes as
constrained multi-dimensional scaling.

The algorithm tries to position the circles so that the separation
between each pair of circles matches the separation required from
the separation equation.
If the two sets are disjoint, however, the algorithm is indifferent to the
relative locations of those circles as long as they do not intersect. The
equivalent applies to subset sets: as long as the circle representing the 
smaller set remains within the larger circle, their locations are free to vary.
In all other cases, the loss function is the residual sums
of squares of the optimal separation of circles, $d$, that we found
in the overlap equation, and the actual distance in the layout we are
currently exploring.

$$
\mathcal{L}(h,k) = \displaystyle \ell{\sum_{1\leq i<j\leq N}}
\begin{cases}
0 & F_i \cap F_j = \emptyset \text{ and } O_{ij} = 0\\
0 & (F_i \subseteq F_j \text{ or } F_i \supseteq F_j) \text{ and } O_{ij}=0\\
\left(\left(h_i-h_j\right)^2+\left(k_i-k_j\right)^2-d_{ij}^2\right)^2  & \text{otherwise} \\
\end{cases}.
$$

The analytical gradient is retrieved as usual by
taking the derivative of the loss function,

$$
\vec{\nabla} f(h_i) = \sum_{j=1}^N
\begin{cases}
\vec{0} & F_i \cap F_j = \emptyset \text{ and } O_{ij} = 0\\
\vec{0} & (F_i \subseteq F_j \text{ or } F_i \supseteq F_j) \text{ and } O_{ij}=0\\
4\left(h_i-h_j\right)\left(\left(h_i-h_j\right)^2+\left(k_i-k_j\right)^2-d_{ij}^2\right) & \text{otherwise}, \\
\end{cases}
$$

where $\vec{\nabla} f(k_i)$ is found as in the gradient with $h_i$
swapped for $k_i$ (and vice versa).

The Hessian for our loss function is given next. However,
because the current release of R suffers from a bug^[The current development version
of R features a fix for this bug; **eulerr** will be updated to
use the analytical Hessian as soon as it is introduced in a stable version of R.] causing the
analytical Hessian to be updated improperly, the current
release of **eulerr** instead relies on the numerical approximation of
the Hessian offered by the optimizer.

$$
\small
\mathbf{H}(h,k) = \ell{\sum_{1\leq i<j\leq N}}
\begin{bmatrix}
4\left(\left(h_i-h_j\right)^2+\left(k_i-k_j\right)^2-d_{ij}^2\right)+8\left(h_i-h_j\right)^2 &
\cdots &
8\left(h_i-h_j\right)\left(k_i-k_j\right)\\
\vdots & \ddots & \vdots \\
8\left(k_i-k_j\right)\left(h_i-h_j\right) &
\cdots &
4\left(\left(h_i-h_j\right)^2+\left(k_i-k_j\right)^2-d_{ij}^2\right)+8\left(k_i-k_j\right)^2
\end{bmatrix}.
$$

Note that the constraints given in loss and gradients
still apply to each element of the Hessian and have been omitted for
convenience only.

We optimize the loss function using the nonlinear optimizer `nlm()` from
the R core package **stats**. The underlying code for `nlm` was
written by @Schnabel_1985. It was ported to R by Saikat DebRoy and the R
Core team [@RCT_2017] from a previous FORTRAN to C
translation by Richard H. Jones. `nlm()` consists of a system
of Newton-type algorithms and performs well for
difficult problems [@Nash_2014].

The initial layout outlined above will sometimes turn up perfect diagrams, but
only reliably so when the diagram is completely determined by its pairwise
intersections. More pertinently, we have not yet considered the higher-order
intersections in our algorithm and neither have we approached the problem
of using ellipses---as we set out to do.

## Final layout

We now need to account for all the sets' intersections and, consequently,
all the overlaps in the diagram. The goal is to map each area uniquely
to a subset of the data from the input and for this purpose we will
use the sets' intersections and the relative complements of these
intersections, for which we will use the shorthand $\omega$. We introduced this
form in the *input* section, but now define it rigorously.

For a family of *N* sets, $F = F_1, F_2, \dots, F_N$, and their $n=2^N-1$
intersections, we define $\omega$ as the intersections of these sets and their
relative complements, such that

$$
\begin{aligned}
\omega_{1} & = F_1 \setminus \bigcap_{i=2}^N F_i  \\
\omega_{2} & = \bigcap_{i=1}^2 F_i \setminus \bigcap_{i=3}^{N} F_i\\
\vdots    & = \vdots \\
\omega_n & = \bigcap_{i=1}^{N}F_i
\end{aligned}
$$

with

$$
\sum_{i = 1}^n \omega_i =  \bigcup_{j=1}^N F_i.
$$

Analogously to $\omega$, we also introduce the $\&$-operator, such that

$$
F_i \& F_j = (F_i \cap F_j)\setminus (F_i \cap F_j)^\textsf{c}.
$$

The fitted diagram's area-equivalents for $\omega$ will be defined as $A$, so that
an exact diagram requires that $\omega_i = A_i$ for $i=1,2,\dots,2^N-1$,
where $N$ is the number of sets in the input.

In our initial configuration, we restricted
ourselves to circles but now extend ourselves also to ellipses. From now on,
we abandon the practice of treating circles separately---they are only
a special case of ellipses, and, hence, everything that applies to an ellipse does so
equally for a circle.

### Intersecting ellipses

We now need the ellipses'
points of intersections. **eulerr**'s approach to this is outlined in
[@Richter-Gebert_2011] and based in *projective*, as opposed to
*Euclidean*, geometry.

To collect all the intersection points, we naturally need only to consider two
ellipses at a time. The canonical form of an ellipse is given by

$$
\frac{\left[ (x-h)\cos{\phi}+(y-k)\sin{\phi} \right]^2}{a^2}+
\frac{\left[(x-h) \sin{\phi}-(y-k) \cos{\phi}\right]^2}{b^2} = 1,
$$

where $\phi$ is the counter-clockwise angle from the positive x-axis to the
semi-major axis $a$, $b$ is the semi-minor axis, and $h, k$ are the x- and
y-coordinates, respectively, of ellipse's center.

```{r, fig.cap = "A rotated ellipse with semimajor axis $a$, semiminor axis $b$, rotation $\\phi$, and center $h,k$."}
h <- 0.7
k <- 0.5
a <- 1
b <- 0.6
phi <- pi/5
n <- 200
ellipse <- RConics::ellipse(c(a, b), c(h, k), phi,  n = n)

xyplot(1 ~ 1, xlim = c(-0.3, 1.7), ylim = c(-0.3, 1.3), asp = "iso",
       xlab = NULL, ylab = NULL,
       par.settings = list(axis.line = list(col = "transparent")),
       scales = list(draw = FALSE),
       panel = function(x, y, ...) {
         # grid
         panel.refline(h = 0)
         panel.refline(v = 0)
         
         # rotation
         arc <- ellipse_arc(c(0.2, 0.2), c(h, k), rng = c(0, phi))
         theta <- ellipse_arc(c(a, b), c(h, k), phi, n = 1)
         
         panel.lines(x = c(h, ellipse[1, 1]), y = c(k, ellipse[1, 2]), lty = 2,
                     col = "grey50")
         panel.lines(x = c(h, theta[1, 1]), y = c(k, theta[1, 2]), lty = 2,
                     col = "grey50")
         panel.lines(arc, col = "grey50")
         panel.text(x = arc[nrow(arc)/2, 1] + 0.06, y = arc[nrow(arc)/2, 2] + 0.02,
                    labels = expression(italic(phi)))
         
         # semiaxes
         pBrackets::grid.brackets(h, k, ellipse[n/2, 1], ellipse[n/2, 2],
                                  col = "grey50")
         pBrackets::grid.brackets(h, k, ellipse[n/4, 1], ellipse[n/4, 2],
                                  h = 0.04, col = "grey50")
         
         panel.text(0.37, 0.12, labels = "a", font = 3)
         panel.text(0.43, 0.68, labels = "b", font = 3)
         
         # center
         panel.text(x = h, y = k + 0.025, pos = 3, label = c("h,k"), font = 3)
         panel.points(x = h, y = k, pch = 21, fill = "white", col = "black")
         
         # ellipse
         panel.lines(ellipse, col = 1)
       })
```

However, because an ellipse
is a conic^[The circle, parabola, and hyperbola are the other types of conics.] it
can be represented in quadric form,

$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
$$

that in turn can be represented as a matrix,

$$
\begin{bmatrix}
A   & B/2 & D/2 \\
B/2 & C   & E/2 \\
D/2 & E/2 & F
\end{bmatrix},
$$

which is the form we need to intersect our ellipses. We now proceed to
form three degenerate conics from a linear combination of the
two ellipses we wish to intersect,
split one of these degenerate conics into two lines, and
intersect one of the ellipses with these lines, yielding 0 to 4
intersection points points.

```{r intersect-1, fig.cap = "The process used to intersect two ellipses, here yielding four points. This figure was inspired by an example from Richter--Gebert 2011.", fig.width = 2, fig.height = 2, fig.show = "hold"}
C1 <- ellipseToConicMatrix(c(8, 2), c(0, 0), -pi/3)
C2 <- ellipseToConicMatrix(c(5, 2), c(1, -2), pi/5)

ellipses <- data.frame(rbind(ellipse(c(8, 2), c(0, 0), -pi/3),
                             ellipse(c(5, 2), c(1, -2), pi/5)))
colnames(ellipses) <- c("x", "y")
ellipses$fac <- rep(c("A", "B"), each = 201)
pp <- intersectConicConic(C1, C2)
x <- pp[1, ]
y <- pp[2, ]
a <- double(6)
b <- double(6)
k <- 1
for (i in 1:3) {
  for (j in (i + 1):4) {
    b[k] <- (y[j] - y[i])/(x[j] - x[i])
    a[k] <- b[k]*(-x[i]) + y[i]
    k <- k + 1
  }
}
a <- a[c(1, 6, 3, 4, 2, 5)]
b <- b[c(1, 6, 3, 4, 2, 5)]

xyplot(
  y ~ x, data = ellipses, type = "l", groups = fac, asp = 1,
  xlab = NULL, ylab = NULL,
  scales = list(draw = FALSE, axes = FALSE),
  par.settings = list(axis.line = list(col = "transparent")),
  panel = function(x, y, ...) {
    panel.abline(a = a[3], b = b[3], col = "#FEFEFE")
    panel.abline(a = a[4], b = b[4], col = "#FEFEFE")
    panel.xyplot(x, y, ..., col = 1)
  })

xyplot(y ~ x, data = ellipses, type = "l", groups = fac, asp = 1,
       xlab = NULL, ylab = NULL,
       scales = list(draw = FALSE, axes = FALSE),
       par.settings = list(axis.line = list(col = "transparent")),
       panel = function(x, y, ...) {
         panel.xyplot(x, y, ..., col = 1)
         for (i in 1:length(a)) {
           panel.abline(a = a[i], b = b[i], col = pal[ceiling(i/2) + 1])
         }
       })

xyplot(
  y ~ x, data = ellipses, type = "l", groups = fac, asp = 1,
  xlab = NULL, ylab = NULL,
  scales = list(draw = FALSE, axes = FALSE),
  par.settings = list(axis.line = list(col = "transparent")),
  panel = function(x, y, ...) {
    panel.xyplot(x, y, ..., col = c("transparent", "black"))
    panel.abline(a = a[3], b = b[3], col = pal[3])
    panel.abline(a = a[4], b = b[4], col = pal[3])
    panel.points(t(pp[1:2, ]), col = 1, pch = 21, fill = "white")
  })
```

### Overlap areas

Using the intersection points of a set of ellipses that we retrieved in,
we can now find the
overlap of these ellipses. We are only interested in the points that are *contained
within all of these ellipses*, which together form a geometric shape consisting of a
convex polygon, the sides of which are made up of straight lines between
consecutive points, and a set of elliptical arcs---one for each
pair of points.

```{r polyarea, fig.cap = "The overlap area between three ellipses is the sum of a convex polygon (in grey) and 2--3 ellipse segments (in blue)."}
x <- c(0, -0.3, 0.2)
y <- c(0, 0.1, 0.3)
ra <- a <- c(0.3, 0.5, 0.4)
rb <- b <- c(0.3, 0.3, 0.6)
phi <- c(-pi/6, 2, -2)

ee <- data.frame(x, y, ra, rb, phi)

tx <- atan2(-b*tan(phi), a)
ty <- atan2(b*tan(pi/2L - phi), a)

xlim <- range(x + a*cos(tx)*cos(phi) - b*sin(tx)*sin(phi),
              x + a*cos(tx + pi)*cos(phi) - b*sin(tx + pi)*sin(phi))
ylim <- range(y + b*sin(ty)*cos(phi) + a*cos(ty)*sin(phi),
              y + b*sin(ty + pi)*cos(phi) + a*cos(ty + pi)*sin(phi))

pp <- matrix(NA, ncol = 4, nrow = 0)

for (i in 1:2) {
  for (j in (i + 1):3) {
    e1 <- ellipseToConicMatrix(c(ra[i], rb[i]), c(x[i], y[i]), phi[i])
    e2 <- ellipseToConicMatrix(c(ra[j], rb[j]), c(x[j], y[j]), phi[j])
    pp <- rbind(pp, cbind(t(intersectConicConic(e1, e2)[1:2, ]), i, j))
  }
}

sel <- logical(nrow(pp))
for (k in 1:nrow(pp)) {
  in_which <- ((pp[k, 1] - x)*cos(phi) + (pp[k, 2] - y)*sin(phi))^2/ra^2 +
    ((pp[k, 1] - x)*sin(phi) - (pp[k, 2] - y)*cos(phi))^2/rb^2 <= 1 + 0.1
  sel[k] <- all(in_which)
}

pp <- pp[sel, ]
mid <- cbind(mean(pp[, 1]), mean(pp[, 2]))
seglines <- matrix(NA, ncol = 2, nrow = 0)

ang <- atan2(pp[, 1] - mid[1], pp[, 2] - mid[2])
ord <- order(ang)

pp <- pp[ord, ]

j <- nrow(pp)
for (i in 1:nrow(pp)) {
  k <- intersect(pp[i, 3:4], pp[j, 3:4])
  start <- atan2(pp[j, 2] - y[k], pp[j, 1] - x[k])
  stop <- atan2(pp[i, 2] - y[k], pp[i, 1] - x[k])
  arc <- ellipse_arc(c(a[k], b[k]), c(x[k], y[k]), theta = phi[k],
                     rng = c(start, stop))
  seglines <- rbind(seglines, arc)
  j <- i
}

els <- eulerr:::ellipse(x, y, ra, rb, phi)
x0 <- c(lapply(els, "[[", "x"), recursive = TRUE)
y0 <- c(lapply(els, "[[", "y"), recursive = TRUE)

xyplot(
  y ~ x, data = ee, asp = "iso",
  xlim = extendrange(xlim, f = 0.01), ylim = extendrange(ylim, f = 0.01),
  scales = list(draw = FALSE), xlab = NULL, ylab = NULL,
  par.settings = list(axis.line = list(col = "transparent")),
  panel = function(x, y, ...) {
    grid.polygon(x0, y0,  id.lengths = rep(200, 3),
                 default.units = "native",
                 gp = grid::gpar(fill = "transparent"))
    panel.polygon(seglines, col = "slategray2")
    panel.polygon(pp[, 1:2], col = "grey90")
    panel.points(mid, pch = 4, col = 1, cex = 1.5)
    panel.points(pp[, 1:2, drop = FALSE], col = 1, pch = 21, fill = "white",
                 cex = 1.5)
  }
)
```

We continue by ordering the points around their centroid. It is then trivial to
find the area of the
polygon section since it is always convex. Now, because each
elliptical segment is formed from the arcs that connect successive
points, we can establish the segments' areas algorithmically [@Eberly_2016].
For each ellipse and its related pair of points (located at angles $\theta_0$ and
$\theta_1$ from the semimajor axis), we proceed to find its area by

1.  centering the ellipse at $(0, 0)$,
2.  normalizing its rotation, which is not needed to compute the area,
3.  integrating the ellipse over [$0,\theta_0$] and [$0,\theta_1$], producing
    elliptical sectors $F(\theta_0)$ and $F(\theta_1)$,
4.  subtracting the smaller ($F(\theta_0$)) of these sectors from the larger ($F(\theta_0$), and
5.  subtracting the triangle section to finally find the segment area,
    $$
    F(\theta_1) - F(\theta_0) - \frac{1}{2}\left|x_1y_0 - x_0y_1\right|,
    $$
    where
    $$
    F(\theta) = \frac{a}{b}\left[ \theta - \arctan{\left(\frac{(b - a)\sin{2\theta}}{b + a +(b - a )\cos{2\theta}} \right)}\right].
    $$

This procedure is illustrated in the following figure. Note that there are
situations where this algorithm is altered, such that when the sector angle
ranges beyond $\pi$---we refer the interested reader to @Eberly_2016.

```{r ellipsesegment, fig.cap = "The elliptical segment in blue is found by first subtracting the elliptical sector from $(a, 0)$ to $\\theta_0$ from the one from $(a, 0)$ to $\\theta_1$ and then subtracting the triangle part (in grey)."}
ellipse <- ellipse(c(1, 0.6), c(0, 0), 0)
i0 <- 15
i1 <- 45

tri <- rbind(cbind(0, 0),
             ellipse[i0, , drop = FALSE],
             ellipse[i1, , drop = FALSE])

xyplot(1~1, xlim = c(-1.2, 1.2), ylim = c(-0.8, 0.8), asp = "iso",
       xlab = NULL, ylab = NULL,
       par.settings = list(axis.line = list(col = "transparent")),
       scales = list(draw = FALSE),
       panel = function(x, y, ...) {
         panel.grid(x = 0, y = 0, h = 1,v = 1)
         panel.lines(ellipse, col = 1)
         panel.polygon(ellipse[i0:i1, ], col = "steelblue1", alpha = 0.5)
         panel.polygon(tri, col = "grey95")
         panel.points(ellipse[c(i0, i1), ], col = 1, pch = 21, fill = "white")
         panel.text(ellipse[c(i0, i1), ],
                    adj = c(-0.5, -0.5),
                    labels = c(expression(italic(theta[0])),
                               expression(italic(theta[1]))))
         pBrackets::grid.brackets(1, 0, 0, 0, h = 0.05,
                                  type = 1)
         pBrackets::grid.brackets(0, 0, 0, 0.6, h = 0.04)
         panel.text(0.5, unit(-0.05, "npc"), labels = "a", font = 3, pos = 1)
         panel.text(-0.05, unit(0.3, "native"), labels = "b", font = 3, pos = 2)
       })
```

Finally, the area of the overlap is then obtained by adding the area of the
polygon and all the elliptical arcs together.

Note that this does not yet give us
the areas that we require, namely $A$: the area-equivalents to the set
intersections and relative complements from our definition of the
intersections. For this, we must
decompose the overlap areas so that each area maps uniquely
to a subspace of the set configuration. This, however, is simply a matter of 
transversing down the
hierarchy of overlaps and subtracting the higher-order overlaps from the
lower-order ones. For a three-set relationship of sets $A$, $B$, and $C$,
for instance, this means subtracting the $A\cap B \cap C$ overlap from
the $A \cap B$ one to retrieve the equivalent of $(A \cap B) \setminus C$.

The exact algorithm may in rare instances^[1 out of approximately 7000 in
our simulations.], break down, the culprit being numerical precision
issues that occur when ellipses are tangent or
completely overlap. In these cases, the algorithm will approximate
the area of the involved overlap by

1.  spreading points across the ellipses using Vogel's
    method.
2.  identifying the points that are inside the intersection via the inequality
    $$
    \begin{aligned}
    &\frac{\left[(x-h)\cos{\phi}+(y-k)\sin{\phi} \right]^2}{a^2} + 
    &\quad \frac{\left[(x-h) \sin{\phi}-(y-k)\cos{\phi}\right]^2}{b^2} < 1,
    \end{aligned}
    $$
    where $x$ and $y$ are the coordinates of the sampled points, and finally
3.  approximating the area by multiplying the proportion of points inside the
    overlap with the area of the ellipse.

With this in place, we are now able to compute the areas of all intersections
and their relative complements, $\omega$, up to numerical precision.

### Final optimization

We feed the initial
layout to the optimizer---once again
we employ `nlm()` from **stats** but now also provide the option to
use ellipses rather than circles, allowing the "circles"
to rotate and the relation between the semiaxes to vary, altogether
rendering five parameters to optimize per set and ellipse (or three if we
restrict ourselves to circles). For each iteration of
the optimizer, the areas of all intersections are analyzed and a measure of loss
returned. The loss we use is the same as in
**venneuler** [@Frederickson_2016], namely the residual sums
of squares.

If the fitted diagram is still inexact after the procedure,
we offer a final step in which we
pass the parameters on to a last-ditch optimizer. The weapon of
choice^[We conducted thorough benchmarking, that we opt not to report here, to decide upon an algorithm for this step.] is
*stress* [@wilkinson_exact_2012], which
is also the loss metric we use in our final optimization step and is
used in **venneuler**, as well as
*diagError* [@Micallef_2014a], which is used by **eulerAPE**.

The stress metric is not easily grasped but
can be transformed into a rough analogue of the correlation coefficient
via $r = \sqrt{1-\text{stress}^2}$.

diagError, meanwhile, is given by

$$
{\max_{i = 1, 2, \dots, n}}\left|
\frac{\omega_i}{\sum_{i=1}^n \omega_i} - \frac{A_i}{\sum_{i=1}^n A_i} \right|,
$$

which is the maximum *absolute* difference of the proportion of any
$\omega$ to the respective unique area of the diagram.

## References
