---
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 seq_along(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 seq_len(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 seq_len(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
