Ellipses

Setup

Packages

library(ggdiagram)
library(ggplot2)
library(dplyr)
library(ggtext)
library(ggarrow)

Base Plot

To avoid repetitive code, we set defaults and make a base plot:

my_font <- "Roboto Condensed"
my_font_size <- 20
my_point_size <- 2


# my_colors <- viridis::viridis(2, begin = .25, end = .5)
my_colors <- c("#3B528B", "#21908C")

theme_set(
  theme_minimal(
    base_size = my_font_size,
    base_family = my_font) +
    theme(axis.title.y = element_text(angle = 0, vjust = 0.5)))

bp <- ggdiagram(
  font_family = my_font,
  font_size = my_font_size,
  point_size = my_point_size,
  linewidth = .5,
  theme_function = theme_minimal,
  axis.title.x =  element_text(face = "italic"),
  axis.title.y = element_text(
    face = "italic",
    angle = 0,
    hjust = .5,
    vjust = .5)) +
  scale_x_continuous(labels = signs_centered,
                     limits = c(-4, 4)) +
  scale_y_continuous(labels = signs::signs,
                     limits = c(-4, 4))

A common way to specify an ellipse is with a center point and the two distances from the center point c to the horizontal and vertical edges, a and b, respectively.

$$\left(\frac{x-c_x}{a}\right)^2+\left(\frac{y-c_y}{b}\right)^2=1$$

a <- 4
b <- 3
c1 <- ob_point(0, 0)
e1 <- ob_ellipse(c1, a = a, b = b)

Code

bp +
  e1 +
  ob_segment(c1,
          ob_point(c(a, 0), c(0, b)),
          color = my_colors,
          label = ob_label(
            label = paste0(c("*a* = ", "*b* = "),
                           c(a, b)),
            angle = 0)) +
  c1

Figure 1: An ellipse can be specified with a center, and semi-major radii.

Foci

A circle has one focus, the center. If a ≠ b, an ellipse has two foci.

bp +
  e1 +
  ob_label("*F*~1~", e1@focus_1, 
           plot_point = TRUE, 
           vjust = 1.2) +
  ob_label("*F*~2~", e1@focus_2, 
           plot_point = TRUE, 
           vjust = 1.2)

Figure 2: An ellipse has two foci

For any point P on the ellipse, the sum of PF₁ and PF₂ is 2a if a > b and 2b if b > a.

deg <- degree(61.5)

bp +
  e1 +
  ob_label("*F*~1~", e1@focus_1, plot_point = TRUE, vjust = 1.2) +
  ob_label("*F*~2~", e1@focus_2, plot_point = TRUE, vjust = 1.2) +
  {p <- e1@point_at(deg)} +
  p@label("*P*", polar_just = ob_polar(deg, 1.5)) +
  ob_segment(e1@focus_1,
          p,
          label = paste0("*PF*~1~ = ",
                         distance(e1@focus_1, p) |>
                           round())) +
  ob_segment(p,
          e1@focus_2,
          label = paste0("*PF*~2~ = ",
                         distance(e1@focus_2, p) |>
                           round())) +
  ob_label("*PF*~1~ + *PF*~2~ = 2*a* = 8",
        center = ob_point(0,4),
        size = 20)

Figure 3: The sum of distances from the foci is constant.

Rotated Ellipses

As seen in fig-ellipserotated, rotate an ellipse by setting the angle property with a degree, radian, or turn object or a numeric value (degrees).

bp +
  ob_ellipse(a  = 2, b = 1, angle = 45)

Figure 4: An ellipse rotated 45 degrees

Setting the angle of an ellipse will rotate it about its center. If you want to rotate it around a different point, use the rotate function, setting the origin parameter to any point you wish to rotate the ellipse around.

bp +
  {p1 = ob_point(2,0)} +
  {e2 <- ob_ellipse(a  = 2, b = 1)} +
  rotate(e2, theta = degree(45), origin = p1, color = "red") 

Figure 5: An ellipse rotated 45 degrees about point (2, 0)

Point on the ellipse at a specific angle

The @point_at property of an ob_ellipse object is a function that can find a point at a specific angle.

e1@point_at(degree(60))
#> 
#> ── <ob_point>
#> # A tibble: 1 × 2
#>       x     y
#>   <dbl> <dbl>
#> 1  1.59  2.75

Code

deg <- degree(60)
bp +
  e1 +
  {p45 <- e1@point_at(deg)} +
  p45@label(polar_just = ob_polar(deg, 1.5)) +
  ob_segment(e1@center, p45) +
  ob_arc(
    center = e1@center,
    radius = 1,
    start = degree(0),
    end = deg,
    label = deg
  )

Figure 6: Point on ellipse that is 45° from the x-axis.

Point on the ellipse using definitional parameter t

The angle expected by the @point_at function is a true angle. However, the parametric equation for ellipses has a parameter t that looks like an angle, but actually has no direct geometric interpretation:

$$\begin{aligned} t&=[0,2\pi)\\ (x,y) &= (a\cos(t),b\sin(t)) \end{aligned}$$

Code

theta <- degree(seq(0, 350, 30))

bp +
  {c1 <- ob_circle(radius = 3.6, color = "gray30")} +
  {e1 <- ob_ellipse(a = 2.8, b = 1)} +
  {p1 <- c1@point_at(theta)} +
  ob_label(theta, p1, polar_just = ob_polar(theta, r = 1.5)) +
  ob_segment(ob_point(), p1, linewidth = .2) +
  {p2 <- e1@point_at(theta, definitional = T, color = "dodgerblue")} +
  ob_segment(ob_point(), p2) + 
  ob_label(theta@degree, p2, polar_just = ob_polar(theta, r = 1.5)) +
  theme_void() + 
  scale_x_continuous(expand = expansion(.12))
#> Scale for x is already present.
#> Adding another scale for x, which will replace the existing scale.

Figure 7: The ellipse’s definitional parameter t does not always line up with angles on a circle

If the definitional point at t is desired:

ob_ellipse(a = 2)@point_at(degree(60), definitional = TRUE)
#> 
#> ── <ob_point>
#> # A tibble: 1 × 2
#>       x     y
#>   <dbl> <dbl>
#> 1     1  1.73

Tangent lines

Like the @point_at property, the @tangent property is a function that will find the tangent line at a specified angle or point.

bp + 
  {e1 <- ob_ellipse(a = 3, b = 2)} + 
  e1@point_at(60, color = "firebrick4") + 
  e1@tangent_at(60, color = "firebrick4")

Figure 8: Tangent lines on an ellipse

The @tangent function can also take a point instead of an angle.

bp + 
  e1 + 
  {p1 <- e1@point_at(60)} +
  e1@tangent_at(p1) 

Figure 9: Tangent line at a specific point on an ellipse

If the point is not on the ellipse, the tangent will be at the point’s projection onto the ellipse:

bp + 
  e1 + 
  {p1 <- ob_point(3, 2, color = "firebrick4")} +
  e1@tangent_at(p1) + 
  projection(p1, e1)

Figure 10: Tangent line at a specific point projected onto an ellipse

Superellipses

The standard formula for an ellipse can be altered such that the squared entities can be raised to any positive number.

$$\left(\frac{x-c_x}{a}\right)^{m_1}+\left(\frac{y-c_y}{b}\right)^{m_2}=1$$

m₂ is set equal to m₁ unless otherwise specified.

If m₁ is 4, and a and b are equal, we can make a squircle, which is a square-ish circle.

bp +
  ob_ellipse(a = 3, b = 3, m1 = 4)

Figure 11: A squircle

If we increase m₁ to a high value like 10, we can create a rectangle with pleasingly rounded corners.

bp +
  ob_ellipse(
    a = 3,
    b = 3,
    m1 = 10,
    color = NA,
    fill = "dodgerblue",
    label = ob_label(
      label = "My<br>Variable",
      fill = NA,
      color = "white",
      size = 70
    )
  )

Figure 12: A superellipse can look like a rectangle with rounded corners.

Connection Paths Among Ellipses

bp + 
  {e1 <- ob_ellipse(ob_point(-2,0), a = 2)} +
  {e2 <- ob_ellipse(ob_point(3,2), b = 2)} +
  connect(e1, e2, resect = 2)

Figure 13: Connections between ellipses

Placing Ellipses

The place function will set an object at a position and distance from another object. Here we set an ellipse to the the right of e1 (i.e., “east” or 0 degrees) with a separation of 2.

bp +
  {e1 <- ob_ellipse(center = ob_point(-2, 0), a = 2)} +
  place(ob_ellipse(b = 2),
        from = e1,
        where = "right",
        sep = 2)

Figure 14: Place an ellipse 2 units to the right of another ellipse.

The sep parameter in the place function is not necessarily the shortest distance between ellipses. Instead, it is the distance between the ellipses on the segment connecting the center points.

deg <- degree(30)

bp + 
  {e1 <- ob_ellipse(
    center = ob_point(-2,-1, color =  "dodgerblue4"), 
    a = 2, 
    b = 1.5)} +
  {e2 <- place(
    ob_ellipse(
      center = ob_point(color = "orchid4"),
      b = 2),
    from = e1, 
    where = deg, 
    sep = 2)} + 
  connect(
    e1,
    e2,
    arrow_head = ggarrow::arrow_head_minimal(),
    linetype = "dashed",
    label = ob_label(2, vjust = 0)
  ) +
  ob_arc(e1@center, end = deg, label = deg) + 
  ob_segment(e1@center, 
             e1@point_at(deg)) + 
  ob_segment(e2@center, 
             e2@point_at(deg + degree(180))) + 
  ob_label("*e*~1~", e1@center) + 
  ob_label("*e*~2~", e2@center)

Figure 15: The separation distance between ellipses is along the path that connects their centers.

You can place many ellipses at once. In Figure 16, 12 ellipses are placed around the central ellipse. Connection paths are then drawn to each ellipse.

# Number of ellipses
k <- 12

# Colors
e_fills <- hsv(
  h = seq(0, 1 - 1 / k, length.out = k), 
  s = .4, 
  v = .6)

bp + 
  {e_0 <- ob_ellipse(
    m1 = 6,
    label = ob_label(
      "*e*~0~",
      size = 40,
      color = "white",
      fill = "gray20"
    ),
    color = NA,
    fill = "gray20"
  )} + 
  {e_x <- place(
    x = ob_ellipse(
      a = .4,
      b = .4,
      m1 = 6,
      label = ob_label(
        paste0("*e*~", seq(k), "~"),
        color = "white",
        fill = e_fills
      ),
      color = NA,
      fill = e_fills
    ),
    from = e_0,
    where = degree(seq(0, 360 - 360 / k, 360 / k)),
    sep = 2
  )} +
  connect(e_0, e_x, resect = 2, color = e_fills) + 
  theme_void()

Figure 16: Many ellipses can be placed at once.

Lines can be placed in relation to ellipses:

bp + 
  {e1 <- ob_ellipse(m1 = 4)} +
  {l1 <- place(
    x = ob_line(),
    from = e1,
    where = {deg1 <- degree(45)},
    sep = {d <- 3}
  )} + 
  connect(
    e1,
    l1,
    label = paste0("Distance = ", d),
    arrow_fins = arrowhead(),
    length_fins = 8,
    length_head = 8,
    resect = 1
  ) + 
  ob_label(
    label = equation(l1),
    center = ob_polar(theta = deg1, 
                      r = e1@point_at(deg1)@r + d),
    angle = l1@angle,
    vjust = 0
  )

Figure 17: A line placed 3 units and 45 degrees from a squircle.