disk-to-plane.typ

#import "@preview/cetz:0.3.2": canvas, draw
#import draw: circle, line, content, arc, rect

#set page(width: auto, height: auto, margin: 8pt)

#let radius = 1
#let axis-length = 2.5
#let arrow-sep = 3.5
#let arrow-width = 1.5
#let x-shift = 8.5

#let dot-style = (fill: gray, radius: 0.08, stroke: none)
#let arrow-style = (
  mark: (end: "stealth", fill: black, scale: 0.7),
  stroke: 0.8pt,
)
#let arc-arrow = (
  mark: (
    end: (
      (pos: 25%, symbol: "barbed", fill: none, scale: 0.7),
      (pos: 75%, symbol: "barbed", fill: none, scale: 0.7),
    ),
  ),
  stroke: 0.8pt,
)

#canvas({
  // Left coordinate system (disk)
  line((-axis-length, 0), (axis-length, 0), ..arrow-style, name: "x-axis")
  content("x-axis.end", $x$, anchor: "west", padding: (0, 3pt, 2pt))
  line((0, -axis-length), (0, axis-length), ..arrow-style, name: "y-axis")
  content("y-axis.end", $y$, anchor: "south", padding: (0, 3pt, 2pt))

  // Transformation arrow and label
  line((arrow-sep, 0), (arrow-sep + arrow-width, 0), ..arrow-style, name: "transform-arrow")
  content(
    (rel: (0, 0), to: "transform-arrow.mid"),
    $z &= S(w)\ &= (w + i) / (i w + 1)$,
    anchor: "south",
    padding: 4pt,
  )

  // Unit disk with gray fill
  circle((0, 0), radius: radius, stroke: 0.8pt, fill: rgb(220, 220, 220).transparentize(60%), name: "disk")
  content((rel: (-0.4, 0.5), to: "disk.center"), $DD^2$)

  // Dots at special points
  circle((0, 0), ..dot-style, name: "w0")
  circle((radius, 0), ..dot-style, name: "w1")
  circle((0, -radius), ..dot-style, name: "w2")
  circle((-radius, 0), ..dot-style, name: "w3")
  circle((0, radius), ..dot-style, name: "w4")

  // Labels for points
  content((rel: (2, 1), to: "w0"), $w_0 = 0$, name: "w0-label")
  line("w0-label", "w0", stroke: gray + 0.5pt)
  content((rel: (0.3, -0.3), to: "w1"), $w_1 = 1$)
  content((rel: (.8, -0.2), to: "w2"), $w_2 = -i$)
  content((rel: (-0.2, -0.3), to: "w3"), $w_3 = -1$)
  content((rel: (0.2, 0.3), to: "w4"), $w_4 = i$, anchor: "west")

  // Blue and red semicircles with arrows at 0.25 and 0.75
  arc(
    (-radius, 0),
    radius: radius,
    start: 270deg,
    stop: 90deg,
    stroke: blue + 1.2pt,
    anchor: "arc-center",
    mark: (
      end: (
        (pos: 25%, symbol: "barbed", fill: blue, scale: 0.7, shorten-to: none),
        (pos: 75%, symbol: "barbed", fill: blue, scale: 0.7, shorten-to: none),
      ),
    ),
    name: "blue-arc",
  )
  arc(
    (radius, 0),
    radius: radius,
    start: -90deg,
    stop: 90deg,
    stroke: red + 0.8pt,
    anchor: "arc-center",
    mark: (
      end: (
        (pos: 25%, symbol: "barbed", fill: red, scale: 0.7, shorten-to: none),
        (pos: 75%, symbol: "barbed", fill: red, scale: 0.7, shorten-to: none),
      ),
    ),
    name: "red-arc",
  )

  // Right coordinate system (upper half-plane)
  line((x-shift - axis-length, 0), (x-shift + axis-length, 0), ..arrow-style, name: "u-axis")
  content("u-axis.end", $u$, anchor: "west", padding: 2pt)
  line((x-shift, -axis-length), (x-shift, axis-length), ..arrow-style, name: "v-axis")
  content("v-axis.end", $v$, anchor: "south", padding: 2pt)

  // Extend blue and red lines to full width
  line(
    (x-shift, 0),
    (x-shift + axis-length, 0),
    stroke: blue + 1.2pt,
    mark: (
      end: (
        (pos: 25%, symbol: "barbed", fill: blue, scale: 0.7, shorten-to: none),
        (pos: 75%, symbol: "barbed", fill: blue, scale: 0.7, shorten-to: none),
      ),
    ),
    name: "pos-real-line",
  )
  line(
    (x-shift, 0),
    (x-shift - axis-length, 0),
    stroke: red + 1.2pt,
    mark: (
      end: (
        (pos: 25%, symbol: "barbed", fill: red, scale: 0.7, shorten-to: none),
        (pos: 75%, symbol: "barbed", fill: red, scale: 0.7, shorten-to: none),
      ),
    ),
    name: "neg-real-line",
  )

  // Upper half-plane with gray fill
  rect(
    (x-shift - axis-length, 0),
    (x-shift + axis-length, axis-length),
    stroke: none,
    fill: rgb(220, 220, 220).transparentize(60%),
    name: "plane",
  )
  content("plane.north-west", $HH$, anchor: "north-west", padding: 4pt)

  // Dots at special points
  circle((x-shift, radius), ..dot-style, name: "z0")
  circle((x-shift + radius, 0), ..dot-style, name: "z1")
  circle((x-shift, 0), ..dot-style, name: "z2")
  circle((x-shift - radius, 0), ..dot-style, name: "z3")

  // Labels for points
  content((rel: (0.2, 0), to: "z0"), $z_0 = i$, anchor: "west")
  content((rel: (0, -0.4), to: "z1"), $z_1 = 1$)
  content((rel: (-1, 0.8), to: "z2"), $z_2 = 0$, name: "z2-label")
  content((rel: (0, -0.4), to: "z3"), $z_3 = -1$)
  content((rel: (0.2, -0.2), to: "v-axis.end"), $z_4 = +i infinity$, anchor: "west")
  line("z2-label", "z2", stroke: gray + 0.5pt)
})