Disk to Plane
Visualization of the Cayley transform S(w) = (w+i)/(iw+1) mapping the unit disk (D²) to the upper half-plane (H). The diagram shows how labeled points and boundary paths map between these domains: the center w₀ = 0 goes to z₀ = i, while the blue and red boundary semicircles through w₃ = −1 and w₁ = 1 unroll onto the negative and positive real axes, with w₄ = i sent to infinity and w₂ = −i to the origin. This conformal transformation is useful in string theory and conformal field theory, where calculations are often simpler in one domain than the other.
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Code
#import "@preview/cetz:0.5.3": canvas, draw
#import draw: arc, circle, content, line, rect
#set page(width: auto, height: auto, margin: 8pt, fill: none)
#set text(size: 12pt)
#let radius = 1
#let axis-length = 2.5
#let arrow-sep = 3.5
#let x-shift = 8.5
#let region-fill = rgb(220, 220, 220).transparentize(60%)
// Special points w_i, z_i and their leader lines
#let point-color = rgb("#777777")
#let arrow-style = (mark: (end: "stealth", fill: black, scale: 0.7), stroke: 0.8pt)
#let direction-marks(paint) = (
end: (25%, 75%).map(pos => (
pos: pos,
symbol: "barbed",
// paths run from -i to i, the marks from w_4 = i down to w_2 = -i
reverse: true,
fill: paint,
scale: 0.7,
shorten-to: none,
)),
)
#canvas({
let axes(x, x-label, y-label, padding) = {
line((x - axis-length, 0), (x + axis-length, 0), ..arrow-style, name: "x-axis")
content("x-axis.end", x-label, anchor: "west", padding: padding)
line((x, -axis-length), (x, axis-length), ..arrow-style, name: "y-axis")
content("y-axis.end", y-label, anchor: "south", padding: padding)
}
axes(0, $x$, $y$, (0, 3pt, 2pt))
// Transformation arrow and label
line((arrow-sep, 0), (arrow-sep + 1.5, 0), ..arrow-style, name: "transform-arrow")
content("transform-arrow.mid", $z &= S(w)\ &= (w + i) / (i w + 1)$, anchor: "south", padding: 4pt)
circle((0, 0), radius: radius, stroke: 0.8pt, fill: region-fill)
content((-0.4, 0.5), $DD^2$)
// Boundary semicircles from w_4 = i down to w_2 = -i through w_3 = -1 (blue) and w_1 = 1 (red)
for (pos, start, paint, thickness) in (
((-radius, 0), 270deg, blue, 1.2pt),
((radius, 0), -90deg, red, 0.8pt),
) {
arc(
pos,
radius: radius,
start: start,
stop: 90deg,
stroke: paint + thickness,
anchor: "arc-center",
mark: direction-marks(paint),
)
}
axes(x-shift, $u$, $v$, 2pt)
// Images of the semicircles: S maps w_3 = -1 to -1 and w_1 = 1 to 1, with w_4 to infinity and w_2 to 0
for (direction, paint) in ((1, red), (-1, blue)) {
line(
(x-shift, 0),
(x-shift + direction * axis-length, 0),
stroke: paint + 1.2pt,
mark: direction-marks(paint),
)
}
// Upper half-plane with gray fill
rect(
(x-shift - axis-length, 0),
(x-shift + axis-length, axis-length),
stroke: none,
fill: region-fill,
name: "plane",
)
content("plane.north-west", $HH$, anchor: "north-west", padding: 4pt)
// Leader lines connect w_0 and z_2 to their offset labels
for (name, pos, offset, label, anchor) in (
("w0", (0, 0), (2, 0.75), $w_0 = 0$, "center"),
("w1", (radius, 0), (0.08, -0.08), $w_1 = 1$, "north-west"),
("w2", (0, -radius), (0.08, -0.08), $w_2 = -i$, "north-west"),
("w3", (-radius, 0), (-0.08, -0.08), $w_3 = -1$, "north-east"),
("w4", (0, radius), (0.2, 0.3), $w_4 = i$, "west"),
("z0", (x-shift, radius), (0.2, 0), $z_0 = i$, "west"),
("z1", (x-shift + radius, 0), (0, -0.4), $z_1 = 1$, "center"),
("z2", (x-shift, 0), (-1, 0.8), $z_2 = 0$, "center"),
("z3", (x-shift - radius, 0), (0, -0.4), $z_3 = -1$, "center"),
) {
circle(pos, fill: point-color, radius: 0.08, stroke: none, name: name)
content((rel: offset, to: pos), label, anchor: anchor, name: name + "-label")
}
content((rel: (0.2, -0.2), to: "y-axis.end"), $z_4 = +i infinity$, anchor: "west")
line("w0-label", "w0", stroke: point-color + 0.5pt)
line("z2-label", "z2", stroke: point-color + 0.5pt)
})