change-of-variables.typ

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

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

#canvas({
  let arrow-style = (mark: (end: "stealth", fill: black, scale: 0.5))
  let plot-size = 6
  let plot-sep = 8

  // Helper to draw coordinate axes
  let draw-axes(origin, label) = {
    // Draw axes
    line(
      (origin.at(0) - plot-size / 2, origin.at(1)),
      (origin.at(0) + plot-size / 2, origin.at(1)),
      ..arrow-style,
      name: label + "-x",
    )
    line(
      (origin.at(0), origin.at(1) - plot-size / 2),
      (origin.at(0), origin.at(1) + plot-size / 2),
      ..arrow-style,
      name: label + "-y",
    )

    // Add axis labels
    content((rel: (0, -0.3), to: label + "-x.end"), eval(lower(label) + "_1", mode: "math"), name: label + "-x-label")
    content((rel: (0.3, 0), to: label + "-y.end"), eval(lower(label) + "_2", mode: "math"), name: label + "-y-label")

    // Add plot label
    content(
      (origin.at(0) - plot-size / 2, origin.at(1) + plot-size / 2),
      text(size: 1.2em)[$#label$],
      anchor: "south-west",
      name: label + "-title",
    )
  }

  // Draw left plot (Z space)
  let z-origin = (0, 0)
  draw-axes(z-origin, "Z")

  // Draw blue square in Z space
  rect(
    (z-origin.at(0), z-origin.at(1)),
    (z-origin.at(0) + 1, z-origin.at(1) + 1),
    fill: blue.transparentize(60%),
    name: "z-square",
  )

  // Draw right plot (X space)
  let x-origin = (plot-sep, 0)
  draw-axes(x-origin, "X")

  // Draw transformed squares in X space
  rect(
    (x-origin.at(0), x-origin.at(1)),
    (x-origin.at(0) + 2, x-origin.at(1) + 2),
    fill: red.transparentize(60%),
    name: "x-square-1",
  )
  rect(
    (x-origin.at(0), x-origin.at(1)),
    (x-origin.at(0) + 2, x-origin.at(1) - 2),
    fill: green.transparentize(60%),
    name: "x-square-2",
  )

  // Draw mapping arrows
  let mid-x = plot-sep / 2
  line(
    (mid-x - 0.3, 2.7),
    (mid-x + 0.3, 2.7),
    ..arrow-style,
    name: "f-arrow",
  )
  content("f-arrow.mid", $f$, name: "f-label", anchor: "south", padding: (bottom: 4pt))
  line(
    (mid-x + 0.3, -2.5),
    (mid-x - 0.3, -2.5),
    ..arrow-style,
    name: "f-inv-arrow",
  )
  content("f-inv-arrow.mid", $f^(-1)$, name: "f-inv-label", anchor: "north", padding: (top: 4pt))

  // Draw dotted transformation arrows with determinant labels
  line(
    "z-square.north-east",
    "x-square-1.north-east",
    ..arrow-style,
    stroke: (dash: "dotted", paint: red),
    name: "det-arrow-1",
  )
  content(
    (rel: (0.2, -0.2), to: "det-arrow-1.mid"),
    [#text(red, $det J_f^(-1) = mat(delim: "|", 2, 0; 0, 2)^(-1) = 1 / 4$)],
    anchor: "north",
    angle: 5deg,
    name: "det-label-1",
  )

  line(
    "z-square.north-east",
    "x-square-2.south-east",
    ..arrow-style,
    stroke: (dash: "dotted", paint: green),
    name: "det-arrow-2",
  )
  content(
    (rel: (0.2, -0.3), to: "det-arrow-2.mid"),
    [#text(green, $det J_f^(-1) = mat(delim: "|", 2, 0; 0, -2)^(-1) = -1 / 4$)],
    anchor: "north",
    angle: -19deg,
    name: "det-label-2",
  )
})