// complexplane.amx — §05 why circles at all.
//
// The closing figure says what the previous three implied: a term of the
// series is a *rotating vector*, and its imaginary part is the sine. So the
// unit circle gets the two projections that make that visible — a drop to the
// real axis (cos θ) and a run to the imaginary axis (sin θ) — plus a live
// readout that names the angle the vector has swept.
//
// The rotation runs counter-clockwise, the same sense as §02 and §03: the
// figure's claim is that this is the *same* pen, and a clockwise turn here
// would contradict the sine the reader just watched it write.

import "../../demos/lib/theme.amx"
import "fourier.amx" as geo

config { resolution: (1280, 720) }

# ComplexPlane
config { colorscheme: "ink", dynamic_layout: true, duration: 11.5 }

let plane_x = 400
let plane_y = geo.cy
// px per unit. The NumberPlane below is sized from this, so the circle of
// radius `unit` really is the unit circle of the grid behind it — sizing the
// two independently is how a "unit circle" ends up drawn at 1.23.
let unit = 160

cp_title: Text, text: "Why circles? Complex exponentials", font_size: 38,
  font_weight: "bold", color: text.primary, anchor: scene.top, offset: (0, 52)

grid: NumberPlane, x_domain: (-1.6, 1.6), y_domain: (-1.6, 1.6),
  size: (3.2 * unit, 3.2 * unit), at: (plane_x, plane_y),
  x_range: (-1, 1, 0.5), y_range: (-1, 1, 0.5),
  stroke: surface.secondary, stroke_width: 1.0

unit_circle: Ellipse, size: (2 * unit, 2 * unit), stroke: accent.info,
  fill_opacity: 0, stroke_width: 2, at: (plane_x, plane_y)

vector: Line, from: (plane_x, plane_y), to: (plane_x + unit, plane_y),
  stroke: accent.primary, stroke_width: 3

// The two projections: this is the sentence "e^(iθ) = cos θ + i sin θ" drawn
// as geometry rather than written down.
drop_v: Line, from: (plane_x + unit, plane_y), to: (plane_x + unit, plane_y),
  stroke: accent.secondary, stroke_width: 1.25, opacity: 0.42
drop_h: Line, from: (plane_x + unit, plane_y), to: (plane_x + unit, plane_y),
  stroke: accent.secondary, stroke_width: 1.25, opacity: 0.42

vec_tip: Ellipse, size: (15, 15), color: accent.primary, at: (plane_x + unit, plane_y)

cp_re: Text, text: "Re", font_size: 19, color: text.muted,
  at: (plane_x + 272, plane_y + 26)
cp_im: Text, text: "Im", font_size: 19, color: text.muted,
  at: (plane_x + 26, plane_y - 250)
cp_read: Text, text: "theta = 0.00 rad", font_size: 21, color: accent.warning,
  at: (plane_x, plane_y + 252)

cp_eq: Text, text: "e^(i theta) = cos(theta) + i sin(theta)", font_size: 27,
  color: accent.primary, anchor: scene.center, offset: (330, -110)
cp_note: Text, text: "one term of the series is one vector, turning at k times the base rate",
  font_size: 20, color: text.secondary, anchor: scene.center, offset: (330, -44),
  text_max_width: 380, text_align: "center"
cp_note2: Text, text: "its radius is a(k), and its height is a(k) sin(k theta)",
  font_size: 20, color: text.secondary, anchor: scene.center, offset: (330, 22),
  text_max_width: 380, text_align: "center"

endnote: Text, text: "sum enough circles and even a square wave is just rotation",
  font_size: 20, color: text.muted, anchor: scene.center, offset: (330, 130),
  text_max_width: 380, text_align: "center"

always {
  let u = clamp((t - 1.2) / 9.0, 0.0, 1.0)
  let th = geo.span * u
  let tx = plane_x + unit * cos(th)
  let ty = plane_y - unit * sin(th)
  vector.to = (tx, ty)
  vec_tip.at = (tx, ty)
  drop_v.from = (tx, ty)
  drop_v.to = (tx, plane_y)
  drop_h.from = (tx, ty)
  drop_h.to = (plane_x, ty)
  cp_read.text = format("theta = {:.2} rad", th)
}

#0.15s
fade-in cp_title [550ms, ease: expo-out]
fade-in grid [600ms, ease: expo-out]
fade-in unit_circle [600ms, ease: expo-out]

#1s
fade-in vector, vec_tip [400ms, ease: expo-out]

#1.6s
stagger [110ms] {
  fade-in drop_v [400ms, ease: expo-out]
  fade-in drop_h [400ms, ease: expo-out]
  fade-in cp_re [400ms, ease: expo-out]
  fade-in cp_im [400ms, ease: expo-out]
  fade-in cp_read [400ms, ease: expo-out]
}

#3.4s
fade-in cp_eq [600ms, ease: expo-out]

#4.6s
fade-in cp_note [500ms, ease: expo-out]

#5.8s
fade-in cp_note2 [500ms, ease: expo-out]

#8.4s
fade-in endnote [600ms, ease: expo-out]