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In an "equation // marchante" watch (Breguet, Blancpain), a kidney-profiled cam turns once // per year and a spring-loaded follower reads its edge, converting the // cam's UNIFORM rotation into the NON-uniform, sign-changing swing between // mean solar time and apparent solar time (-14m15s mid-February to // +16m23s early November, crossing zero four times a year). // // The cam profile itself is never generated live. MONTH_EOT below holds 12 // real, almanac-sourced equation-of-time values (minutes) at 1-of-month // intervals; wrapping December back into January makes a 13th, closing // point. A periodic Catmull-Rom spline through those 12 anchors (each // evaluated with its wrapped neighbours) is resampled at 5 points per // segment — 60 points total, EOT_LUT — once, at module load. That table IS // the cam: index i is the follower's radial reading at the angle the disc // would be at i/60 of a full turn. Per rAF frame the only work done is // picking two adjacent LUT entries by the current phase and lerping // between them — no trig, no live curve evaluation. // // Real annual period: 1 year. Rendered period: 11s, documented here as a // ~2.87-million-times compression (365.25 days / 11s) purely for card // legibility — not a simulated calendar, just the same closed curve run // fast. The follower's on-screen offset is the LUT value scaled by the // LARGER of the curve's two real extremes (|+16.23| minutes), so the // positive swing reaches the full +-22% of the divider's half-length while // the (smaller-magnitude) negative swing falls proportionally short of // it — the asymmetry is inherited from the real numbers, never forced. // Because the underlying anchors are unevenly spaced in "when it crosses // zero", the resulting sweep is not a sine: the drop from most-negative to // zero is visibly quicker than the climb from zero to most-positive, and // the four zero-crossing brighten-pulses per 11s cycle land at uneven // intervals — the detail that reads as "cam", not "wave". // // Colour: the rule line sits at a mid-strength mix of --foreground (35%) // so it's visible without being a full-strength rule (a --border stroke, // at light theme's ~1.1:1 contrast, would make the follower's motion // unreadable — the whole point of this component). Each zero-crossing // briefly lifts that same --foreground opacity by 8% for 400ms via a // restarted CSS animation — never --ns-accent, which is reserved for // interaction chrome and never appears here since this is a passive // divider with no interaction. An optional low-opacity ghost of the cam's // own kidney outline (rendered from the identical LUT, as a closed SVG // path) fades in on hover/focus purely as a decorative aid — position: // absolute, so it can never affect layout. // --------------------------------------------------------------------------- export interface EquationKidneyCamProps { className?: string; } // -- Real, almanac-sourced equation-of-time values (minutes), 1st-of-month, // Jan through Dec. Standard published approximation; the true extremes // (-14m15s mid-Feb, +16m23s early Nov) fall between these monthly anchors // and are recovered by the spline, not hardcoded. -------------------------- const MONTH_EOT = [-3.4, -13.6, -12.6, -4.1, 2.9, 2.4, -3.6, -6.3, -0.1, 10.1, 16.4, 11.2]; const ANCHOR_COUNT = MONTH_EOT.length; const SAMPLES_PER_SEGMENT = 5; const LUT_SIZE = ANCHOR_COUNT * SAMPLES_PER_SEGMENT; // 60 function catmullRom(p0: number, p1: number, p2: number, p3: number, t: number): number { const t2 = t * t; const t3 = t2 * t; return 0.5 * (2 * p1 + (-p0 + p2) * t + (2 * p0 - 5 * p1 + 4 * p2 - p3) * t2 + (-p0 + 3 * p1 - 3 * p2 + p3) * t3); } function buildLut(): number[] { const lut: number[] = []; for (let seg = 0; seg < ANCHOR_COUNT; seg++) { const p0 = MONTH_EOT[(seg - 1 + ANCHOR_COUNT) % ANCHOR_COUNT]; const p1 = MONTH_EOT[seg]; const p2 = MONTH_EOT[(seg + 1) % ANCHOR_COUNT]; const p3 = MONTH_EOT[(seg + 2) % ANCHOR_COUNT]; for (let s = 0; s < SAMPLES_PER_SEGMENT; s++) { const t = s / SAMPLES_PER_SEGMENT; lut.push(catmullRom(p0, p1, p2, p3, t)); } } return lut; } const EOT_LUT = buildLut(); const LUT_MAX_ABS = EOT_LUT.reduce((m, v) => Math.max(m, Math.abs(v)), 0); let MIN_INDEX = 0; let minVal = Infinity; for (let i = 0; i < EOT_LUT.length; i++) { const v = EOT_LUT[i]; if (v < minVal) { minVal = v; MIN_INDEX = i; } } // -- ghost cam outline, built once from the same LUT: a closed polar path // (angle = i/60 of a turn, radius = base + LUT value * scale). Decorative // only; never rendered visibly at rest, only revealed faintly on hover. -- const GHOST_VB = 64; const GHOST_CX = GHOST_VB / 2; const GHOST_CY = GHOST_VB / 2; const GHOST_BASE_R = 14; const GHOST_R_SCALE = 8 / LUT_MAX_ABS; const GHOST_PATH = (() => { let d = ""; for (let i = 0; i < LUT_SIZE; i++) { const angle = (i / LUT_SIZE) * Math.PI * 2 - Math.PI / 2; const r = GHOST_BASE_R + EOT_LUT[i] * GHOST_R_SCALE; const x = GHOST_CX + Math.cos(angle) * r; const y = GHOST_CY + Math.sin(angle) * r; d += i === 0 ? `M ${x.toFixed(2)} ${y.toFixed(2)}` : ` L ${x.toFixed(2)} ${y.toFixed(2)}`; } return `${d} Z`; })(); const CAM_PERIOD_MS = 11000; // real period: 1 year — an ~2.87M:1 compression const MAX_OFFSET_PCT = 22; // +-22% of the divider's half-length const PULSE_MS = 400; const HEIGHT_PX = 28; // fixed card-scale height; every other dimension below derives from it const TICK_HEIGHT_PX = Math.round(HEIGHT_PX * 0.5); export function EquationKidneyCam({ className = "" }: EquationKidneyCamProps) { const rootRef = useRef(null); const lineRef = useRef(null); const tickRef = useRef(null); useEffect(() => { const root = rootRef.current; const line = lineRef.current; const tick = tickRef.current; if (!root || !line || !tick) return; const reduced = window.matchMedia("(prefers-reduced-motion: reduce)").matches; const sampleAt = (idxFloat: number): number => { const i0 = Math.floor(idxFloat) % LUT_SIZE; const i1 = (i0 + 1) % LUT_SIZE; const frac = idxFloat - Math.floor(idxFloat); return EOT_LUT[i0] + (EOT_LUT[i1] - EOT_LUT[i0]) * frac; }; const applyOffset = (value: number) => { const pct = (value / LUT_MAX_ABS) * MAX_OFFSET_PCT; tick.style.left = `${50 + pct}%`; }; const firePulse = () => { line.style.animation = "none"; // force reflow so re-setting the animation below restarts it void line.offsetWidth; line.style.animation = `ns-eqcam-pulse ${PULSE_MS}ms ease-out`; }; if (reduced) { // freeze on the real-world most-negative point (mid-February // analogue, -14m15s) — the cam's tightest inward point and the most // visually distinct resting position, per the reduced-motion rule. applyOffset(EOT_LUT[MIN_INDEX]); return; } let raf = 0; let visible = true; let startTime = 0; let prevSign = Math.sign(EOT_LUT[0]) || 1; const loop = (now: number) => { raf = 0; if (!visible) return; if (startTime === 0) startTime = now; const elapsed = now - startTime; const phase = (elapsed % CAM_PERIOD_MS) / CAM_PERIOD_MS; const idxFloat = phase * LUT_SIZE; const value = sampleAt(idxFloat); applyOffset(value); const sign = value === 0 ? prevSign : Math.sign(value); if (sign !== prevSign) firePulse(); prevSign = sign; raf = requestAnimationFrame(loop); }; // -- no paint before the first token/geometry read: only start once // the divider is actually on screen, and pause the rAF loop (not the // logical clock — startTime is wall-clock, so re-entering resumes the // correct phase, not a reset) while it scrolls off. ------------------- const io = new IntersectionObserver((entries) => { visible = entries[0]?.isIntersecting ?? true; if (visible && !raf) raf = requestAnimationFrame(loop); }); io.observe(root); return () => { cancelAnimationFrame(raf); io.disconnect(); }; }, []); return (

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