feat(mq_editor): switch curve interpolation to Lagrange through all control points

Replace cubic Bezier with Lagrange interpolation so P1/P2 are actual
points on the curve. Eval uses stored t1/t2 as arbitrary knot positions.
fitBezier keeps least-squares but with Lagrange basis at u=1/3,2/3.
Remove endpoint companion drag (no longer tangent handles).
TODO: support arbitrary number of inner control points.

handoff(Claude): Lagrange interpolation replaces Bezier in mq_editor curve eval/fit.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

4 files changed, 54 insertions, 62 deletions

diff --git a/tools/mq_editor/editor.js b/tools/mq_editor/editor.js
index 0854ec0..a7d0879 100644
--- a/tools/mq_editor/editor.js
+++ b/tools/mq_editor/editor.js
@@ -25,8 +25,7 @@ class PartialEditor {
 
     // Private state
     this._selectedIndex  = -1;
-    this._dragPointIndex    = -1;
-    this._dragCompanionOff  = null; // {dt, dv} offset of companion handle relative to anchor
+    this._dragPointIndex = -1;
     this._amp = { tMin: 0, tMax: 1, ampTop: 1 };
 
     this._setupButtons();
@@ -496,12 +495,7 @@ class PartialEditor {
       for (let i = 0; i < 4; ++i) {
         if (Math.hypot(this._tToX(curve['t' + i]) - x, this._ampToY(curve['a' + i]) - y) <= 8) {
           if (this.onBeforeChange) this.onBeforeChange();
-          this._dragPointIndex   = i;
-          this._dragCompanionOff = null;
-          if (i === 0)
-            this._dragCompanionOff = { da: curve.a1 - curve.a0 };
-          else if (i === 3)
-            this._dragCompanionOff = { da: curve.a2 - curve.a3 };
+          this._dragPointIndex = i;
           canvas.style.cursor  = 'grabbing';
           e.preventDefault();
           return;
@@ -516,11 +510,6 @@ class PartialEditor {
         const curve = this.partials[this._selectedIndex].freqCurve;
         const i     = this._dragPointIndex;
         curve['a' + i] = Math.max(0, this._yToAmp(y));
-        if (this._dragCompanionOff) {
-          const off = this._dragCompanionOff;
-          if (i === 0) { curve.a1 = curve.a0 + off.da; }
-          else          { curve.a2 = curve.a3 + off.da; }
-        }
         this._renderAmpEditor();
         if (this.viewer) this.viewer.render();
         e.preventDefault();
diff --git a/tools/mq_editor/mq_extract.js b/tools/mq_editor/mq_extract.js
index ff38d63..18897fb 100644
--- a/tools/mq_editor/mq_extract.js
+++ b/tools/mq_editor/mq_extract.js
@@ -526,7 +526,10 @@ function trackIsoContour(stftCache, seedTime, seedFreq, params) {
   };
 }
 
-// Fit cubic bezier to trajectory using least-squares for inner control points
+// Fit interpolating curve to trajectory via least-squares for inner control point values.
+// Inner knots fixed at u=1/3 and u=2/3 (t = t0+dt/3, t0+2*dt/3).
+// The curve passes through all 4 control points (Lagrange interpolation).
+// TODO: support arbitrary number of inner control points
 function fitBezier(times, values) {
   const n = times.length - 1;
   const t0 = times[0], v0 = values[0];
@@ -534,43 +537,35 @@ function fitBezier(times, values) {
   const dt = t3 - t0;
 
   if (dt <= 1e-9 || n < 2) {
-    // Linear fallback for too few points or zero duration
     return {t0, v0, t1: t0 + dt / 3, v1: v0 + (v3 - v0) / 3, t2: t0 + 2 * dt / 3, v2: v0 + 2 * (v3 - v0) / 3, t3, v3};
   }
 
-  // Least squares solve for v1, v2
-  // Bezier: B(u) = (1-u)^3*v0 + 3(1-u)^2*u*v1 + 3(1-u)*u^2*v2 + u^3*v3
-  // Target_i = val_i - (1-u)^3*v0 - u^3*v3
-  // Model_i = A_i*v1 + B_i*v2
-  // A_i = 3(1-u)^2*u
-  // B_i = 3(1-u)*u^2
+  // Lagrange basis with inner knots at u1=1/3, u2=2/3
+  // l1(u) = u*(u-2/3)*(u-1) / ((1/3)*(1/3-2/3)*(1/3-1))  = 13.5*u*(u-2/3)*(u-1)
+  // l2(u) = u*(u-1/3)*(u-1) / ((2/3)*(2/3-1/3)*(2/3-1))  = -13.5*u*(u-1/3)*(u-1)
+  // l0(u) = (u-1/3)*(u-2/3)*(u-1) / ((-1/3)*(-2/3)*(-1)) = -4.5*(u-1/3)*(u-2/3)*(u-1)
+  // l3(u) = u*(u-1/3)*(u-2/3) / ((2/3)*(1/3))             = 4.5*u*(u-1/3)*(u-2/3)
+  // Least-squares: minimize Σ(l1*v1 + l2*v2 - target_i)^2
+  // target_i = values[i] - l0*v0 - l3*v3
 
   let sA2 = 0, sB2 = 0, sAB = 0, sAT = 0, sBT = 0;
 
   for (let i = 0; i <= n; ++i) {
-    const u = (times[i] - t0) / dt;
-    const u2 = u * u;
-    const u3 = u2 * u;
-    const invU = 1.0 - u;
-    const invU2 = invU * invU;
-    const invU3 = invU2 * invU;
-
-    const A = 3 * invU2 * u;
-    const B = 3 * invU * u2;
-    const target = values[i] - (invU3 * v0 + u3 * v3);
-
-    sA2 += A * A;
-    sB2 += B * B;
-    sAB += A * B;
-    sAT += A * target;
-    sBT += B * target;
+    const u  = (times[i] - t0) / dt;
+    const l0 = -4.5 * (u - 1/3) * (u - 2/3) * (u - 1);
+    const l1 =  13.5 * u * (u - 2/3) * (u - 1);
+    const l2 = -13.5 * u * (u - 1/3) * (u - 1);
+    const l3 =   4.5 * u * (u - 1/3) * (u - 2/3);
+    const A = l1, B = l2;
+    const target = values[i] - l0 * v0 - l3 * v3;
+    sA2 += A * A; sB2 += B * B; sAB += A * B;
+    sAT += A * target; sBT += B * target;
   }
 
   const det = sA2 * sB2 - sAB * sAB;
   let v1, v2;
 
   if (Math.abs(det) < 1e-9) {
-    // Fallback to simple 1/3, 2/3 heuristic if matrix is singular
     const idx1 = Math.round(n / 3);
     const idx2 = Math.round(2 * n / 3);
     v1 = values[idx1];
diff --git a/tools/mq_editor/utils.js b/tools/mq_editor/utils.js
index 96d807c..2c6b2f5 100644
--- a/tools/mq_editor/utils.js
+++ b/tools/mq_editor/utils.js
@@ -1,29 +1,48 @@
 // Shared utilities for mq_editor
 
-// Evaluate cubic bezier curve at time t (robust: handles dt<=0)
+// Evaluate interpolating curve at time t via Lagrange interpolation.
+// The curve passes through all 4 control points at their stored time positions.
+// Knot positions: u_k = (t_k - t0) / (t3 - t0), k=0..3.
+// TODO: support arbitrary number of inner control points
 function evalBezier(curve, t) {
   const dt = curve.t3 - curve.t0;
   if (dt <= 0) return curve.v0;
   let u = (t - curve.t0) / dt;
   u = Math.max(0, Math.min(1, u));
-  const u1 = 1.0 - u;
-  return u1*u1*u1 * curve.v0 +
-         3*u1*u1*u * curve.v1 +
-         3*u1*u*u  * curve.v2 +
-         u*u*u     * curve.v3;
+  const u1 = (curve.t1 - curve.t0) / dt;
+  const u2 = (curve.t2 - curve.t0) / dt;
+  const d0 = (-u1) * (-u2) * (-1);
+  const d1 = u1 * (u1 - u2) * (u1 - 1);
+  const d2 = u2 * (u2 - u1) * (u2 - 1);
+  const d3 = (1 - u1) * (1 - u2);
+  if (Math.abs(d0) < 1e-9 || Math.abs(d1) < 1e-9 || Math.abs(d2) < 1e-9 || Math.abs(d3) < 1e-9)
+    return curve.v0 + (curve.v3 - curve.v0) * u;
+  const l0 = (u - u1) * (u - u2) * (u - 1) / d0;
+  const l1 = u * (u - u2) * (u - 1) / d1;
+  const l2 = u * (u - u1) * (u - 1) / d2;
+  const l3 = u * (u - u1) * (u - u2) / d3;
+  return l0 * curve.v0 + l1 * curve.v1 + l2 * curve.v2 + l3 * curve.v3;
 }
 
-// Evaluate amplitude component of unified bezier curve at time t
+// Evaluate amplitude component of interpolating curve at time t
 function evalBezierAmp(curve, t) {
   const dt = curve.t3 - curve.t0;
   if (dt <= 0) return curve.a0;
   let u = (t - curve.t0) / dt;
   u = Math.max(0, Math.min(1, u));
-  const u1 = 1.0 - u;
-  return u1*u1*u1 * curve.a0 +
-         3*u1*u1*u * curve.a1 +
-         3*u1*u*u  * curve.a2 +
-         u*u*u     * curve.a3;
+  const u1 = (curve.t1 - curve.t0) / dt;
+  const u2 = (curve.t2 - curve.t0) / dt;
+  const d0 = (-u1) * (-u2) * (-1);
+  const d1 = u1 * (u1 - u2) * (u1 - 1);
+  const d2 = u2 * (u2 - u1) * (u2 - 1);
+  const d3 = (1 - u1) * (1 - u2);
+  if (Math.abs(d0) < 1e-9 || Math.abs(d1) < 1e-9 || Math.abs(d2) < 1e-9 || Math.abs(d3) < 1e-9)
+    return curve.a0 + (curve.a3 - curve.a0) * u;
+  const l0 = (u - u1) * (u - u2) * (u - 1) / d0;
+  const l1 = u * (u - u2) * (u - 1) / d1;
+  const l2 = u * (u - u1) * (u - 1) / d2;
+  const l3 = u * (u - u1) * (u - u2) / d3;
+  return l0 * curve.a0 + l1 * curve.a1 + l2 * curve.a2 + l3 * curve.a3;
 }
 
 // Get canvas-relative {x, y} from a mouse event
diff --git a/tools/mq_editor/viewer.js b/tools/mq_editor/viewer.js
index c69d9e7..d7b5ac1 100644
--- a/tools/mq_editor/viewer.js
+++ b/tools/mq_editor/viewer.js
@@ -626,14 +626,8 @@ class SpectrogramViewer {
       if (this.selectedPartial >= 0 && this.selectedPartial < this.partials.length) {
         const ptIdx = this.hitTestControlPoint(x, y, this.partials[this.selectedPartial]);
         if (ptIdx >= 0) {
-          const curve = this.partials[this.selectedPartial].freqCurve;
-          let companionOff = null;
-          if (ptIdx === 0)
-            companionOff = { dt: curve.t1 - curve.t0, dv: curve.v1 - curve.v0 };
-          else if (ptIdx === 3)
-            companionOff = { dt: curve.t2 - curve.t3, dv: curve.v2 - curve.v3 };
           if (this.onBeforeChange) this.onBeforeChange();
-          this.dragState = { pointIndex: ptIdx, companionOff };
+          this.dragState = { pointIndex: ptIdx };
           canvas.style.cursor = 'grabbing';
           e.preventDefault();
           return;
@@ -655,11 +649,6 @@ class SpectrogramViewer {
         const i = this.dragState.pointIndex;
         partial.freqCurve['t' + i] = t;
         partial.freqCurve['v' + i] = v;
-        if (this.dragState.companionOff) {
-          const off = this.dragState.companionOff;
-          if (i === 0) { partial.freqCurve.t1 = t + off.dt; partial.freqCurve.v1 = v + off.dv; }
-          else          { partial.freqCurve.t2 = t + off.dt; partial.freqCurve.v2 = v + off.dv; }
-        }
         this.render();
         e.preventDefault();
         return;