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>

1 files changed, 31 insertions, 12 deletions

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