FE-1512: Place arc weight labels on the Adaptive Bezier curve

[kube]

🌟 What is the purpose of this PR?

With the Arcs rendering setting on Adaptive Bezier (the default), arc weight labels were drawn at the straight-line midpoint between the arc's endpoints, despite the code's comment claiming the bezier midpoint. Long arcs sweep far from that line, so the "× n" label floated in empty space with no visible connection to its arc. The label now sits on the curve.

Before — the "× 2" floats at the straight-line midpoint, in empty space:

After — the same label sits on its Infection → Infected curve (same net, same framing):

🔗 Related links

• FE-1512

🚫 Blocked by

• FE-1510: Render the Petrinaut canvas centered in a single initial render #9352 — stacked on claude/fe-1510-canvas-single-initial-render; only the last commit belongs to this PR until that merges.

🔍 What does this change?

• getCustomArcPath in arc.tsx computes the label position as the cubic bezier point at t = 0.5 — (P0 + 3·CP1 + 3·CP2 + P3) / 8 — from the same control points that define the path. The Square and Bezier renderings already got this right via React Flow's path helpers and are untouched.

Pre-Merge Checklist 🚀

🚢 Has this modified a publishable library?

This PR:

• modifies an npm-publishable library and I have added a changeset file(s)

📜 Does this require a change to the docs?

The changes in this PR:

• are internal and do not require a docs change

The visual-settings page describes the rendering styles, not label placement.

🕸️ Does this require a change to the Turbo Graph?

The changes in this PR:

• do not affect the execution graph

🛡 What tests cover this?

• None directly — the function is private to the arc component. Verified in the running app: for every weighted arc, the label's translate now lies on the rendered path (measured 0.7px off via getPointAtLength sampling; it was ~100px off for long arcs before).

❓ How to test this?

1. yarn workspace @hashintel/petrinaut dev, open a net with a weight > 1 arc (e.g. Petrinaut → Handle Spike With Sir).
2. Viewport settings → Arcs rendering → Adaptive Bezier (default).
3. Drag the connected nodes far apart vertically: the "× 2" label stays on the curve instead of drifting into empty space.

🤖 Generated with Claude Code

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Project	Deployment	Actions	Updated (UTC)
hash	Ready	Preview	Aug 27, 2026 12:05pm
petrinaut	Ready	Preview	Aug 27, 2026 12:05pm
petrinaut-docs	Ready	Preview	Aug 27, 2026 12:05pm

1 Skipped Deployment

Project	Deployment	Actions	Updated (UTC)
hashdotdesign-tokens	Ignored	Preview	Aug 27, 2026 12:05pm

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PR Summary

Low Risk
Visual-only label positioning in one private path helper; no auth, data, or simulation logic changes.

Overview
Fixes Adaptive Bezier arc rendering so weight labels (e.g. × 2) align with the visible curve instead of the straight line between endpoints.

In getCustomArcPath, label coordinates now use the cubic Bézier midpoint at t = 0.5 — (P0 + 3·CP1 + 3·CP2 + P3) / 8 — using the same control points as the path. The unused dy term is dropped. Smoothstep and standard Bezier modes are unchanged (React Flow already returns correct label positions). A patch changeset documents the @hashintel/petrinaut release.

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