A cycloid, a straight line and a specified power curve connect the same endpoints. Gravity and height determine speed; numerical segment integration times the comparison tracks, while the cycloid uses its analytic time, and the film positions each marker from its own elapsed-time table. Track geometry, travel-time integration and marker positions derive from one model. The race can explain why shape matters without using arbitrary easing to manufacture a winner.

Uses PopNow for
Music & voice · Images
Needs
Node.js · FFmpeg · Chrome

Generation is optional when you have suitable input material.

v1.0.0 · Updated Oct 26, 2026 · 132 KB · Use it for personal and client work; please don't resell or share the package.

What you get

  1. An editable 20-second 720p film and native portrait project, silent.

  2. Local HTML playback, H.264 export, a real-frame poster and contact sheet.

  3. Validated project inputs and protected export with a SHA receipt.

Good for

  • Brachistochrone animation
  • Cycloid race video

Not for

  • Compares three specified ideal tracks and ignores friction, drag and rolling inertia.
  • The numerical curves have finite sampling error; it is not an arbitrary variational equation solver.

How it works

Includes a local renderer

  1. Choose the story inputs

    Set the vertical drop, cycloid finish angle and uniform gravity. These are idealized frictionless assumptions, not measured results from a physical track.

  2. Build and inspect the local preview

    Validate the project, then build the portable HTML player. Edit the source JSON and rebuild to change the actual geometry or motion.

  3. Export and review the film

    Check the common start and finish, arrival times and the stated gravity. Compare the analytic cycloid time with the reported value after a structural input change.

How the workflow runs

How Variational Curve Race works

A cycloid, a straight line and a specified power curve connect the same endpoints. Gravity and height determine speed; numerical segment integration times the comparison tracks, while the cycloid uses its analytic time, and the film positions each marker from its own elapsed-time table.

What's inside the skill

What you can change

  • Drop height changes track dimensions and travel times.
  • Finish angle changes the cycloid and common horizontal endpoint.
  • Gravity changes all computed times.
  • Title can frame the lesson without changing its assumptions.

What to review before delivery

  • The self-check verifies that a structural input changes frame geometry and rejects a documented invalid case.
  • Output protection rejects collisions with source files and existing sidecars; the exporter checks actual decoded frame count and format.
  • Check the common start and finish, arrival times and the stated gravity. Compare the analytic cycloid time with the reported value after a structural input change. This visual review is manual.

See the package files

These are selected working files from this edition, with their actual role. The complete download also contains instructions and applicable license notices.

  • The specific mechanism, frame geometry and input validation.

    scripts/composition.mjs
  • Protected MP4 export, decoded-format checks and a hash receipt.

    scripts/render.mjs
  • The complete editable wide example; portrait has a separate project.

    examples/project.json
  • Normal, changed-structure and invalid-input checks.

    scripts/selftest.mjs
  • Input ranges, assumptions and manual art review.

    references/inputs.md

Why use this skill

Start with this package

Track geometry, travel-time integration and marker positions derive from one model. The race can explain why shape matters without using arbitrary easing to manufacture a winner.

What PopNow adds

Use PopNow for the missing Music & voice, Images in this workflow. Keep adequate material you already own, then bring new assets back into the editing and assembly steps described here.

Compare the options

A prompt, your own skill, or this workflow?

Start with a prompt

Your agent can create from a plain request. The brief, visual rules, asset choices, output format and review process still need to be established for that project.

Write your own skill

You choose and maintain the reusable instructions, defaults, file structure and checks. This is useful when your process needs different rules or integrations.

A workflow created for PopNow

This workflow was built for PopNow. Its source files, included dependencies and applicable notices are part of the package; the scope described below belongs to this edition.

Guide reviewed Oct 25, 2026

What will this cost?

The mathematical film needs no generated footage. Optional narration or restrained movement sounds can be quoted through PopNow after the assumptions and wording are approved. Local rendering uses the user’s installed tools and computer.

Generation is optional when you have suitable input material. Free skill. New media is quoted before it runs.

The skill package

Free to install under your account's download allowance. The package license covers your own and client work; redistribution is excluded.

Existing assets and local work

Reusing suitable assets, editing files and running local assembly do not call a PopNow generation model. Your computer and any required runtime still do the work.

New AI images, clips or sound

Quoted for the actual model, inputs, count, duration and resolution. Extra alternatives, revisions or optional enhancement passes can add generation requests.

Your agent and hosting

Any subscription, hosted computer, deployment or other provider fee belongs to that service. Installing this skill does not include those services.

Check the cost of your version first

Your assistant can inspect the available models and quote the exact request first. You choose which optional passes to include and approve spending before generation.

Ask your assistant

  • Use the PopNow skill "Variational Curve Race" for my project. List the inputs you can reuse, quote only the missing media, show the quoted total and flag any unpriced steps. Separate optional enhancements, and wait for my approval before starting anything paid.

How PopNow billing works

Connections using a PopNow API key, including API-key MCP connections, spend from the separate API balance in USD. A ChatGPT or Claude connection authorized through OAuth spends website credits. These balances are separate; the quote shown by your connection is the approval basis.

Download allowances and current account conditions apply. A free package does not mean unlimited free generation.

How to use it

You say

  • Use the PopNow skill "Variational Curve Race" (variational-curve-race) for this: Render the supplied example, then make a second film with heightMetres set to 6. Show the changed result and quote any optional media before generation.

Then try

  • Render a native portrait version and inspect the framing.

  • Keep the local artwork and quote an optional soundtrack through PopNow.

What to have ready

  • Set the vertical drop, cycloid finish angle and uniform gravity. These are idealized frictionless assumptions, not measured results from a physical track.
  • Node.js 22.12+, Chrome or Chromium, FFmpeg and ffprobe.
  • The new film examples use ASCII text. Accents and other scripts need a suitable font plus input and layout adjustments.

Similar skills

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Questions

What actually makes this animation work?

A cycloid, a straight line and a specified power curve connect the same endpoints. Gravity and height determine speed; numerical segment integration times the comparison tracks, while the cycloid uses its analytic time, and the film positions each marker from its own elapsed-time table.

What can I change without generating new media?

Drop height changes track dimensions and travel times. Finish angle changes the cycloid and common horizontal endpoint. Gravity changes all computed times. Title can frame the lesson without changing its assumptions.

What costs extra?

The mathematical film needs no generated footage. Optional narration or restrained movement sounds can be quoted through PopNow after the assumptions and wording are approved. Local rendering uses your own computer and installed dependencies.

What is already packaged in Variational Curve Race?

The specific mechanism, frame geometry and input validation. Protected MP4 export, decoded-format checks and a hash receipt. The complete editable wide example; portrait has a separate project. Normal, changed-structure and invalid-input checks. Input ranges, assumptions and manual art review.

Which parts of Variational Curve Race can add a charge?

The mathematical film needs no generated footage. Optional narration or restrained movement sounds can be quoted through PopNow after the assumptions and wording are approved. Local rendering uses the user’s installed tools and computer. Connections using a PopNow API key, including API-key MCP connections, spend from the separate API balance in USD. A ChatGPT or Claude connection authorized through OAuth spends website credits. These balances are separate; the quote shown by your connection is the approval basis.

How does Variational Curve Race differ from writing a prompt or my own skill?

Track geometry, travel-time integration and marker positions derive from one model. The race can explain why shape matters without using arbitrary easing to manufacture a winner.