Fine Arc — Numerical Methods Studio

A full-stack numerical methods solver. Enter a problem (or upload/describe one in plain English) and get a step-by-step solution — root finding, interpolation, integration, and ODEs — with charts and a downloadable Word/PDF report.

Live app: https://fine-arc.vercel.app

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Features

• 11 numerical methods, all with full step-by-step working:
  • Root finding — Bisection, False Position, Newton-Raphson, Secant
  • Interpolation — Lagrange, Newton's Forward Difference, Newton's Backward Difference, Newton's Divided Difference
  • Integration — Trapezoidal, Simpson's 1/3, Simpson's 3/8
  • ODEs — Euler's Method, Runge-Kutta 4th Order (RK4)
• Smart Interpolation — pick equal or unequal interval, Fine Arc automatically selects the right formula (Newton Forward/Backward for equal spacing depending on where the target sits, Lagrange for unequal spacing)
• General Solver — describe a problem in plain English, or upload a file, and the parser extracts the data and picks a method
• Interactive "Teach me this" tutorial — built dynamically from the problem you just solved (not a fixed example), covering Lagrange, Newton Forward/Backward, Bisection, Newton-Raphson, and Secant
• Charts for every method — interpolation curve, ODE trajectory, root-finding convergence, and (for root-finding/integration) an actual sampled f(x) curve with a tangent/secant line or shaded integration region
• Auth — email/password (JWT), Google Sign-In, or continue as a guest; solving works without an account, sign-in unlocks a per-user solve counter and profile panel
• Export — download any solution as a formatted Word document or PDF, including difference tables and term-by-term breakdowns
• Light/dark theme, animated UI (Framer Motion), KaTeX-rendered math

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Stack

Backend — FastAPI (Python 3.13), SQLite via SQLAlchemy, sympy/numpy for the solvers, python-docx/reportlab for exports, bcrypt + PyJWT for auth.

Frontend — React 19 + Vite, Framer Motion, KaTeX, Recharts, lucide-react.

Deployment — backend on Render, frontend on Vercel, connected via CORS + an VITE_API_URL environment variable.

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Project structure

fine-arc/
├── backend/
│   ├── app/
│   │   ├── main.py              # FastAPI app, routes, CORS config
│   │   ├── core/
│   │   │   ├── solvers.py       # All 11 numerical methods
│   │   │   ├── security.py      # Password hashing, JWT
│   │   │   ├── deps.py          # Auth dependencies (required / optional)
│   │   │   └── mailer.py        # Email sending (SMTP)
│   │   ├── db/
│   │   │   ├── database.py      # SQLAlchemy engine/session
│   │   │   └── models.py        # User model
│   │   ├── routers/auth.py      # /api/auth/* endpoints
│   │   ├── schemas/auth.py      # Pydantic request/response models
│   │   └── utils/
│   │       ├── parser.py        # Plain-English problem parsing
│   │       ├── exporter.py      # DOCX/PDF generation
│   │       └── file_extractor.py
│   ├── tests/                   # pytest suite for solvers + parser
│   └── requirements.txt
└── frontend/
    ├── src/
    │   ├── App.jsx               # Main app shell
    │   ├── api/client.js         # Backend API wrapper
    │   ├── context/AuthContext.jsx
    │   └── components/
    │       ├── auth/AuthPage.jsx
    │       ├── SmartInterpolation.jsx
    │       ├── ChatInterface.jsx     # General solver chat UI
    │       ├── SolutionRenderer.jsx  # Renders a solved result
    │       ├── SolutionChart.jsx     # Recharts visualizations
    │       ├── InterpolationTutorial.jsx
    │       └── ProfilePanel.jsx
    └── public/                  # Logo, favicon, static assets

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Running locally

Backend

cd backend
python -m venv .venv
# Windows: .venv\Scripts\activate
# Mac/Linux: source .venv/bin/activate
pip install -r requirements.txt

Create backend/.env (see backend/.env.example for the full list):

JWT_SECRET_KEY=<any random string>
GOOGLE_CLIENT_ID=<your Google OAuth client ID>
SMTP_HOST=...
SMTP_PORT=...
SMTP_USERNAME=...
SMTP_PASSWORD=...
SMTP_FROM_EMAIL=...
SMTP_FROM_NAME=...

Then run:

python -m uvicorn app.main:app --reload --port 8000

Frontend

cd frontend
npm install

Create frontend/.env (see frontend/.env.example):

VITE_GOOGLE_CLIENT_ID=<same Google OAuth client ID as the backend>

VITE_API_URL doesn't need to be set locally — it defaults to http://localhost:8000/api when absent.

npm run dev

Visit http://localhost:5173.

Tests

cd backend
pytest tests/

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Deployment notes

• Backend (Render): root directory backend, build command pip install -r requirements.txt, start command uvicorn app.main:app --host 0.0.0.0 --port $PORT. Set all the .env variables above as environment variables in the Render dashboard, plus FRONTEND_URL set to the deployed frontend's URL (needed for CORS).
• Frontend (Vercel): root directory frontend, framework auto-detected as Vite. Set VITE_GOOGLE_CLIENT_ID and VITE_API_URL (pointing at the Render backend + /api) as environment variables.
• Google OAuth: the Google Cloud Console OAuth client needs both the local (http://localhost:5173) and deployed frontend URL listed under Authorized JavaScript origins (and Authorized redirect URIs), or Google Sign-In will fail with origin_mismatch on the deployed site.
• Database: SQLite, created automatically on first run. Render's free tier disk is not persistent across redeploys — fine for a demo, but a managed Postgres database is recommended before relying on it for real user data long-term.
• Render's free tier spins the backend down after 15 minutes of inactivity; the first request after idle can take 30–60 seconds to respond.

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Security notes

• Never commit backend/.env or frontend/.env — see .gitignore. .env.example files in both folders document the required variable names without real values.
• If a secret is ever accidentally exposed (e.g. pasted somewhere it shouldn't be), rotate it — generate a new value and update it everywhere it's configured (local .env files and the Render dashboard).