# curvat.io — Visual Geometry & Dynamic Calculus

> "Think in curves." Curvat.io is a modern, zero-build, client-side web application for interactive geometry and dynamic calculus exploration, designed for students, educators, and researchers.

## Overview
- **Name**: curvat.io (formerly OpenGraph)
- **Tagline**: Think in curves. | Visual Geometry & Dynamic Calculus
- **Publisher**: EduNetSystems (https://edunetsystems.com)
- **License**: 100% Free Educational License (No accounts, no subscriptions, no paywalls)
- **Architecture**: Zero-build pure ES6+ vanilla module architecture. 100% client-side calculation using JSXGraph, MathJS, MathLive, and KaTeX.

## Key URLs
- Main Application: https://curvat.io/
- AI Assistants & WebMCP: https://curvat.io/agents
- Visual Calculus Gallery: https://curvat.io/gallery
- Student Privacy Policy: https://curvat.io/privacy
- Accessibility Statement: https://curvat.io/accessibility
- Terms & Educational License: https://curvat.io/terms

## Privacy & Educational Architecture
- **Account Requirement**: None. 100% anonymous usage.
- **Client-Side Security**: All equation evaluations, derivatives, integrals, and coordinate geometry execute locally on the user's device. Curvat does not transmit formulas to AI models or remote compute engines; connected AI assistants process equations under their own provider's terms.
- **Classroom & LMS Privacy**: When embedded inside Learning Management Systems (Canvas, Blackboard, Moodle, Google Classroom, Schoology) or accessed with `?embed=true`, analytics tracking is completely disabled and zero advertising or tracking cookies are created, supporting FERPA- and COPPA-conscious classroom use.

## Accessibility (WCAG 2.1 Level AA)
- Full physical keyboard navigation with LaTeX shortcuts (`/` for fractions, `^` for exponents).
- Accessible on-screen virtual keyboard with elevation physics for mobile/touch screens.
- Dashed stroke toggle for non-color shape differentiation (color vision deficiency support).
- ARIA live region announcements for coordinate plane updates.
- Respects `prefers-reduced-motion`.

## Embedding in LMS & Courses
Curvat.io graphs can be embedded into any LMS via iframe:
```html
<iframe src="https://curvat.io/?embed=true" width="100%" height="500" style="border:1px solid #E2DCD2;border-radius:12px;" allowfullscreen title="curvat.io Interactive Graph"></iframe>
```

## Curriculum Presets
- Tangent Line & Extrema (Cubic): `https://curvat.io/?preset=tangent-extrema`
- Riemann Sums & Definite Integrals: `https://curvat.io/?preset=riemann-sums`
- Definite Integral & Bounded Area: `https://curvat.io/?preset=definite-integral`
- Interactive Unit Circle & Trigonometry: `https://curvat.io/?preset=unit-circle`
- Trigonometric Radian Scale (π/2): `https://curvat.io/?preset=trig-radian-scale`
- Conic Sections & Native Implicit Curves: `https://curvat.io/?preset=conic-implicit`
- Dynamic Lagrange Polynomial Interpolation: `https://curvat.io/?preset=lagrange-geometry`
- Vertical Line Test for Functions: `https://curvat.io/?preset=vertical-line-test`

## Assistant & Automation Integration (window.__curvat)

Curvat.io provides an agent-friendly, browser-first integration facade for personal assistants, AI tutors, and browser automation:

- **Positioning**: *"Interactive calculus and geometry for people and their assistants."*
- **Architecture**: 100% client-side calculation on the user's device with zero backend compute. See Privacy Details (https://curvat.io/privacy) for wider site telemetry and measurement practices.
- **Privacy Boundary**: Standard fragment links (`#g=...`) are computed entirely client-side and never reach server logs. Optional query imports (`?expr=...`) travel in the initial HTTP GET request to hosting server logs before client-side hydration replaces the URL with a canonical `#g=` fragment.

### 1. Repeatable Expression URLs
Load curves directly via repeated `expr` parameters with optional four-value bounds (`xmin`, `xmax`, `ymin`, `ymax`):
- Two explicit curves: `https://curvat.io/?expr=x%5E2&expr=2x%2B1`
- Implicit circle with bounds: `https://curvat.io/?expr=x%5E2%2By%5E2%3D25&xmin=-7&xmax=7&ymin=-7&ymax=7`
- Dynamic parameter slider: `https://curvat.io/?expr=a%3D2&expr=y%3Da*x%5E2`

### 2. Browser Automation Facade (`window.__curvat`)
The global facade (`apiVersion: 1`) provides atomic operations and optimistic revision safety:
```javascript
// Check capabilities
const caps = window.__curvat.getCapabilities();

// Inspect state and current revision
const state = window.__curvat.getState();

// Atomic graph replacement (requires expectedRevision)
await window.__curvat.applyGraph({
  expressions: ['x^2', '2x+1'],
  bounds: { xmin: -6, xmax: 6, ymin: -6, ymax: 6 }
}, { expectedRevision: state.revision });

// Update a single expression card
await window.__curvat.updateEntry('entry-2', { raw: '3x-1' }, { expectedRevision: window.__curvat.revision });

// Generate canonical share link
const links = await window.__curvat.getShareLinks();
console.log(links.data.shareUrl, links.data.embedUrl);

// Export high-resolution PNG or SVG image
const img = await window.__curvat.exportImage({ format: 'png', scale: 2 });

// Guarded undo of last assistant batch
await window.__curvat.undoLastAgentChange({ expectedRevision: window.__curvat.revision });
```

### 3. WebMCP Browser Assistant Tools
When supported by the browser (via `modelContext.registerTool`), curvat.io automatically exposes 8 native assistant tools (`get_capabilities`, `get_state`, `validate_graph`, `apply_graph`, `update_entry`, `get_share_links`, `export_image`, `undo_last_agent_change`). Full documentation, tool schemas, and safety guidelines: https://curvat.io/agents.

## Contact & Support
- Email: privacy@edunetsystems.com / accessibility@edunetsystems.com / jtredden@gmail.com
- Organization: EduNetSystems