Build a single-page static website in HTML, CSS, and JavaScript that lets the user click to place vertices of a polygon inside a square canvas, then immediately renders a seamless tessellation of that polygon (or fills gaps if it doesn’t tessellate perfectly). Your implementation should meet the following requirements:## 1. Core Libraries and Frameworks- Use plain JavaScript (ES6+) and the HTML5 Canvas API—no build tools.- Include these helper libraries via CDN:- earcut.js for triangulating arbitrary polygons<script src="https://unpkg.com/earcut@2.2.3/dist/earcut.min.js"></script> - d3-delaunay for Voronoi/Delaunay gap-fill generation<script src="https://unpkg.com/d3-delaunay@6.0.2/dist/d3-delaunay.min.js"></script> - turf.js for geometric operations (point-in-polygon, area)<script src="https://unpkg.com/@turf/turf@6.5.0/turf.min.js"></script>## 2. UI & Input Method- Display an 800×600 <canvas> with a 1px border.- On mousedown, add a vertex at the mouse location.- Draw connecting line segments as the user clicks.- On double-click or clicking the first vertex again, “close” the polygon and start tessellation.## 3. Tessellation Detection Algorithm### 3.1 Angle-Sum Test (O(n))For each vertex i of an n‐sided polygon:1. Compute interior angle at i:$$\theta_i = \pi - \bigl(\arctan2(y_{i+1}-y_i, x_{i+1}-x_i) - \arctan2(y_{i-1}-y_i, x_{i-1}-x_i)\bigr)$$2. Sum all θ around a hypothetical shared vertex:$$\sum \theta_i \stackrel{?}{=} 2\pi$$3. If integer‐multiple combinations of identical angles can sum to 2π (e.g. 3×120°, 4×90°), mark as regular tessellation.### 3.2 Edge-Matching Test (O(n²))Pseudocode:function edgesMatch(poly): edges = list of {length, angle} for each side for each edge e in edges: for each other edge f: if |e.length - f.length| < ε and |normalize(e.angle + f.angle) - π| < ε: mark pair as matchable return all edges paired?## 4. Tessellation Generation- Translation only for regular cases:- Tile the polygon by copying at integer multiples of its bounding‐box width/height.- Rotation/reflection for more complex repeats:- Allowed rotations: 90°, 120°, 180°- Reflections: horizontal/vertical mirror- Pseudocode to place tiles:for x from -W to canvas.width+W step W: for y from -H to canvas.height+H step H: ctx.save() ctx.translate(x, y) if rotateAngle: ctx.rotate(rotateAngle) if reflect: ctx.scale(-1, 1) drawPolygon(vertices) ctx.restore() - W,H = width/height of shape’s axis‐aligned bounding box.## 5. Gap-Filling StrategyIf tessellation detection fails:1. Detect gaps via scanline sampling (step = 5px): record contiguous unfilled segments.2. Choose fill method:- Geometric: triangulate each gap region with earcut → draw triangles with 30% opacity.- Organic: compute Delaunay triangulation of gap centroids via d3-delaunay → fill cells with light shading.3. Render connectors: draw minimal line segments bridging nearest polygon edges if gaps remain large.## 6. Performance & UX- Throttle redraws: use requestAnimationFrame for background updates; skip if no new vertex added.- Offscreen canvas: pre-render one tile grid to an OffscreenCanvas for speed, then blit to main canvas.- Interactive feedback: while placing points, show a semi-transparent preview tessellation at 25% opacity.End of Prompt
I'll create a comprehensive polygon tessellation tool that meets all your requirements. This will be a single HTML file with embedded CSS and JavaScript that implements the tessellation detection algorithms and gap-filling strategies you specified.
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This implementation provides a comprehensive polygon tessellation tool with the following features:
Tessellation Detection:
Performance Optimizations:
The tool provides real-time feedback about tessellation capability and uses different visual strategies for perfect tessellations versus gap-filled patterns. Try creating squares, triangles, or hexagons to see perfect tessellations, or irregular shapes to see the gap-filling algorithm in action!
Create a minimal single-page static website with just an HTML5 Canvas for polygon creation and the entire page background showing the live tessellation pattern. Fix the core tessellation detection algorithm with these specific improvements:1. Simplified UIRemove all buttons and status text - just canvas centered on pageCanvas size: 400×400px with thin borderEntire body background shows the tessellation patternPage should have no scrollbars, tessellation fills viewport completely2. Corrected Tessellation Detection AlgorithmReplace the current testAngleSum() with this fixed logic:javascript// CORRECTED: Simple angle-based tessellation test function canTessellateRegular() { if (this.vertices.length < 3) return false; // Calculate one interior angle (assuming regular polygon) const n = this.vertices.length; const interiorAngle = (n - 2) * Math.PI / n; // Check if interior angle divides evenly into 2π const tolerance = 0.05; // ~3 degrees const arrangements = [3, 4, 6, 8, 12]; // common vertex arrangements for (const k of arrangements) { if (Math.abs(k * interiorAngle - 2 * Math.PI) < tolerance) { return { type: 'regular', arrangement: k, angle: interiorAngle }; } } return false; } 3. Simple Shape-Specific DetectionAdd this shape classification before the general algorithm:javascriptfunction detectSpecialShapes() { const n = this.vertices.length; // Quick tests for known tessellating shapes switch(n) { case 3: return { type: 'triangle', tessellates: true }; case 4: if (this.isRectangle()) return { type: 'rectangle', tessellates: true }; if (this.isParallelogram()) return { type: 'parallelogram', tessellates: true }; break; case 6: if (this.isRegularHexagon()) return { type: 'hexagon', tessellates: true }; break; } return null; } // Helper: Check if 4-sided polygon is rectangle-ish function isRectangle() { if (this.vertices.length !== 4) return false; const edges = this.getEdgeLengths(); const angles = this.getInteriorAngles(); // Check opposite sides equal + all angles ~90° const sideA = edges[0], sideB = edges[1], sideC = edges[2], sideD = edges[3]; const tolerance = 0.1; return Math.abs(sideA - sideC) < tolerance && Math.abs(sideB - sideD) < tolerance && angles.every(angle => Math.abs(angle - Math.PI/2) < tolerance); } 4. Corrected Background Tessellation RenderingReplace the current tessellation rendering with simple translation-only tiling:javascriptfunction renderBackgroundTessellation() { if (!this.vertices.length || !this.tessellationResult) return; const bbox = this.getBoundingBox(); const tileWidth = bbox.width + 5; // small gap const tileHeight = bbox.height + 5; // Cover entire viewport const startX = -Math.ceil(window.innerWidth / tileWidth) * tileWidth; const startY = -Math.ceil(window.innerHeight / tileHeight) * tileHeight; const endX = window.innerWidth + tileWidth; const endY = window.innerHeight + tileHeight; // Create CSS background pattern const canvas = document.createElement('canvas'); canvas.width = tileWidth; canvas.height = tileHeight; const ctx = canvas.getContext('2d'); // Draw one tile const offsetVertices = this.vertices.map(v => ({ x: v.x - bbox.x + 2, y: v.y - bbox.y + 2 })); this.drawPolygon(ctx, offsetVertices, { fill: 'rgba(120, 150, 255, 0.6)', stroke: '#666', lineWidth: 1 }); // Apply as CSS background const dataURL = canvas.toDataURL(); document.body.style.backgroundImage = url(${dataURL}); document.body.style.backgroundRepeat = 'repeat'; } 5. Minimal Event HandlingSimplify the interaction to just polygon creation:Click to add vertices (max 12 vertices)Auto-close after 3+ vertices when you click near the first vertexAuto-analyze tessellation on polygon closeClear polygon with spacebar key6. Performance RequirementsReal-time updates: Background tessellation should update immediately when polygon closesSmooth interaction: No lag when placing verticesMemory efficient: Reuse canvas objects, don't create new ones each frame7. Visual Stylecssbody { margin: 0; padding: 0; height: 100vh; overflow: hidden; display: flex; justify-content: center; align-items: center; background-color: #f5f5f5; } #canvas { border: 2px solid #333; background: white; box-shadow: 0 4px 8px rgba(0,0,0,0.2); } Key Changes from v1:Fixed tessellation detection - properly tests if interior angles can arrange around a vertex to sum to 360°Shape-specific detection - immediately recognizes triangles, rectangles, parallelograms as tessellatingSimplified rendering - uses basic translation tiling instead of complex rotation logicBackground tessellation - entire page background shows the pattern, not just canvas areaMinimal UI - removes all clutter, focuses on core interactionThis should finally detect tessellation correctly for basic shapes like triangles, squares, and hexagons!Also add buttons for testing / demonstration purposes that will load vertexes for regular polygons that we know can tessellate, so that the algorithm can be demonstrated
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Build a single‐page static site (plain ES6+ JavaScript + HTML5 Canvas) that lets the user click to place vertices of an arbitrary simple polygon inside a square canvas and then instantly fills the entire page background with a seamless tiling of that polygon. Use exactly these libraries via CDN:turf.js for geometry (point-in-polygon, convex hull)earcut.js for triangulation if needed (for gap filling only)No shape‐specific code paths—just one general tiling engine that works for all input shapes, including regular polygons.Algorithm requirements (pseudocode):Compute the polygon’s minimum bounding parallelogram via convex‐hull + rotating‐calipers.texthull = turf.convexHull(vertices) { widthVec, heightVec, polyBBox } = rotatingCalipers(hull) // widthVec, heightVec are orthogonal unit vectors along minimal bbox // polyBBox.width, .height are extents along those vectors Build a 2D lattice of tile origins:textfor i from -N to N: for j from -M to M: origin = polyBBox.origin + iwidthVecpolyBBox.width + jheightVecpolyBBox.height tileTransforms.push({ translate: origin, rotate: 0, reflect: false }) tileTransforms.push({ translate: origin + widthVec*(polyBBox.width/2) + heightVec*(polyBBox.height/2), rotate: π, reflect: false }) This 2-point lattice automatically aligns opposite edges.For each transform in tileTransforms:textctx.save() ctx.translate(transform.translate.x, transform.translate.y) if (transform.reflect) ctx.scale(-1,1) ctx.rotate(transform.rotate) drawPolygon(vertices) ctx.restore() If any pixel remains unfilled (detect via low‐resolution sampling), triangulate gaps via earcut and fill.On polygon closure, recompute tileTransforms and update CSS body { background: url(...) repeat; } using an offscreen pattern canvas of size polyBBox.width×polyBBox.height.Performance: throttle to 15 FPS during drag, recompute only on polygon close. No separate “if regular” branches—this single lattice + optional 180° flip pattern handles triangles, squares, hexagons, and any shape. Provide implementations for rotating‐calipers, lattice generation, and gap‐fill fallback.