How to fill vector polygons with triangulation
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VectorArtist_Dev
Posted on January 23, 2024 • Advanced
🔺 Vector polygon filling challenge
I’m working on a graphics engine in Scratch and need to implement polygon filling using triangulation. I have a list of vector points that define complex shapes, including concave polygons with indentations.
The basic fan triangulation approach works for simple convex shapes, but fails when I have concave polygons - it creates triangles that extend outside the intended shape boundaries.
How can I properly triangulate complex polygons to fill them correctly? Any algorithms or techniques that work well in Scratch? 🤔
GeometryAlgorithm_Expert
Replied 3 hours later • ⭐ Best Answer
Excellent question @VectorArtist_Dev! Polygon triangulation is indeed tricky for concave shapes. Let me break down the proper approach:
🔺 Triangulation Algorithm Flow
Here’s how proper polygon triangulation works:
🔧 Implementation Strategy
1. Polygon Classification
First, determine if your polygon is convex or concave:
// Check if polygon is convex define is convex (points list) set [convex v] to [true] repeat (length of [points v]) set [i v] to ((i) mod (length of [points v])) set [j v] to (((i) + [1]) mod (length of [points v])) set [k v] to (((i) + [2]) mod (length of [points v])) // Calculate cross product set [cross v] to (cross product of vectors) if <(cross) < [0]> then set [convex v] to [false] end end
2. Simple Fan Triangulation (Convex Only)
For convex polygons, use the simple approach:
// Fan triangulation for convex polygons define fan triangulation (points) set [fixed point v] to [1] repeat ((length of [points v]) - [2]) set [triangle v] to [] add (item (fixed point) of [points v]) to [triangle v] add (item ((counter) + [1]) of [points v]) to [triangle v] add (item ((counter) + [2]) of [points v]) to [triangle v] // Draw or store triangle draw triangle (triangle) end
3. Ear Clipping Algorithm (Concave Polygons)
For concave polygons, implement ear clipping:
// Ear clipping algorithm define ear clipping (points) set [remaining points v] to (copy of [points v]) repeat until <(length of [remaining points v]) = [3]> set [ear found v] to [false] repeat (length of [remaining points v]) if <is ear (counter)> then // Create triangle set [triangle v] to [] add (item ((counter) - [1]) of [remaining points v]) to [triangle v] add (item (counter) of [remaining points v]) to [triangle v] add (item ((counter) + [1]) of [remaining points v]) to [triangle v] draw triangle (triangle) delete (counter) of [remaining points v] set [ear found v] to [true] stop this script end end end // Draw final triangle draw triangle (remaining points)
4. Ear Detection Function
The key is properly detecting “ears” (valid triangles):
// Check if vertex forms a valid ear define is ear (vertex index) set [prev v] to (item ((vertex index) - [1]) of [points v]) set [curr v] to (item (vertex index) of [points v]) set [next v] to (item ((vertex index) + [1]) of [points v]) // Check if triangle is oriented correctly if <(cross product) < [0]> then return [false] end // Check if any other point is inside triangle repeat (length of [points v]) if <not <(counter) = (vertex index)>> then if <point in triangle (item (counter) of [points v])> then return [false] end end end return [true]
🎯 Point-in-Triangle Test
Essential helper function for ear clipping:
// Check if point is inside triangle define point in triangle (point) (triangle) set [area1 v] to (triangle area (point) (vertex 1) (vertex 2)) set [area2 v] to (triangle area (point) (vertex 2) (vertex 3)) set [area3 v] to (triangle area (point) (vertex 3) (vertex 1)) set [total area v] to (triangle area (vertex 1) (vertex 2) (vertex 3)) if <((area1) + (area2) + (area3)) = (total area)> then return [true] else return [false] end
⚡ Performance Optimization Tips
- Pre-sort vertices: Order points counter-clockwise for consistent results
- Cache calculations: Store cross products and areas to avoid recalculation
- Use spatial indexing: For large polygons, implement quadtree or grid-based optimization
- Handle edge cases: Check for degenerate triangles and collinear points
This approach will correctly handle both convex and concave polygons, including those tricky indented shapes! 🎨
VectorArtist_Dev
Replied 2 hours later
@GeometryAlgorithm_Expert This is incredibly detailed! 🤩 Thank you so much!
I implemented the ear clipping algorithm and it works perfectly for my concave polygons. The point-in-triangle test was exactly what I was missing. My graphics engine can now handle complex shapes without any artifacts!
One question - how do you handle polygons with holes? Is there a way to extend this approach? 🕳️
GeometryAlgorithm_Expert
Replied 1 hour later
@VectorArtist_Dev Great question about holes! 🕳️ For polygons with holes, you can:
🔗 Bridge Method
- Connect each hole to the outer boundary with a “bridge”
- This converts the polygon with holes into a simple polygon
- Then apply normal ear clipping
// Bridge holes to outer boundary define bridge holes (outer polygon) (holes list) repeat (length of [holes list v]) set [hole v] to (item (counter) of [holes list v]) set [bridge point v] to (find closest point on outer boundary) // Insert hole vertices into outer polygon insert hole vertices at bridge point end
This maintains the triangulation properties while handling complex shapes! 🎯
Vibelf_Community
Pinned Message • Moderator
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