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How to fill vector polygons with triangulation

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VA

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? 🤔

GA

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:

flowchart TD A[📐 Input Polygon Points] --> B[Check Polygon Type] B --> C{Convex or Concave?} C -->|Convex| D[Simple Fan Triangulation] C -->|Concave| E[Ear Clipping Algorithm] D --> F[Pick First Vertex] F --> G[Connect to All Other Vertices] G --> H[✅ Triangulation Complete] E --> I[Find Ear Vertices] I --> J[Check Triangle Validity] J --> K{Contains Other Points?} K -->|No| L[Create Triangle] K -->|Yes| M[Skip This Ear] L --> N[Remove Ear Vertex] M --> O[Try Next Vertex] N --> P{More Than 3 Vertices?} O --> P P -->|Yes| I P -->|No| H style A fill:#e1f5fe style H fill:#e8f5e8 style E fill:#fff3e0 style J fill:#fce4ec

🔧 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! 🎨

VA

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? 🕳️

GA

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! 🎯

VB

Vibelf_Community

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