How do you triangulate a polygon?
1) For each vertex in the polygon, compute the angle between the two linked edges. 2) Sort vertices by decreasing angle relative to the interior of the polygon. 3) If there is less than 3 vertices in the set, we’re done. 4) Take the last vertex in the set and output the triangle formed by it and its two neighbours.
What is triangulation algorithm?
Introduction. Computing the triangulation of a polygon is a fundamental algorithm in computational geometry. for triangulating simple polygons having no holes (The code has since then been extended to handle holes). It is an incremental randomized algorithm whose expected complexity is O(n log*n).
How many ways can you triangulate a polygon?
42 possible
The 42 possible triangulations for a convex heptagon (7-sided convex polygon).
How do you triangulate a convex polygon?
A triangulation of a convex polygon is formed by drawing diagonals between non-adjacent vertices (corners), provided you never intersect another diagonal (except at a vertex), until all possible choices of diagonals have been used (see Figure 1).
Can all polygons be triangulated?
Triangulations. Theorem: Any polygon can be decomposed into triangles. Proof: The key idea of the proof goes by the method of mathematical induction. Let n = the number vertices = the number of sides in the polygon.
Why do we need to triangulate a polygon?
Because of this, it’s often the case that complex shapes are made up of simpler shapes, such as triangles, circles, and rectangles, for which it’s easy to determine whether a point lies in the complex shape or to determine that shape’s bounding box.
How do you triangulate a polygon with holes?
Let a polygon P with h holes have n vertices total, counting vertices on the holes as well as on the outer boundary. Then a triangulation of P has t = n + 2h – 2 triangles, and a quadrilateralization has q = n/2 + h ā 1 quadrilaterals. ort = n+2h-2. Since q = t/2, q=n/2 + h-l.
How do you prove all polygons have triangulation?
Triangulation theorem Every simple polygon admits a triangulation, and any triangulation of a simple polygon with n n n vertices consists of n ā 2 n-2 nā2 triangles. Proof: This can be proven by an induction over n n n. When n = 3 n=3 n=3, this is trivially true because it is a triangle itself.
What is the triangulation of polygon with sides n?
How to calculate minimum cost of polygon triangulation?
Minimum Cost Polygon Triangulation. A triangulation of a convex polygon is formed by drawing diagonals between non-adjacent vertices (corners) such that the diagonals never intersect. The problem is to find the cost of triangulation with the minimum cost. The cost of a triangulation is sum of the weights of its component triangles.
Which is the best algorithm for triangulating polygons?
Methods of triangulation include greedy algorithms [O’Rourke 1994], convex hull differences [Tor and Middleditch 1984] and horizontal decompositions [Seidel 1991] . This Gem describes an implementation based on Seidel’s algorithm ( op. cit.) for triangulating simple polygons having no holes (The code has since then been extended to handle holes) .
How is a triangulation of a convex polygon formed?
A triangulation of a convex polygon is formed by drawing diagonals between non-adjacent vertices (corners) such that the diagonals never intersect. The problem is to find the cost of triangulation with the minimum cost. The cost of a triangulation is sum of the weights of its component triangles.
How do you triangulate an array of triangles?
The triangulation code is invoked through the main interface routine, int triangulate_polygon (n, vertices, triangles); with an n -sided polygon given for input (the vertices are specified in canonical anticlockwise order with no duplicate points). The output is an array of n-2 triangles (with vertices also in anticlockwise order).