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● •Mean value coordinates
•Dividing polygons into triangles Polygon Decomposition
For polygons with more than three vertices, we
usually decompose them into triangles
P6
P7 Simple for convex polygons.
P5
Concave more difficult.
P4 P0 P1 P3
P2 Polygon Decomposition:
Algorithm
Start from the left and form the leftmost triangle:
•Find leftmost vertex (smallest x) – A
•Compose possible triangle out of A and the two
adjacent vertices B and C
•Check to ensure that no other polygon point P is
inside of triangle ABC
•If all other polygon points are outside of ABC
then cut it off from polygon and proceed with
next leftmost triangle Polygon decomposition (2)
•The left most vertex A
•A triangle is formed by A and the two
adjacent B and C
•Check if all the other vertices are outside the
triangle Polygon decomposition (3)
If a vertex is inside, split the polygon by the
inside vertex and point A, proceed as before. Polygon decomposition (4)
This edge may split the polygon into two.
Then, recurse the method with each polygon
(ABCDE and AEFGH in the example below) Summary
Rasterization
•Line Rasterization  Midpoint algorithm
•Triangle Rasterization  barycentric coordinates
•Mean value coordinates
•Dividing polygons into triangles Reading for rasterization
Scanline algorithm
Foley et al., Chapter 3.5, 3.6
Baricentric coordinates
www.cs.caltech.edu/courses/cs171/barycentric.pdf
Mean value coordinates for closed triangular meshes,
SIGGRAPH 2005
Polygon decomposition
http://www.siggraph.
org/education/materials/HyperGraph/scanline/outprims/
polygon1.htm *...
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 Spring '14

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