Filling Graphical Shapes

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Filling Graphical Shapes
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We know how to draw outlines

• Can we just fill the inside?
• but how do we know the difference between the
  inside and outside?
• Can we determine if a point is inside a shape?

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A Filling Anomaly
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One ApproachThe Odd-Even Rule
0

• Choose an arbitrary point
• Draw ray to a distant point
• Dont intersect any vertices
• Whats a distant point?
• Count edges crossed
• Odd count means interior
• Even count means exterior

1
2
1
3
1
2
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Odd-Even Result
1
2
1
1
3
1
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Another approachNonzero Winding Rule
0

• Choose a point
• Draw ray to a distant point
• Dont intersect any vertices
• Consider edges crossed (right hand rule)
• Subtract 1 when ray to edge is clockwise
• Add 1 when ray to edge is counter-clockwise
• Nonzero count means interior
• Count Winding number

-1
-2
-1
-1
-1
0
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Nonzero Winding Example
-1
-2
-1
-1
-1
-1
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Computing Winding Number

• Compute cross product
• Between ray and edge
• Sign of z value determines direction
• z implies CCW (add 1 to winding number)

See text, page 128
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Cross Product Example
-1 means u crosses E clockwise
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Cross Product Simplification
negative means clockwise
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Computing Winding Number

• Compute dot product
• Use perpendicular to ray vs. edge
• If ray is given by (ux, uy), perpendicular is (-
  uy, ux)
• Sign of product determines direction
• product implies right-to-left (add 1 to
  winding number)

See text, page 128
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Dot Product Example
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Dot Product Simplification
Same result as from cross product!
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Inside-Outside Tests Summary

• Odd-even rule
• Generalized from scan-line fill
• May produce unusual results if edges intersect
• Nonzero winding number rule
• Alternate way of determining interior