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Processing Polygons

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ProcessingPolygons

A polygon is a many-sided planar figure composed of vertices and edges.

Vertices are represented by points (x,y).

Edges are represented as line segments which connect two points, (x1,y1) and (x2,y2).

P = { (xi , yi ) } i=1,n

E1

(x1,y1)

(x2,y2)

E2

E3

(x3,y3)

Convex Polygon - For any two point s P1, P2 inside the polygon, all points on the line segment which connects P1 and P2 are inside the polygon.

All points P = uP1 + (1-u)P2, u in [0,1] are inside the polygon provided that P1 and P2 are inside the polygon.

Concave Polygon - A polygon which is not convex.

How do we know a point is "inside" a polygon?

Inside test: A point P is inside a polygon if and only if a scanline intersects the polygon edges an odd number of times moving from P in either direction.

1

3

2

1

Problem when scan line crosses a vertex:

Does the vertex count as two points?

Or should it count as one point?

When crossing a vertex, if the vertex is a local maximum or minimum (in y) then count it twice, else count it once.

or

or

Count twice

Count once

Fill the polygon 1 scanline at a time

Determine which pixels on each scanline are inside the polygon and set those pixels to the appropriate value.

Look only for those pixels at which changes occur.

For each scan-line:

1.Find the intersections of the scan line with all edges of the polygon.

2.Sort the intersections by increasing x-coordinate.

3.Fill in all pixels between pairs of intersections.

Possible Problems

1.Horizontal edges ==> Ignore

2.Vertices ==> If local max or min, then count twice, else count once. (This is implemented by shortening an edge by one pixel.)

3.Calculating intersections is slow.

8

6

4

2

For scan-line number 7 the sorted list of x-coordinates is (1,3,7,9)

Therefore fill pixels with x-coordinates 1-3 and 7-9.

Not all edges intersect each scan line.

Many edges intersected by scan line i will also be intersected by scan line i+l

Formula for scan line s is y = s,

for an edge is y = mx + b

Their intersection of a scan line and an edge is

s = mxs + b therefore xs = (s-b)/m

For scan line s + 1,

xs+1 = (s+1 - b)/m

= xs + 1/m