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Rasterizing Polygons. Lecture 29 Wed, Dec 7, 2005. Rasterizing Polygons. Given a the vertices of a polygon, in viewport coordinates, which pixels should be shaded? How do we determine them efficiently?. Rasterizing Polygons. Rasterizing Polygons.

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Rasterizing polygons

Rasterizing Polygons

Lecture 29

Wed, Dec 7, 2005


Rasterizing polygons1
Rasterizing Polygons

  • Given a the vertices of a polygon, in viewport coordinates, which pixels should be shaded?

  • How do we determine them efficiently?



Rasterizing polygons3
Rasterizing Polygons

  • Obviously, any pixel lying entirely within the polygon should be shaded.

  • What about the pixels that are partially within the polygon?

    • Rule: Shade them if their center is within the polygon.

  • What about the pixels whose center is exactly on the edge?



Neighboring polygons
Neighboring Polygons

  • When two polygons share an edge, which one “owns” the pixels on the edge?

  • Rule

    • A polygon owns all pixels whose centers are within its interior.

    • A polygon owns all pixels whose centers lie on one of its left edges.


Neighboring polygons1
Neighboring Polygons

  • Fill in all pixels whose centers are within the polygon.


Neighboring polygons2
Neighboring Polygons

  • Fill in all pixels whose centers are on a left edge of the polygon.


A rasterization algorithm
A Rasterization Algorithm

  • To rasterize a polygon, first determine its bounding box.


A rasterization algorithm1
A Rasterization Algorithm

  • To rasterize a polygon, first determine its bounding rectangle.


A rasterization algorithm2
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.


A rasterization algorithm3
A Rasterization Algorithm

  • When an edge is encountered,

    • If we were on the outside, we move to the inside.

    • If we were on the inside, we move to the outside.

  • Thus, we stop or resume shading accordingly.


A rasterization algorithm4
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 0


A rasterization algorithm5
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 1


A rasterization algorithm6
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 2


A rasterization algorithm7
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 3


A rasterization algorithm8
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 4


A rasterization algorithm9
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 5


A rasterization algorithm10
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 6


A rasterization algorithm11
A Rasterization Algorithm

  • Then scan each row of pixels in the bounding rectangle, left to right, bottom to top.

row 7


A rasterization algorithm12
A Rasterization Algorithm

  • The following algorithm is designed to allow rapid shading of the pixels.

    • As the vertices are given, create a list of the vertices. (The order matters!)

      • {(1, 0), (4, 3), (6, 1), (12, 1), (11, 8), (7, 8), (6, 5), (4, 8), (0, 7), (1, 0)}.

    • From the list of vertices, form an edge table.

      • {{(1, 0), (4, 3)}, {(4, 3), (6, 1)}, …, {(0, 7), (1, 0)}}.


A rasterization algorithm13
A Rasterization Algorithm

  • Organization of the edge table.

    • Eliminate any horizontal edges.

    • Sort the edges in the edge table by the y-coordinate of the lower endpoint.

  • Begin scanning with the bottom scan line.


The active edge table
The Active Edge Table

  • Create the active edge table (AET).

    • For each edge in the edge table whose lower endpoint is on the scan line,

      • Create an active-edge-table entry.

      • Add it to the active edge table.

      • Delete the edge from the edge table.


The active edge table1
The Active Edge Table

  • Organization of an active-edge-table entry:

    • y-coordinate of upper endpoint.

    • Reciprocal of the slope.

    • x-intercept with the horizontal line ½ unit above the current scan line.


The active edge table2
The Active Edge Table

Active

edges

Scan

line


The active edge table3
The Active Edge Table

(7, -1/7, 13/14)

Scan

line

(3, 1, 1-1/2)


The active edge table4
The Active Edge Table

  • Sort the AET entries by their x-intercepts.

  • The AET must contain an even number of entries.

    • Why?

  • Shade pixels from the 1st to the 2ndx-intercepts, 3rd to 4thx-intercepts, etc., in the AET.


The active edge table5
The Active Edge Table

(7, -1/7, 13/14)

(no shading on

this scan line)

Scan

line

(3, 1, 1-1/2)


The active edge table6
The Active Edge Table

  • Update the AET.

    • Increment the scan line number.

    • Delete from the AET any entries for which the upper endpoint is on the scan line.

    • Update the x-intercepts of all AET entries.

      • Add the reciprocal slope to the x-intercept.

    • Create and add entries from the edge table for edges whose lower endpoint is on the scan line.


A rasterization algorithm14
A Rasterization Algorithm

(7, -1/7, 13/14)

Scan

line

(3, 1, 1-1/2)


A rasterization algorithm15
A Rasterization Algorithm

(7, -1/7, 11/14)

Scan

line

(8, -1/7, 11-13/14)

(3, -1, 5-1/2)

(3, 1, 2-1/2)


A rasterization algorithm16
A Rasterization Algorithm

(7, -1/7, 11/14)

Scan

line

(8, -1/7, 11-13/14)

(3, -1, 5-1/2)

(3, 1, 2-1/2)


A rasterization algorithm17
A Rasterization Algorithm

(7, -1/7, 9/14)

Scan

line

(8, -1/7, 11-11/14)

(3, -1, 4-1/2)

(3, 1, 3-1/2)


A rasterization algorithm18
A Rasterization Algorithm

(7, -1/7, 9/14)

Scan

line

(8, -1/7, 11-11/14)

(3, -1, 4-1/2)

(3, 1, 3-1/2)


A rasterization algorithm19
A Rasterization Algorithm

(7, -1/7, 1/2)

Scan

line

(8, -1/7, 11-9/14)


A rasterization algorithm20
A Rasterization Algorithm

(7, -1/7, 1/2)

Scan

line

(8, -1/7, 11-9/14)


A rasterization algorithm21
A Rasterization Algorithm

(7, -1/7, 5/14)

Scan

line

(8, -1/7, 11-1/2)


A rasterization algorithm22
A Rasterization Algorithm

(7, -1/7, 5/14)

Scan

line

(8, -1/7, 11-1/2)


A rasterization algorithm23
A Rasterization Algorithm

(8, -2/3, 5-5/6)

(8, 1/3, 6-1/6)

(7, -1/7, 3/14)

Scan

line

(8, -1/7, 11-5/14)


A rasterization algorithm24
A Rasterization Algorithm

(8, -2/3, 5-2/3)

(8, 1/3, 6-1/6)

(7, -1/7, 3/14)

Scan

line

(8, -1/7, 11-5/14)


A rasterization algorithm25
A Rasterization Algorithm

(8, -2/3, 5)

(8, 1/3, 6-1/2)

(7, -1/7, 1/14)

Scan

line

(8, -1/7, 11-3/14)


A rasterization algorithm26
A Rasterization Algorithm

(8, -2/3, 5)

(8, 1/3, 6-1/2)

(7, -1/7, 1/14)

Scan

line

(8, -1/7, 11-3/14)


A rasterization algorithm27
A Rasterization Algorithm

(8, -2/3, 5)

(8, 1/3, 6-1/2)

(8, 4, 2)

Scan

line

(8, -1/7, 11-3/14)


A rasterization algorithm28
A Rasterization Algorithm

(8, -2/3, 5)

(8, 1/3, 6-1/2)

(8, 4, 2)

Scan

line

(8, -1/7, 11-3/14)






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