Graphics6 bresenham circlesandpolygons

Computer Graphics 4: Bresenham Line Drawing Algorithm, Circle Drawing & Polygon Filling

Contents ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Bresenham Line Algorithm ,[object Object],[object Object],Jack Bresenham worked for 27 years at IBM before entering academia. Bresenham developed his famous algorithms at IBM in the early 1960s

The Big Idea ,[object Object],2 3 4 5 2 4 3 5 For example, from position (2, 3) we have to choose between (3, 3) and (3, 4) We would like the point that is closer to the original line ( x k , y k ) ( x k +1, y k ) ( x k +1, y k +1)

Deriving The Bresenham Line Algorithm ,[object Object],The y coordinate on the mathematical line at x k + 1 is: y y k y k+ 1 x k + 1 d lower d upper

[object Object],[object Object],[object Object],Deriving The Bresenham Line Algorithm (cont…)

[object Object],[object Object],Deriving The Bresenham Line Algorithm (cont…)

The Bresenham Line Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object]

The Bresenham Line Algorithm (cont…) ,[object Object],[object Object],[object Object]

Bresenham Example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Bresenham Example (cont…) 17 16 15 14 13 12 11 10 18 29 27 26 25 24 23 22 21 20 28 30 k p k (x k+1 ,y k+1 ) 0 1 2 3 4 5 6 7 8 9

Bresenham Exercise ,[object Object]

Bresenham Exercise (cont…) 17 16 15 14 13 12 11 10 18 29 27 26 25 24 23 22 21 20 28 30 k p k (x k+1 ,y k+1 ) 0 1 2 3 4 5 6 7 8

Bresenham Line Algorithm Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

A Simple Circle Drawing Algorithm ,[object Object],[object Object],[object Object]

A Simple Circle Drawing Algorithm (cont…)

A Simple Circle Drawing Algorithm (cont…) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Eight-Way Symmetry ,[object Object],(x, y) (y, x) (y, -x) (x, -y) (-x, -y) (-y, -x) (-y, x) (-x, y)

Mid-Point Circle Algorithm ,[object Object],[object Object],The mid-point circle algorithm was developed by Jack Bresenham, who we heard about earlier. Bresenham’s patent for the algorithm can be viewed here .

Mid-Point Circle Algorithm (cont…) ,[object Object],[object Object],[object Object],[object Object],( x k +1, y k ) ( x k +1, y k -1) ( x k , y k )

Mid-Point Circle Algorithm (cont…) ,[object Object],[object Object],[object Object]

Mid-Point Circle Algorithm (cont…) ,[object Object],[object Object],[object Object],[object Object]

The Mid-Point Circle Algorithm ,[object Object],[object Object],[object Object],[object Object]

The Mid-Point Circle Algorithm (cont…) ,[object Object],[object Object],[object Object],[object Object]

Mid-Point Circle Algorithm Example ,[object Object]

Mid-Point Circle Algorithm Example (cont…) 9 7 6 5 4 3 2 1 0 8 9 7 6 5 4 3 2 1 0 8 10 10 k p k (x k+1 ,y k+1 ) 2x k+1 2y k+1 0 1 2 3 4 5 6

Mid-Point Circle Algorithm Exercise ,[object Object]

Mid-Point Circle Algorithm Example (cont…) k p k (x k+1 ,y k+1 ) 2x k+1 2y k+1 0 1 2 3 4 5 6 7 8 9 10 11 12 9 7 6 5 4 3 2 1 0 8 9 7 6 5 4 3 2 1 0 8 10 10 13 12 11 14 15 13 12 14 11 16 15 16

Mid-Point Circle Algorithm Summary ,[object Object],[object Object],[object Object]

Filling Polygons ,[object Object],[object Object],[object Object]

Scan-Line Polygon Fill Algorithm 2 4 6 8 10 Scan Line 0 2 4 6 8 10 12 14 16

Scan-Line Polygon Fill Algorithm ,[object Object],[object Object],[object Object],[object Object]

Scan-Line Polygon Fill Algorithm (cont…)

Line Drawing Summary ,[object Object],[object Object],[object Object],[object Object]

Mid-Point Circle Algorithm (cont…) 6 2 3 4 1 5 4 3

Mid-Point Circle Algorithm (cont…) M 6 2 3 4 1 5 4 3

Blank Grid 9 7 6 5 4 3 2 1 0 8 9 7 6 5 4 3 2 1 0 8 10 10

Graphics6 bresenham circlesandpolygons

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