NOTE DDA Line Drawing Algorithm y y x

2 If the scan line is crossing a vertex, treat it as two intersection

If the scan line is crossing a vertex, treat it as two intersection points

Edge Table Add & Sort !! Active Edge List

Scan Line Polygon Filling Construct the Edge Table (ET) excluding horizontal edge; Active Edge

Boundary Polygon Filling • Proceed to Neighboring Pixels – 4 -Connected – 8 -Connected

Inside/Outside Test • Self-Intersections – Odd-Even rule – Nonzero winding number rule

Anti-aliasing - Supersampling • Subpixels increase resolution • Maximum intensity level: 9 or more

Anti-aliasing - Filtering Box Filter Cone Filter Gaussian Filter

Anti-aliasing – Area sampling 22 21 20 10 11 12

Transformation Matrices using Homogeneous Coordinates 2 D Affine Transformation Translation Rotation Scaling Shear Mirror

Transformation Matrices using Homogeneous Coordinates • Basic 2 D transformations as 3 x 3

Transformation Matrices using Homogeneous Coordinates • Reflection with respect to an axis X Axis

Transformation Matrices using Homogeneous Coordinates • Combined transformations – By matrix multiplication • Efficiency

• Pivot Point Rotation • Fixed Point Scaling

• Pivot Point Rotation y (0, 1) x (0, 0) yref= – 1

Point Clipping • Is the point (x, y) inside the clipping window? wy 2

Cohen-Sutherland Line Clipping • Classify some lines quickly – AND of bit codes of

Sutherland-Hodgeman Polygon Clipping • Clip to each window boundary at a time

Sutherland-Hodgeman Polygon Clipping • Clip to each window boundary at a time P 2

Viewport Transformation • Transform 2 D geometric primitives – From screen coordinate (projection coordinates)

Viewport Transformation • Window-to-viewport mapping wy 2 Window (wx, wy) wy 1 wx 2

Slides: 48

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NOTE

DDA Line Drawing Algorithm y y x x

Bresenham’s Line Drawing Algorithm

Scan Line Polygon Filling C B A

2 If the scan line is crossing a vertex, treat it as two intersection points 2 2

2 If the scan line is crossing a vertex, treat it as two intersection points Except the slops of neighboring edges are increasing/decreasing monotonically. 1 2 2 2

If the scan line is crossing a vertex, treat it as two intersection points Except the slops of neighboring edges are increasing/decreasing monotonically. Separate these edges Is it really necessary?

Edge Table Active Edge List

Edge Table Add & Sort !! Active Edge List

Edge Table Sort !! Active Edge List

Edge Table Add & Sort !! Active Edge List

Edge Table Active Edge List

Edge Table Sort !! Active Edge List

Edge Table Active Edge List

Edge Table Empty Active Edge List

Scan Line Polygon Filling Construct the Edge Table (ET) excluding horizontal edge; Active Edge List (AEL) = null; for y = Ymin to Ymax Merge-sort ET[y] into AEL by x value Fill between pairs of x in AEL for each edge in AEL if edge. ymax = y remove edge from AEL else edge. x = edge. x + dx/dy sort AEL by x value end scan_fill

Boundary Polygon Filling • Proceed to Neighboring Pixels – 4 -Connected – 8 -Connected

Inside/Outside Test • Self-Intersections – Odd-Even rule – Nonzero winding number rule

Anti-aliasing - Supersampling • Subpixels increase resolution • Maximum intensity level: 9 or more 1 22 21 20 10 11 12 2 4 2 1

Anti-aliasing - Filtering Box Filter Cone Filter Gaussian Filter

Anti-aliasing – Area sampling 22 21 20 10 11 12

Transformation Matrices using Homogeneous Coordinates 2 D Affine Transformation Translation Rotation Scaling Shear Mirror

Transformation Matrices using Homogeneous Coordinates • Basic 2 D transformations as 3 x 3 matrices Translation Rotation Scaling Shear

Transformation Matrices using Homogeneous Coordinates • Reflection with respect to an axis X Axis y Y Axis 1 1 2 3 2’ 3’ 1’ 2 x y 3 Origin y 1’ 3’ 2’ x 1’ 3’ 3 1 2 2’ x

Transformation Matrices using Homogeneous Coordinates • Combined transformations – By matrix multiplication • Efficiency with pre-multiplication – Matrix multiplication is associative

• Pivot Point Rotation • Fixed Point Scaling

• Pivot Point Rotation y (0, 1) x (0, 0) yref= – 1 (1, 0) y (0, 1) (0, 0) xref= – 1 hx= 0. 5 (1, 1) yref= -1 y (1, 1) (0. 5, 0) yref= – 1 y hy= 0. 5 (1, 1) xref= -1 (0, 1. 5) (0, 0. 5) (1, 0) x xref= – 1 (2, 1) x (1. 5, 0) (1, 2) (1, 1) x

Point Clipping • Is the point (x, y) inside the clipping window? wy 2 Inside = (x>=wx 1) && (x<=wx 2) && (x, y) (y>=wy 1) && (y<=wy 2); wy 1 wx 2

Cohen-Sutherland Line Clipping • Classify some lines quickly – AND of bit codes of two endpoints (0 if not fully outside) P 7 0001 1001 0101 Bit 4 P 1 P 4 1000 P 2 1010 P 5 Bit 1 P 3 P 8 0100 0000 P 6 0010 P 9 0110 Bit 2 Bit 3

Sutherland-Hodgeman Polygon Clipping • Clip to each window boundary at a time

Sutherland-Hodgeman Polygon Clipping • Clip to each window boundary at a time P 2 P 1 Window Boundary Inside P 5 P 3 P 4 Outside

Sutherland-Hodgeman Polygon Clipping • Clip to each window boundary at a time P 2 P 1 Window Boundary P’ P” P 5 P 3 P 4 Inside Outside

Sutherland-Hodgeman Polygon Clipping • Clip to each window boundary at a time P 2 Window Boundary P’ P 1 P” Inside Outside

Viewport Transformation • Transform 2 D geometric primitives – From screen coordinate (projection coordinates) – To image coordinate (device coordinates) Screen Image Viewport

Viewport Transformation • Window-to-viewport mapping wy 2 Window (wx, wy) wy 1 wx 2 Screen Coordinates vy 2 Viewport (vx, vy) vy 1 vx 2 Image Coordinates vx = vx 1 + (wx – wx 1) * (vx 2 – vx 1) / (wx 2 – wx 1); vy = vy 1 + (wy – wy 1) * (vy 2 – vy 1) / (wy 2 – wy 1);