ECM2410: Graphics and Animation

Rasterisation

Part I: Simple Lines

Rasterisation 1/16

Rendering a scene

User space Device space

Compose Render

Affine transformation

Transform

Compose from primitives (lines, ellipses, etc) in user space.

Transform affine transformations:lines→ lines;ellipses→ ellipses

Render primitives on device.Rasterisation Line Rasterisation 2/16

Rendering a scene

User space Device space

Compose Render

Affine transformation

Transform

Compose from primitives (lines, ellipses, etc) in user space.

Transform affine transformations:lines→ lines;ellipses→ ellipses

Render primitives on device.Rasterisation Line Rasterisation 2/16

Rendering a scene

User space Device space

Compose Render

Affine transformation

Transform

Compose from primitives (lines, ellipses, etc) in user space.

Transform affine transformations:lines→ lines;ellipses→ ellipses

Render primitives on device.Rasterisation Line Rasterisation 2/16

Rendering a scene

User space Device space

Compose Render

Affine transformation

Transform

Compose from primitives (lines, ellipses, etc) in user space.

Transform affine transformations:lines→ lines;ellipses→ ellipses

Render primitives on device.Rasterisation Line Rasterisation 2/16

Rendering a scene

User space Device space

Compose Render

Affine transformation

Transform

Compose from primitives (lines, ellipses, etc) in user space.

Transform affine transformations:lines→ lines;ellipses→ ellipses

Render primitives on device.Rasterisation Line Rasterisation 2/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Rasterisation

Vector displaysPen plottersTektronix cathode ray tubes

Raster displaysMonitorsTelevisionsLaser, ink-jet, dot matrix printersLiquid crystal displaysArrays of LEDs

Rasterisation is the conversion of ideal, mathematicalgraphics primitives onto raster displays.

Drawing linesDrawing circles and ellipsesFilling areas

Hearn & Baker, Chapter 2 for displays.Rasterisation Line Rasterisation 3/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Framebuffer

Raster display is a 2D array of picture elements: pixelsFramebuffer: in-memory image of the display.

00

1 2

12

Screen coordinates of pixelsare centered at the integers.

Pixels are individually set orcleared (random access).Sequential readout byphysical device.

Graphical objects arecontinuous.Framebuffer is discrete.

Rasterisation / Framebuffer Line Rasterisation 4/16

Rasterisation

Rasterisationis the conversion of ideal, mathematical graphicsprimitives onto raster displays.

Efficiencywe seek efficient algorithms for computing thediscrete approximation to continuous description.

Scan Conversionrasterisation line by line of the raster display.

Rasterisation / Framebuffer Line Rasterisation 5/16

Rasterisation

Rasterisationis the conversion of ideal, mathematical graphicsprimitives onto raster displays.

Efficiencywe seek efficient algorithms for computing thediscrete approximation to continuous description.

Scan Conversionrasterisation line by line of the raster display.

Rasterisation / Framebuffer Line Rasterisation 5/16

Rasterisation

Rasterisationis the conversion of ideal, mathematical graphicsprimitives onto raster displays.

Efficiencywe seek efficient algorithms for computing thediscrete approximation to continuous description.

Scan Conversionrasterisation line by line of the raster display.

Rasterisation / Framebuffer Line Rasterisation 5/16

Rasterisation

Rasterisationis the conversion of ideal, mathematical graphicsprimitives onto raster displays.

Efficiencywe seek efficient algorithms for computing thediscrete approximation to continuous description.

Scan Conversionrasterisation line by line of the raster display.

Rasterisation / Framebuffer Line Rasterisation 5/16

Rasterisation

Rasterisationis the conversion of ideal, mathematical graphicsprimitives onto raster displays.

Efficiencywe seek efficient algorithms for computing thediscrete approximation to continuous description.

Scan Conversionrasterisation line by line of the raster display.

Rasterisation / Framebuffer Line Rasterisation 5/16

Rasterisation

Rasterisationis the conversion of ideal, mathematical graphicsprimitives onto raster displays.

Efficiencywe seek efficient algorithms for computing thediscrete approximation to continuous description.

Scan Conversionrasterisation line by line of the raster display.

Rasterisation / Framebuffer Line Rasterisation 5/16

Rasterisation

Rasterisationis the conversion of ideal, mathematical graphicsprimitives onto raster displays.

Efficiencywe seek efficient algorithms for computing thediscrete approximation to continuous description.

Scan Conversionrasterisation line by line of the raster display.

Rasterisation / Framebuffer Line Rasterisation 5/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Line Segments

Problem: Draw a line from (Xstart ,Ystart) to (Xend ,Yend )Simplified: Draw a line from (0, 0) to (Xend ,Yend )

(Xend > 0 and Yend > 0)

(0,0)

(19,4)

Grid marks pixel centres

Example: Line from (0, 0) to (Xend ,Yend )= (19, 4)

Lines Line Rasterisation 6/16

Which pixels to illuminate?

Illuminate every pixel the line intersects

Illuminate the vertically closest pixel

Lines / Which Pixel? Line Rasterisation 7/16

Which pixels to illuminate?

Illuminate every pixel the line intersects

Illuminate the vertically closest pixel

Lines / Which Pixel? Line Rasterisation 7/16

Which pixels to illuminate?

Illuminate every pixel the line intersects

Illuminate the vertically closest pixel

Lines / Which Pixel? Line Rasterisation 7/16

Which pixels to illuminate?

Illuminate every pixel the line intersects

Illuminate the vertically closest pixel

Lines / Which Pixel? Line Rasterisation 7/16

Which pixels to illuminate?

Illuminate every pixel the line intersects

Illuminate the vertically closest pixel

Lines / Which Pixel? Line Rasterisation 7/16

Gradient

Gradient of line

y = m x + c

m =Yend − Ystart

Xend − Xstart

Simplified line from (0, 0) to (Xend ,Yend ):

y = m x , m =Yend

Xend

Lines / Gradient Line Rasterisation 8/16

Gradient

Gradient of line

y = m x + c

m =Yend − Ystart

Xend − Xstart

Simplified line from (0, 0) to (Xend ,Yend ):

y = m x , m =Yend

Xend

Lines / Gradient Line Rasterisation 8/16

Gradient

Gradient of line

y = m x + c

m =Yend − Ystart

Xend − Xstart

Simplified line from (0, 0) to (Xend ,Yend ):

y = m x , m =Yend

Xend

Lines / Gradient Line Rasterisation 8/16

Gradient

Gradient of line

y = m x + c

m =Yend − Ystart

Xend − Xstart

Simplified line from (0, 0) to (Xend ,Yend ):

y = m x , m =Yend

Xend

Lines / Gradient Line Rasterisation 8/16

Gradient

Gradient of line

y = m x + c

m =Yend − Ystart

Xend − Xstart

Simplified line from (0, 0) to (Xend ,Yend ):

y = m x , m =Yend

Xend

Lines / Gradient Line Rasterisation 8/16

Gradient

Gradient of line

y = m x + c

m =Yend − Ystart

Xend − Xstart

Simplified line from (0, 0) to (Xend ,Yend ):

y = m x , m =Yend

Xend

Lines / Gradient Line Rasterisation 8/16

Gradient

Gradient of line

y = m x + c

m =Yend − Ystart

Xend − Xstart

Simplified line from (0, 0) to (Xend ,Yend ):

y = m x , m =Yend

Xend

Lines / Gradient Line Rasterisation 8/16

Naïve Midpoint Algorithm

Illuminate pixel vertically closest to the line

4/198/19 16/19

12/19

double m = Yend / Xend;int x = 0;double y = 0.0;

while ( x <= Xend ) {y = m * x;iy = round(y);setpixel(x, iy);x++;

}

DDA Digital Differential Analyser. Hearn & Baker, p86.

Floating point operations and round() are computationallyexpensive.

Lines / Naïve Algorithm Line Rasterisation 9/16

Naïve Midpoint Algorithm

Illuminate pixel vertically closest to the line

4/198/19 16/19

12/19

double m = Yend / Xend;int x = 0;double y = 0.0;

while ( x <= Xend ) {y = m * x;iy = round(y);setpixel(x, iy);x++;

}

DDA Digital Differential Analyser. Hearn & Baker, p86.

Floating point operations and round() are computationallyexpensive.

Lines / Naïve Algorithm Line Rasterisation 9/16

Naïve Midpoint Algorithm

Illuminate pixel vertically closest to the line

4/198/19 16/19

12/19

double m = Yend / Xend;int x = 0;double y = 0.0;

while ( x <= Xend ) {y = m * x;iy = round(y);setpixel(x, iy);x++;

}

DDA Digital Differential Analyser. Hearn & Baker, p86.

Floating point operations and round() are computationallyexpensive.

Lines / Naïve Algorithm Line Rasterisation 9/16

Naïve Midpoint Algorithm

Illuminate pixel vertically closest to the line

4/198/19 16/19

12/19

double m = Yend / Xend;int x = 0;double y = 0.0;

while ( x <= Xend ) {y = m * x;iy = round(y);setpixel(x, iy);x++;

}

DDA Digital Differential Analyser. Hearn & Baker, p86.

Floating point operations and round() are computationallyexpensive.

Lines / Naïve Algorithm Line Rasterisation 9/16

Naïve Midpoint Algorithm

Illuminate pixel vertically closest to the line

4/198/19 16/19

12/19

double m = Yend / Xend;int x = 0;double y = 0.0;

while ( x <= Xend ) {y = m * x;iy = round(y);setpixel(x, iy);x++;

}

DDA Digital Differential Analyser. Hearn & Baker, p86.

Floating point operations and round() are computationallyexpensive.

Lines / Naïve Algorithm Line Rasterisation 9/16

Naïve Midpoint Algorithm

Illuminate pixel vertically closest to the line

4/198/19 16/19

12/19

double m = Yend / Xend;int x = 0;double y = 0.0;

while ( x <= Xend ) {y = m * x;iy = round(y);setpixel(x, iy);x++;

}

DDA Digital Differential Analyser. Hearn & Baker, p86.

Floating point operations and round() are computationallyexpensive.

Lines / Naïve Algorithm Line Rasterisation 9/16

Remove round()

Line coherence: As x 7→ x + 1 then y 7→ y + m

4/1916/19

12/19

8/19

At each step: move right one pixeland up by 4

19 .

midpoint indicates pixel centreabove illuminated pixel.

When y crosses verticalmidpoint between two centresmove up and compute newmidpoint target.

double m = Yend / Xend;int x = 0;int iy = 0;double y = 0.0;double midpoint = 0.5;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 1.0;

}}

Lines / Eliminating Floats Line Rasterisation 10/16

Remove round()

Line coherence: As x 7→ x + 1 then y 7→ y + m

4/1916/19

12/19

8/19

At each step: move right one pixeland up by 4

19 .

midpoint indicates pixel centreabove illuminated pixel.

When y crosses verticalmidpoint between two centresmove up and compute newmidpoint target.

double m = Yend / Xend;int x = 0;int iy = 0;double y = 0.0;double midpoint = 0.5;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 1.0;

}}

Lines / Eliminating Floats Line Rasterisation 10/16

Remove round()

Line coherence: As x 7→ x + 1 then y 7→ y + m

4/1916/19

12/19

8/19

At each step: move right one pixeland up by 4

19 .

midpoint indicates pixel centreabove illuminated pixel.

When y crosses verticalmidpoint between two centresmove up and compute newmidpoint target.

double m = Yend / Xend;int x = 0;int iy = 0;double y = 0.0;double midpoint = 0.5;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 1.0;

}}

Lines / Eliminating Floats Line Rasterisation 10/16

Remove round()

Line coherence: As x 7→ x + 1 then y 7→ y + m

4/1916/19

12/19

8/19

At each step: move right one pixeland up by 4

19 .

midpoint indicates pixel centreabove illuminated pixel.

When y crosses verticalmidpoint between two centresmove up and compute newmidpoint target.

double m = Yend / Xend;int x = 0;int iy = 0;double y = 0.0;double midpoint = 0.5;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 1.0;

}}

Lines / Eliminating Floats Line Rasterisation 10/16

Remove round()

Line coherence: As x 7→ x + 1 then y 7→ y + m

4/1916/19

12/19

8/19

At each step: move right one pixeland up by 4

19 .

midpoint indicates pixel centreabove illuminated pixel.

When y crosses verticalmidpoint between two centresmove up and compute newmidpoint target.

double m = Yend / Xend;int x = 0;int iy = 0;double y = 0.0;double midpoint = 0.5;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 1.0;

}}

Lines / Eliminating Floats Line Rasterisation 10/16

Remove round()

Line coherence: As x 7→ x + 1 then y 7→ y + m

4/1916/19

12/19

8/19

At each step: move right one pixeland up by 4

19 .

midpoint indicates pixel centreabove illuminated pixel.

When y crosses verticalmidpoint between two centresmove up and compute newmidpoint target.

double m = Yend / Xend;int x = 0;int iy = 0;double y = 0.0;double midpoint = 0.5;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 1.0;

}}

Lines / Eliminating Floats Line Rasterisation 10/16

Remove round()

Line coherence: As x 7→ x + 1 then y 7→ y + m

4/1916/19

12/19

8/19

At each step: move right one pixeland up by 4

19 .

midpoint indicates pixel centreabove illuminated pixel.

When y crosses verticalmidpoint between two centresmove up and compute newmidpoint target.

double m = Yend / Xend;int x = 0;int iy = 0;double y = 0.0;double midpoint = 0.5;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 1.0;

}}

Lines / Eliminating Floats Line Rasterisation 10/16

Integers only

m =Yend

Xendy = m x =

Yend xXend

Multiply m, y and midpoint bytwice total change in x ;(i.e. by 2Xend )

m = 2 Yend y = 2 Yendx

2432

y=816

19 19 57midpoint19 57

int m = 2 * Yend;int x = 0;int iy = 0;int y = 0;int midpoint = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 2 * Xend;

}}

Lines / Eliminating Floats Line Rasterisation 11/16

Integers only

m =Yend

Xendy = m x =

Yend xXend

Multiply m, y and midpoint bytwice total change in x ;(i.e. by 2Xend )

m = 2 Yend y = 2 Yendx

2432

y=816

19 19 57midpoint19 57

int m = 2 * Yend;int x = 0;int iy = 0;int y = 0;int midpoint = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 2 * Xend;

}}

Lines / Eliminating Floats Line Rasterisation 11/16

Integers only

m =Yend

Xendy = m x =

Yend xXend

Multiply m, y and midpoint bytwice total change in x ;(i.e. by 2Xend )

m = 2 Yend y = 2 Yendx

2432

y=816

19 19 57midpoint19 57

int m = 2 * Yend;int x = 0;int iy = 0;int y = 0;int midpoint = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 2 * Xend;

}}

Lines / Eliminating Floats Line Rasterisation 11/16

Integers only

m =Yend

Xendy = m x =

Yend xXend

Multiply m, y and midpoint bytwice total change in x ;(i.e. by 2Xend )

m = 2 Yend y = 2 Yendx

2432

y=816

19 19 57midpoint19 57

int m = 2 * Yend;int x = 0;int iy = 0;int y = 0;int midpoint = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 2 * Xend;

}}

Lines / Eliminating Floats Line Rasterisation 11/16

Integers only

m =Yend

Xendy = m x =

Yend xXend

Multiply m, y and midpoint bytwice total change in x ;(i.e. by 2Xend )

m = 2 Yend y = 2 Yendx

2432

y=816

19 19 57midpoint19 57

int m = 2 * Yend;int x = 0;int iy = 0;int y = 0;int midpoint = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 2 * Xend;

}}

Lines / Eliminating Floats Line Rasterisation 11/16

Integers only

m =Yend

Xendy = m x =

Yend xXend

Multiply m, y and midpoint bytwice total change in x ;(i.e. by 2Xend )

m = 2 Yend y = 2 Yendx

2432

y=816

19 19 57midpoint19 57

int m = 2 * Yend;int x = 0;int iy = 0;int y = 0;int midpoint = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 2 * Xend;

}}

Lines / Eliminating Floats Line Rasterisation 11/16

Integers only

m =Yend

Xendy = m x =

Yend xXend

Multiply m, y and midpoint bytwice total change in x ;(i.e. by 2Xend )

m = 2 Yend y = 2 Yendx

2432

y=816

19 19 57midpoint19 57

int m = 2 * Yend;int x = 0;int iy = 0;int y = 0;int midpoint = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y += m;

if (y > midpoint) {iy++;midpoint += 2 * Xend;

}}

Lines / Eliminating Floats Line Rasterisation 11/16

Bresenham’s algorithm

Eliminate midpoint by adjusting y tostart at the midpoint and decreasingto 0.

y=19

3 −5 => 332511

Hearn & Baker, page 88

int m = 2 * Yend;int x = 0;int iy = 0;int y = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y -= m;

if (y < 0) {iy++;y += 2 * Xend;

}}

Lines / Bresenham’s Algorithm Line Rasterisation 12/16

Bresenham’s algorithm

Eliminate midpoint by adjusting y tostart at the midpoint and decreasingto 0.

y=19

3 −5 => 332511

Hearn & Baker, page 88

int m = 2 * Yend;int x = 0;int iy = 0;int y = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y -= m;

if (y < 0) {iy++;y += 2 * Xend;

}}

Lines / Bresenham’s Algorithm Line Rasterisation 12/16

Bresenham’s algorithm

Eliminate midpoint by adjusting y tostart at the midpoint and decreasingto 0.

y=19

3 −5 => 332511

Hearn & Baker, page 88

int m = 2 * Yend;int x = 0;int iy = 0;int y = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y -= m;

if (y < 0) {iy++;y += 2 * Xend;

}}

Lines / Bresenham’s Algorithm Line Rasterisation 12/16

Bresenham’s algorithm

Eliminate midpoint by adjusting y tostart at the midpoint and decreasingto 0.

y=19

3 −5 => 332511

Hearn & Baker, page 88

int m = 2 * Yend;int x = 0;int iy = 0;int y = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y -= m;

if (y < 0) {iy++;y += 2 * Xend;

}}

Lines / Bresenham’s Algorithm Line Rasterisation 12/16

Bresenham’s algorithm

Eliminate midpoint by adjusting y tostart at the midpoint and decreasingto 0.

y=19

3 −5 => 332511

Hearn & Baker, page 88

int m = 2 * Yend;int x = 0;int iy = 0;int y = Xend;

while (x <= Xend) {setpixel(x, iy);x++;y -= m;

if (y < 0) {iy++;y += 2 * Xend;

}}

Lines / Bresenham’s Algorithm Line Rasterisation 12/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Caveats

Needs adjustment for:

General starting pointm > 1

Negative slopes

Lines with different slope may have different brightness.Alleviated with anti-aliasing(give pixels different brightness).

References

Hearn & Baker, pp. 84-94Foley & van Dam

Lines / Bresenham’s Algorithm Line Rasterisation 13/16

Bresenham Circle Algorithm

Choose pixel withcentre closest tocircle.

Work on a singleoctant and drawothers by symmetry.

Can be written ininteger arithmetic.

Details in Hearn &Baker, Foley et al.

Lines / Bresenham Circle Algorithm Line Rasterisation 14/16

Bresenham Circle Algorithm

Choose pixel withcentre closest tocircle.

Work on a singleoctant and drawothers by symmetry.

Can be written ininteger arithmetic.

Details in Hearn &Baker, Foley et al.

Lines / Bresenham Circle Algorithm Line Rasterisation 14/16

Bresenham Circle Algorithm

Choose pixel withcentre closest tocircle.

Work on a singleoctant and drawothers by symmetry.

Can be written ininteger arithmetic.

Details in Hearn &Baker, Foley et al.

Lines / Bresenham Circle Algorithm Line Rasterisation 14/16

Bresenham Circle Algorithm

Choose pixel withcentre closest tocircle.

Work on a singleoctant and drawothers by symmetry.

Can be written ininteger arithmetic.

Details in Hearn &Baker, Foley et al.

Lines / Bresenham Circle Algorithm Line Rasterisation 14/16

Bresenham Circle Algorithm

Choose pixel withcentre closest tocircle.

Work on a singleoctant and drawothers by symmetry.

Can be written ininteger arithmetic.

Details in Hearn &Baker, Foley et al.

Lines / Bresenham Circle Algorithm Line Rasterisation 14/16

Bresenham Circle Algorithm

Choose pixel withcentre closest tocircle.

Work on a singleoctant and drawothers by symmetry.

Can be written ininteger arithmetic.

Details in Hearn &Baker, Foley et al.

Lines / Bresenham Circle Algorithm Line Rasterisation 14/16

Bresenham Circle Algorithm

Choose pixel withcentre closest tocircle.

Work on a singleoctant and drawothers by symmetry.

Can be written ininteger arithmetic.

Details in Hearn &Baker, Foley et al.

Lines / Bresenham Circle Algorithm Line Rasterisation 14/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Column Replication

Efficient, easily implemented.Perceived thickness varieswith slope.Gaps at joins.

Rectangular Pen

Ends are thicker than middle.Perceived thickness varieswith slope.Pixels are set multiple times.

Lines / Thick Primitives Line Rasterisation 15/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16

Thick Primitives

Circular Pen

Thickness independent of slope.

Pixels are set multiple times.

Can be made efficient by settingonly boundary pixels.

Boundary Fill

Fill the area between boundaries.

Polygon with edges parallel toprimitive.

Precise control over joins(butt, bevel, cap ends)

Lines, circles become polygons.

Expensive.

Lines / Thick Primitives Line Rasterisation 16/16