Dr' Scott Schaefer

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Scan Conversion of Lines

• Dr. Scott Schaefer

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Displays Cathode Ray Tube
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Displays Liquid Crystal Displays
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Displays Plasma
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Displays Pixels
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Problem

• Given two points (P, Q) on the screen (with
  integer coordinates) determine which pixels
  should be drawn to display a unit width line

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Problem

• Given two points (P, Q) on the screen (with
  integer coordinates) determine which pixels
  should be drawn to display a unit width line

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Problem

• Given two points (P, Q) on the screen (with
  integer coordinates) determine which pixels
  should be drawn to display a unit width line

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Special Lines - Horizontal
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Special Lines - Horizontal

• Increment x by 1, keep y constant

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Special Lines - Vertical
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Special Lines - Vertical

• Keep x constant, increment y by 1

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Special Lines - Diagonal
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Special Lines - Diagonal

• Increment x by 1, increment y by 1

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How about Arbitrary Lines?
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Arbitrary Lines
Slope-intercept equation for a line
m slope
b y-intercept
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Arbitrary Lines
Slope-intercept equation for a line How can
we compute the equation from (xL,yL), (xH,yH)?
m slope
b y-intercept
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Arbitrary Lines
Slope-intercept equation for a line How can
we compute the equation from (xL,yL), (xH,yH)?
m slope
b y-intercept
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Arbitrary Lines

• Assume
• Other lines by swapping x, y or negating
• Take steps in x and determine where to fill y

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Simple Algorithm

• Start from (xL, yL) and draw to (xH, yH)
• where xLlt xH
• ( x0, y0 ) ( xL, yL )
• For ( i 0 i lt xH-xL i )
• DrawPixel ( xi, Round ( yi ) )
• xi1 xi 1
• yi1 m xi1 b

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Simple Algorithm

• Start from (xL, yL) and draw to (xH, yH)
• where xLlt xH
• ( x0, y0 ) ( xL, yL )
• For ( i 0 i lt xH-xL i )
• DrawPixel ( xi, Round ( yi ) )
• xi1 xi 1
• yi1 m ( xi 1 ) b

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Simple Algorithm

• Start from (xL, yL) and draw to (xH, yH)
• where xLlt xH
• ( x0, y0 ) ( xL, yL )
• For ( i 0 i lt xH-xL i )
• DrawPixel ( xi, Round ( yi ) )
• xi1 xi 1
• yi1 m xi m b

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Simple Algorithm

• Start from (xL, yL) and draw to (xH, yH)
• where xLlt xH
• ( x0, y0 ) ( xL, yL )
• For ( i 0 i lt xH-xL i )
• DrawPixel ( xi, Round ( yi ) )
• xi1 xi 1
• yi1 yi m

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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Example
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Simple Algorithm - Problems

• Floating point operations
• Rounding
• Can we do better?
• ( x0, y0 ) ( xL, yL )
• For ( i 0 i lt xH-xL i )
• DrawPixel ( xi, Round ( yi ) )
• xi1 xi 1
• yi1 yi m

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Midpoint Algorithm

• Given a point just drawn, determine whether we
  move E or NE on next step

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Midpoint Algorithm

• Given a point just drawn, determine whether we
  move E or NE on next step
• Is the line above or below ?
• Below move E
• Above move NE

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Midpoint Algorithm Implicit Forms

• How do we tell if a point is above or below the
  line?

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Midpoint Algorithm Implicit Forms

• How do we tell if a point is above or below the
  line?

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Midpoint Algorithm Implicit Forms

• How do we tell if a point is above or below the
  line?

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Midpoint Algorithm Implicit Forms

• How do we tell if a point is above or below the
  line?

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Midpoint Algorithm Implicit Forms

• How do we tell if a point is above or below the
  line?

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Midpoint Algorithm Implicit Forms

• How do we tell if a point is above or below the
  line?

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Midpoint Algorithm Implicit Forms

• How do we tell if a point is above or below the
  line?

Properties f(x,y)0 (x,y) on the line f(x,y)lt0
(x,y) below the line f(x,y)gt0 (x,y) above the
line
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Midpoint Algorithm Implicit Forms
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Midpoint Algorithm Implicit Forms
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Midpoint Algorithm Implicit Forms
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Midpoint Algorithm Implicit Forms
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Midpoint Algorithm Implicit Forms
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Midpoint Algorithm Implicit Forms
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Midpoint Algorithm

• Need value of to determine
  E or NE
• Build incremental algorithm
• Assume we have value of
• Find value of if E chosen
• Find value of if NE chosen

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Midpoint Algorithm

• If E was chosen, find

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Midpoint Algorithm

• If E was chosen, find

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Midpoint Algorithm

• If NE was chosen, find

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Midpoint Algorithm

• If NE was chosen, find

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Midpoint Algorithm

• What about starting value?

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Midpoint Algorithm

• What about starting value?

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Midpoint Algorithm

• What about starting value?

is on the line!
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Midpoint Algorithm

• What about starting value?

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Midpoint Algorithm

• What about starting value?

Multiplying coefficients by 2 doesnt change sign
of f
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Midpoint Algorithm Summary

• xxL yyL
• dxxH - xL dyyH - yL
• sumdx - 2dy
• draw(x,y)
• while ( x lt xH)
• if ( sum lt 0 )
• sum 2dx
• y
• x
• sum - 2dy
• draw(x,y)

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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm Example
xxL yyL dxxH - xL dyyH -
yL sumdx - 2dy draw(x,y) while ( x lt xH) if (
sum lt 0 ) sum 2dx y x sum -
2dy draw(x,y)
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Midpoint Algorithm

• Only integer operations
• Exactly the same as the more commonly found
  Bresenhams line drawing algorithm
• Extends to other types of shapes (circles)

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Circle, ellipse, etc

• Need implicit forms
• - circle
• - ellipse
• Equations for incremental calculations
• Initial positions
• More details in section 3.8-3.11 of the textbook