1. Chapter 3 Graphics Output Primitives
Computer Graphics
Chapter 3
Graphics Output Primitives

2. Display screen과 Frame Buffer
Frame buffer Display screen
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3. Computer Vision & Pattern Recognition Lab.
1. Point
Point plotting : converting a single coordinate position
setPixel(x,y)
getPixel(x,y,color)
addr(x,y) = addr(0,0) + y(xmax+1) +x
addr(x+1,y) = addr(x,y) + 1
addr(x, y+1) = addr(x,y) + (xmax+1)
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4. Computer Vision & Pattern Recognition Lab.
2. Line
Line drawing
Calculating intermediate positions along the line path between two specified endpoint positions
Output device is directed to fill in these positions between the endpoints
Digital devices
- Plotting discrete points between the two endpoints
⇒ jaggies
- Calculate from the equation of the line
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5. 3. Line-Drawing Algorithms
Problem: Given start and end points,
     -. Find sequence of pixels (integer coor.) as close to ideal line as possible (Given point drawing software)
     -. Issues: Correctness(accuracy) and Effectiveness(Speed)
Slope-intercept equation
For each x, compute y (if m<1). or
For each y, compute x (if m>1).
=> Floating point multiplication, addition, and rounding
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6. 3. Line-Drawing Algorithms
DDA Algorithm
Digital Differential Analyzer
Scan-conversion line algorithm
Calculate either ∆x or ∆y
Left endpoint → right endpoint
⇒ slope ≤ 1 ; yk+1 = yk + m (∆x=1)
⇒ slope > 1 ; xk+1 = xk + 1/m(∆y=1)
Right endpoint → left endpoint
⇒ slope ≤ 1 ; yk+1 = yk - m (∆x=-1)
⇒ slope > 1 ; xk+1 = xk - 1/m(∆y=-1)
Floating point addition & rounding
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7. 3. Line-Drawing Algorithms
Midpoint algorithm (Bresenham’s line algorithm)
An accurate and efficient raster line-generating algorithm
Use only incremental integer calculations that can be adapted to display circles and curves
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8. 3. Line-Drawing Algorithms
Assume : 0 < m < 1
Given (xk, yk), choose NE or E 
⇒  Is M below the line ?
Let
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9. 3. Line-Drawing Algorithms
Let decision variable
Approach:
           (1) test the sign of pk
           (2) select E or NE
           (3) update pk+1 using pk
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10. 3. Line-Drawing Algorithms
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11. 3. Line-Drawing Algorithms
To avoid , let
only integer addition
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12. 3. Line-Drawing Algorithms
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13. 3. Line-Drawing Algorithms
Note:
-. Generalization
-. Special case can be handled separately
Horizontal lines (Δy = 0)
Vertical lines (Δx = 0)
Diagonal lines with |Δx| = |Δy|
-. H/W dependent address generation ⇒ fast
-. Additional Issues
Use of symmetry ⇒ faster      
Independence of endpoint order       
Anti-aliasing
Parallel algorithm
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14. 4. Circle generating algorithms
Problem: Given center & radius, generate a circle.
Integer coordinates
Accuracy and Efficiency
Properties of circles
Center : (xc, yc), distance : r
(x - xc)2 + (y - yc)2 = r2
y = yc ± √ r2 – (x - xc)2
Multiplication, square root
Problem
At each step, considerable computation
Unequally spacing
Another way to eliminate the unequal spacing
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15. 4. Circle generating algorithms
Parametric polar form
x = xc + r·cosθ
y = yc + r·sinθ
Multiplication, trigonometric calculation
For a continuous boundary,
set the angular step size at 1/r
Considering the symmetry of circles
Quadrant
Octants
Position (x, y) on a one-eighth circle sector is mapped into
the seven circle points in the other octants of the xy plane
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16. 4. Circle generating algorithms
Midpoint circle algorithm
Problem: Given center & radius, generate a circle.
Assume: center at the origin
Consider only 2nd octant and use 8-way symmetry.
Given (xk, yk), choose E (xk+1, yk) or SE (xk+1, yk-1)
⇒  Is Midpoint inside the circle?
fcirc(x, y) = x2 + y2 – r2
fcirc(x, y) = 0, if (x, y) is on the circle boundary
< 0, if (x, y) is inside the circle boundary
> 0, if (x, y) is outside the circle boundary
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17. 4. Circle generating algorithms
Decision function : pk = fcirc(xk+1, yk-1/2)
Approach : (1) test the sign of pk
       (2) select E or SE
       (3) update pk+1 using pk
pk = fcirc (xk+1, yk-1/2) = (xk+1)2 + (yk-1/2)2 – r2
If pk<0, select E (xk+1= xk +1, yk +1 = yk )
pk+1 = fcirc (xk +2, yk-1/2)
= (xk+2)2 + (yk-1/2)2 – r2
= pk+ 2xk + 3 <= (xk = xk+1 -1)
= pk+ 2xk
Otherwise, select SE (xk+1= xk +1, yk +1 = yk -1)
pk+1 = fcirc (xk+2, yk-3/2) = (xk+2)2 + (yk-3/2)2 – r2
= xk2 + 4xk yk2 – 3yk + 9/4 – r2
= pk+ 2xk – 2yk + 5 <= (xk = xk+1–1, yk = yk+1+1)
= pk+ 2(xk +1 – yk +1 ) + 1
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18. 4. Circle generating algorithms
Initial decision function : p0 = fcirc(1, r-1/2)
= 1 + (r -1/2)2 – r2
= 5/4 – r
If r is specified as an integer, p0 = 1- r
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19. Computer Vision & Pattern Recognition Lab.
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20. 5. Ellipse generating algorithms
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21. Computer Vision & Pattern Recognition Lab.
6. Other Curves
Conic section
Ax2 + By2 + Cxy + Dx + Ey + F = 0
B – 4AC < 0 : generates an ellipse
= 0 : generates a parabola
> 0 : generates s hyperbola
Polynomials and spline curves
Polynomial function of n-th degree in x
y = ∑akxk
Useful in a number of graphics applications
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22. 7. Pixel addressing & object geometry
Maintaining geometric properties of displayed objects
Pixel area : referenced by the integer coordinates of
its lower left corner
Adjust the length of the displayed line by omitting one of the endpoint pixel
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23. 7. Pixel addressing & object geometry
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24. 8. Filled-Area Primitives
Inside-outside tests
Odd-even rule
If # of crossings is odd, interior.
If # of crossings is even, exterior.
        (Make sure there is no crossing at vertex)
Nonzero winding number rule
Step 1: Count directional crossing of PP' to each polygon edge
              (Use vector product.)
  If right to left, increment winding #.
  If left to right, decrement winding #.
Step 2: If winding # is nonzero, interior
(more versatile than the odd-even rule)
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