1. Lecture 05: Mid-point Ellipse algorithm Dr. Manal Helal – Fall 2014
Computer Graphics CC416
Lecture 05: Mid-point Ellipse algorithm
Dr. Manal Helal – Fall 2014

2. Ellipse Generation Algorithm
An ellipse is an elongated circle
Therefore, elliptical curves can be generated by modifying circle-drawing procedures to take into account the different dimensions of an ellipse along the major and minor axes
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3. Start with the basic equation of a circle
Divide both sides by r2
The general equation of an ellipse can be stated as rx ry xc yc
(x,y)
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4. Properties of Ellipses
An ellipse is defined as the set of points such that the sum of the distances from two fixed positions (foci) is the same for all points d1 d2
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5. So
Expressing distances d1, and d2, in terms of the focal coordinates F1 = (x1, y1) and F2 = (x2, y2), we have d1 d2
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6. Symmetry considerations can be used to further reduce computations.
An ellipse in standard position is symmetric between quadrants, but unlike a circle, it is not symmetric between the two octants of a quadrant.
Thus, we must calculate pixel positions along the elliptical arc throughout one quadrant, then we obtain positions in the remaining three quadrants by symmetry
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7. Mid-Point Ellipse Algorithm
The approach is similar to that used in displaying a circle
Given parameters rx, ry, and (xc, yc), we determine points (x, y) for an ellipse in standard position centered on the origin
Then the points are shifted so the ellipse is centered at (xc, yc)
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8. This is done according to the slope of the tangent
The mid-point ellipse method is applied throughout the first quadrant in two parts
This is done according to the slope of the tangent
When slope < -1, take unit steps in the x direction
When slope > -1, take unit steps in the y direction
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9. the decision parameter will be according to
From equation
When (xc, yc) = (0,0)
the decision parameter will be according to
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10. So, the next pixel along the ellipse path according to the sign of the ellipse function evaluated at the midpoint between the two candidate pixels
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11. Starting at (0,ry)
take unit steps in the x direction until reaching the boundary between region 1 and region 2
switch to unit steps in the y direction over the remainder of the curve in the first quadrant
At each step, test the value of the slope using
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12. At the boundary between region 1 and region 2, dy/dx = - 1 and
Therefore, we move out of region 1 whenever
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13. Assuming position (xk, yk) has been selected at the previous step
Recall
Assuming position (xk, yk) has been selected at the previous step
the next position along the ellipse path is determined by evaluating the decision parameter
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14. If p1k < 0, the midpoint is inside the ellipse and the pixel on scan line y, is closer to the ellipse boundary.
Otherwise, the mid position is outside or on the ellipse boundary, and we select the pixel on scan line yk - 1.
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15. where yk+1 is either yk or yk - 1, depending on the sign of p1k
At the next sampling position (xk+1+1 = xk + 2), the decision parameter for region 1 is evaluated as
[(xk+1) + 1]2 = (xk+1)2 + 2(xk+1) + 1
[(Yk+1-(1/2))]2 = Yk+12 - Yk+1 + (1/4)
P1k+1 = ry2(xk+1)2 + 2ry2 (xk+1) + ry2 +(rx2 Yk+12 - rx2Yk+1 + rx2 (1/4)) - rx2ry2
where yk+1 is either yk or yk - 1, depending on the sign of p1k
P1k
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16. So decision parameters are incremented by the following amounts:
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17. In region 1, the initial value of the decision parameter is obtained by evaluating the ellipse function at the start position (x, y) = (0, ry) or
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18. In region 2, we sample at unit steps in the negative y direction
The midpoint is now taken between horizontal pixels at each step
The decision parameter is evaluated as
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19. If p2k > 0, the mid-position is outside the ellipse boundary, and we select the pixel at xk
If p2k ≤ 0, the midpoint is inside or on the ellipse boundary, and we select pixel position xk+1
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20. When we enter region 2, the initial position (x0, y0) is taken as the last position selected in region 1 and the initial decision parameter in region 2 is then
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21. To simplify the calculation of p20, we could select pixel positions in counterclockwise order starting at (rx, 0).
Unit steps would then be taken in the positive y direction up to the last position selected in region 1
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22. Where transformation methods can be used to reorient the ellipse
The midpoint algorithm can be adapted to generate an ellipse in nonstandard position
Where transformation methods can be used to reorient the ellipse
Non Standard position
Standard position
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23. Midpoint Ellipse Algorithm
Input rx, ry and ellipse center (xc, yc), and obtain the first point on an ellipse centered on the origin as
Calculate the initial value of the decision parameter in region 1 as
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24. At each x position in region 1, starting at k = 0, perform the following test: If p1k < 0, the next point along the ellipse centered on (0, 0) is (xk+1, yk) and Otherwise, the next point along the circle is (xk + 1, yk – 1)
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25. With and continue until
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26. Calculate the initial value of the decision parameter in region 2 using the last point (x0, y0) calculated in region 1 as
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27. using the same incremental calculations for x and y as in region 1.
At each yk position in region 2, starting at k = 0, perform the following test: If p2k> 0, the next point along the ellipse centered on (0, 0) is (xk, yk - 1) and Otherwise, the next point along the circle is (xk + 1, yk - 1) and
using the same incremental calculations for x and y as in region 1.
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28. Determine symmetry points in the other three quadrants.
Move each calculated pixel position (x, y) onto the elliptical path centered on (xc, yc) and plot the coordinate values:
x = x + xc y = y + yc
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29. Example Given rx = 8 and ry = 6, rx2 = 64 and ry2 = 36
Initial values and increments for the decision parameter calculations are
For region 1:
The initial point for the ellipse centered on the origin is (x0, y0) = (0,6), and the initial decision parameter value is
Applying the following when p1k < 0
Otherwise:
Loop until
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30. We now move out of region 1, since
Successive decision parameter values and positions along the ellipse path are calculated using the midpoint method as
We now move out of region 1, since
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31. For region 2 rx = 8 and ry = 6, rx2 = 64 and ry2 = 36
the initial point is (x0, y0) = (7,3) and the initial decision parameter is
The remaining positions along the ellipse path in the first quadrant are then calculated as:
If If p2k> 0:
Else: k
p2k
Xk+1
Yk+1
2r2yXk+1
2r2xYk+1
-23 8 2
72*8 = 576
128*2 = 256 1
( )= 297
576
128
( ) = 809 -
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33. More about ellipses Source: http://en.wikipedia.org/wiki/Ellipse
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