2. Lecture outline
n Ellipse-generating algorithm
n Properties of ellipse
n Midpoint ellipse algorithm
2
3. Ellipse Generating Algorithm
n Ellipse – an elongated circle.
n A modified circle whose radius varies from a
maximum value in one direction to a minimum value
in the perpendicular direction.
n A precise definition in terms of
distance from any point on the
d1
ellipse to two fixed position, F1 P=(x,y)
d2
called the foci of the ellipse. F2
n The sum of these two distances
is the same value for all points
on the ellipse. 3
4. Ellipse Generating Algorithm
n If the distance of the two focus positions from any
point P=(x, y) on the ellipse are labeled d1 and d2,
the general equation of an ellipse:
d1 + d2 = constant
n Expressing distance d1 and d2 in terms of the focal
coordinates F1=(x1, y1) and F2=(x2, y2), we have
(x - x1 )2 + ( y - y1 )2 + (x - x2 )2 + ( y - y2 )2 = constant
4
5. Ellipse Generating Algorithm
However, we will only consider ‘standard’ ellipse:
2 2
ry
æ x - xc ö æ y - yc ö
yc
ç
ç r ÷ +ç
÷ ç r
÷ =1
rx
÷
è x ø è y ø
xc
5
6. Symmetry
n An ellipse only has a 4-
way symmetry.
(x,y)
(-x,y) ry
rx
(x,-y)
(-x,-y)
6
7. Equation of an ellipse
n Consider an ellipse centered at the origin,
(xc,yc)=(0,0): 2 2
æxö æyö
ç ÷ + ç ÷ =1
çr ÷ çr ÷
è xø è yø
f e ( x, y ) = ry x + rx y - rx ry
2 2 2 2
Ellipse function 2 2
…and its properties:
fellipse(x,y) < 0 if (x,y) is inside the ellipse
fellipse(x,y) = 0 if (x,y) is on the ellipse
fellipse(x,y) > 0 if (x,y) is outside the ellipse
7
8. Midpoint Ellipse Algorithm
n Ellipse is different from circle.
n Similar approach with circle,
different in sampling direction. Slope =
-1
n Region 1: ry
n Sampling is at x direction
n Choose between (xk+1, yk), or rx
(xk+1, yk-1) Region 1
n Move out if 2r2yx >= 2r2xy Region 2
n Region 2:
n Sampling is at y direction
n Choose between (xk, yk-1), or
(xk+1, yk-1) 8
9. Decision parameters
Region 1: Region 2:
p1k = f e ( xk + 1, yk - 1 2 ) p2k = fe ( xk + 12 , yk -1)
p1k < 0: p2k <= 0:
n midpoint is inside • midpoint is inside/on
n choose pixel (xk+1, yk) • choose pixel (xk+1, yk-1)
p1k >= 0: p2k > 0:
n midpoint is outside/on • midpoint is outside
n choose pixel (xk+1, yk-1) • choose pixel (xk, yk-1)
9
10. Midpoint Ellipse Algorithm
1. Input rx, ry and ellipse center (xc, yc). Obtain the first point on an
ellipse centered on the origin (x0, y0) = (0, ry).
2. Calculate initial value for decision parameter in region 1 as:
2 2 2
p1 0 = r y - rx r y + 1
4 rx
3. At each xk in region 1, starting from k = 0, test p1k :
2 2
If p1k < 0, next point (xk+1, yk) and p1k +1 = p1k + 2 ry xk +1 + ry
2 2 2
else, next point (xk+1, yk-1) and p1k +1 = p1k + 2 ry xk +1 - 2 rx y k +1 + ry
with 2ry2xk+1 = 2ry2xk + 2ry2, 2rx2yk+1 = 2rx2yk – 2rx2
and repeat step 1 to 3 until 2ry2x ³ 2rx2y 10
11. Midpoint Ellipse Algorithm
4. Initial value for decision parameter in region 2:
p 2 0 = ry
2
(x 0 + 1 )2 + rx 2 ( y 0 - 1 )2 - rx 2 r y 2
2
where (x0, y0) is the last position calculate in region 1
5. At each yk in region 2, starting from k = 0, test p2k:
2 2
If p2k > 0, next point is (xk, yk-1) and p 2 k +1 = p 2 k - 2rx y k +1 + rx
2 2 2
else, next point is (xk+1, yk-1) and p 2 k +1 = p 2 k + 2ry xk +1 - 2rx yk +1 + rx
continue until y=0
11
12. Midpoint Ellipse Algorithm
6. For both region determine symmetry points in the other 3
quadrants
7. 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
12
13. Try this out!
Given input ellipse k p1k (xk+1,yk+1) 2ry2 xk+1 2 rx2yk+1
parameter rx=8 and ry=6, 0 -332 (1,6) 72 768
determine pixel positions 1
along the ellipse path in 2
the first quadrant using 3
the midpoint ellipse 4
algorithm 5
6
2ry2x = ?
k p2k (xk+1,yk+1) 2rv2 xk+1 2 rx2yk+1
2rx2y = 2rx2ry = ?
0
p10 = ?
1
2
13
