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Unit 5: Implementing Ellipse Algorithm

5. A prediction (xk+1, yk–½) is set at the middle between the four candidate pixels. Notes
(a) True (b) False

5.3 Ellipse Area, Sector Area and Segment Area

The ellipse area, sector area and segment area are defined as follows:

5.3.1 Ellipse Area
Consider an ellipse that is cantered at the origin, with its axes aligned to the coordinate axes.
If the semi-axis length along the x-axis is a and the semi-axis length along the y-axis is B, then
the ellipse is defined by a locus of points that reassure the implicit polynomial Equation (1a),
or alternatively defined parametrically as in Equation (1b):
x 2 + y 2 = 1 (1a)
A 2 B 2
x = A◊ cos( ) 0 t
t ¸
y =◊ cos( t) ˝ ££ 2p (1b)
B
˛
The area of such an ellipse can be found using the parameterized form given in Equation (1b)
with the Gauss-Green formula:
1 2 t
Area = 1 t Ú [x(t)· y(t) – x(t)· y(t)]dt (2a)
2
1 2p
= 0 Ú [A· cost(t)· B· cos(t) – B· sin(t)· (–A)· sin(t)]dt
2
◊
◊
AB 2p AB 2p
= [cos () 2 t dt = dt (2b)
2
t + sin( )]
2 Ú 0 2 Ú 0
=  · A· B (3)
For more general ellipses, e.g., ellipses that have been rotated and/or translated, the area
determined by Equation (3) is still applicable, provided the semi-axis lengths are known. Other
methods are also available for calculating ellipse areas.
5.3.2 Ellipse Sector Area
The ellipse sector between two points (x 1 , y 1 ) and (x 2 , y 2 ) on the ellipse is the area that is swept
out by a vector from the origin to the ellipse, beginning at the first point (x 1 , y 1 ), as the vector
pass through the ellipse in a counter-clockwise direction from (x 1 , y 1 ) to (x 2 , y 2 ). An example is
shown in Figure. (See Figure 5.1)

Figure 5.1: Ellipse Sector Area

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