Page 78 - DCAP313_LAB_ON_COMPUTER_GRAPHICS

Lab on Computer Graphics

Notes The area of an ellipse sector between two points on the ellipse is the area swept out by a vector
from the origin to the first point as the vector travels along the ellipse in a counter-clockwise
direction to the second point. The area of an ellipse sector can be determined with the Gauss-
Green formula, using the parametric angles θ 1 and θ 2 .
The Gauss-Green formula applied to an ellipse (Equation (2b)) can be used to determine the
area of an ellipse sector as shown in Figure 5.1:

◊
◊
Sector Area = AB q 2 dt = q ( 2 - q ) AB (4)
1
2 Ú q 1 2
The parametric angle θ corresponding to a point (x, y) on the ellipse is formed between the positive
x-axis, with positive θ in the counter-clockwise direction. Using the ellipse parameterizations,
and the principal-valued opposite trigonometric functions that return angles in the in the range
0 ≤ θ ≤ for θ = across (z), and in the range –/2 ≤ θ ≤/2 for θ = arcsin (z), ellipse parametric
angles can be found with the relations.
Quadrant II (x ≤ 0 and y ≥ 0) Quadrant I (x ≥ 0 and y ≥ 0)
θ = arccos(x/A) θ = arccos(x/A)

=  – arcsin(y/B) = arcsin(y/B)
Quadrant III (x < 0 and y < 0) Quadrant IV (x ≥ 0 and y < 0)
θ = 2 – arccos(x/A) θ = 2 – arccos(x/A)

=  – arcsin(y/B) = 2 + arcsin(y/B)
For calculating a sector area, the Gauss-Green formula is sensitive to the direction of integration.
For the larger goal of determining ellipse overlap areas, we follow the convention that the sector
area is calculated from the first point (x 1 , y 1 ) to the second point (x 2 , y 2 ) in a counter-clockwise
direction. For example, if the points (x 1 , y 1 ) and(x 2 , y 2 ) of Figure 5.1 were to have their labels
switched, then the ellipse sector defined by the new points will have an area that is complementary
to that of the sector in Figure 5.1, as shown in Figure(See Figure 5.2)

Figure 5 2: The Ellipse Sector Area is Calculated From the First Point (x 1 , y 1 )
to the Second Point (x 2 , y 2 ) in a Counter-clockwise Direction

The definitions will always produce an angle in the range 0 ≤ θ < 2 for any point on the ellipse;
as such, with the point orientations shown in Figure 5.2, the corresponding angles will be ordered
as θ 1 > θ 2 . The first angle can be transformed by subtracting 2 to restore the condition that θ 1 <
θ 2 and the sector area formula (Equation (4) can then be used, with the integration angle from
(θ 1 – 2) through θ 2 .

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