**ELLIPSE GENERATING ALGORITHMS**

**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.

If the distances to the two foci from any point P = (x,
y) on the ellipse are labeled d

_{1}and d_{2}then the general equation of an ellipse can be stated as
d

_{1}+ d_{2}= constant
Expressing distances d

_{1}and d_{2 }in terms of the focal coordinates F_{1}= (x_{1}, y_{1})
and F

_{2}= (x_{2}, y_{2}), we have
√ ((x -
x

_{1})^{2}+ (y - y_{1})^{2}) + √ ((x - x_{2})^{2}+ (y - y_{2})^{2}) = constant
By squaring this equation, isolating the remaining
radical, and then squaring again, we can rewrite the general ellipse equation
in the form Ax

^{2}+ By^{2}+ Cxy + Dx + Ey + F = 0
The major axis is the straight line segment extending
from one side of the ellipse to the other through the foci. The minor axis
spans the shorter dimension of the ellipse, bisecting the major axis at the
half way position (ellipse center) between the two foci.

If r

_{x}labels the semi major axis, and r_{y}labels the semi minor axis, then the equation of the ellipse can be written as in terms of the ellipse center coordinates and parameters r_{x}and r_{y}as
(x-x

_{c})^{2}/r_{x}^{2}+ (y-y_{c})^{2}/r_{y}^{2 }= l
Using polar coordinates r and 9, we can also describe the
ellipse in standard position with the parametric equations:

x = x

_{c}+ r_{x}cosθ
y = y

_{c}+r_{y}sinθ
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, the elliptical arc throughout one quadrant, then we
obtain positions in the remaining three quadrants by symmetry.

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