Page 42 - Computer Graphics

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Ellipse Algorithm

Properties of ellipse

Ellipse is defined as the locus of a point in a plane which moves in a plane in such
a manner that the ratio of its distance from a fixed point called focus in the same

plane to its distance from a fixed straight line called directrix is always constant,
which should always be less than unity.

If the distance to the two foci from any point P=(x,y) on the ellipse are
labeled d1 and d2 then the general equation of the ellipse can be stated
as- d1+d2=constant.

For expressing the distances d1 and d2 in terms of focal coordinates F1 and F2 we
2
have:- Ax +By +Cxy+Dx+Ey+F=0 where A, B, C, D,E, and F are evaluated in
2
terms of focal coordinates and dimensions of the major and minor axes of the
ellipse.

Midpoint ellipse algorithm

The midpoint ellipse method is applied throughout the first quadrant in two
parts. Now let us take the start position at (0,ry) and step along the ellipse path in
clockwise order throughout the first quadrant.

Ellipse function can be defined as:

2 2
2
2
2 2
fellipse(x,y)=ry x +rx y -rx ry
According to this there are some properties which have been generated that are:
1. fellipse(x,y)<0 which means (x,y) is inside the ellipse boundary.

2. fellipse(x,y)>0 which means (x,y) is outside the ellipse boundary.

3. fellipse(x,y)=0 which means (x,y) is on the ellipse boundary.

(-x,y) (x,y)

(-x,-y) (x,-y)