ELLIPSE GENERATING ALGORITHMS

1

DEFINITION

Ellipse is an elongated circle.

Elliptical curves can be generated by

modifying circle-drawing procedures to

take into account the different dimensions of

an ellipse along the major and minor axes.

2

PROPERTIES OF ELLIPSES

Distances from any point on the ellipse to two

fixed positions called the foci of the ellipse.

Sum of these two distances is the same values

for all points on the ellipse. d1+d2=constant

3

Expressing distances d1 and d2 in terms of the focal coordinates F1=(x1,y2) and F2=(x2,y2)

sqrt((x-x1)2+(y-y1)2)+sqrt((x-x2)2+(y-y2)2)=constant

The general ellipse equation in the form Ax2+By2+Cxy+Dx+Ey+F=0

Coefficients A,B,C,D,E, and F are evaluated in terms of the focal coordinates and the dimensions of the major and minor axes of the ellipse.

4

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, perpendicularly bisecting the major axis at the halfway position (ellipse center) between the two foci.

5

((x-xc)/rx)2+((y-yc)/ry)2=1

Polar coordinates r and θx=xc+rxcos θ y=yc+rxsin θ

6

MIDPOINT ELLIPSE ALGORITHM

The Ellipse function isfellipse (x,y)=ry

2x2+rx2y2 –rx

2 ry2

which has the following properties: fellipse (x,y)

<0, if (x,y) is inside the ellipse boundary =0, if(x,y) is on ellipse boundary >0, if(x,y) is outside the ellipse boundary

7

REGION 1 AND 2 CAN BE PROCESSED IN VARIOUS WAYS

STEP 1 : Start at position (0,ry) and step clockwise along the elliptical path in the first quadrant shifting from unit steps in x to unit steps in y when the slope becomes less than -1 STEP2 : Start at (rx,0) and select points in a counter clockwise order. Shifting from unit steps in y to unit steps in x when the slope becomes greater than -1.0 2.2 Using parallel processors calculate pixel positions in the two regions simultaneouslySTEP 3: Start at (0,ry) step along the ellipse path in clockwise order throughout the first quadrant ellipse function (xc,yc)=(0,0)

fellipse (x,y)=ry2x2+rx

2y2 –rx2 ry

2 8

STEP 4: Check the properties

Ellipse function fellipse (x,y) serves as the decision parameter in the midpoint algorithm. Starting at (0,ry)

The ellipse slope is calculated dy/dx= -(2ry2x/2rx

2y) At the boundary between region 1 and region 2 dy/dx

= -1.0 2ry

2x=2rx2y

Move out of region 1 ; 2ry2x>=2rx

2y

9

To determine the next position along the ellipse path by evaluating the decision parameter at this mid point

P1k = fellipse (xk+1,yk - 1/2)

= ry2 (xk+1)2 + rx

2 (yk-1/2)2 – rx2 ry

2

if P1k <0,

Midpoint is inside the ellipse Otherwise the midpoint is outside

Next sampling position (xk+1+1=xk+2) the decision parameter for region 1 is calculated as

p1k+1 = fellipse(xk+1 +1,yk+1 -½ )

=ry2[(xk +1) + 1]2 + rx

2 (yk+1 -½)2 - rx2 ry

2

OR p1k+1 = p1k +2 ry2(xk +1) + ry

2 + rx2 [(yk+1 -½)2 - (yk -

½)2] Where yk+1 is yk or yk-1 depending on the sign of P1k. 10

Decision parameters are incremented by the following amounts increment

= { 2 ry2(xk +1) + ry

2 if p1k <0 }

{ 2 ry2(xk +1) + ry

2 - 2rx2 yk+1 if p1k ≥ 0 }

Increments for the decision parameters can be calculated using only addition and subtraction as in the circle algorithm.

The terms 2ry2x and 2rx2y can be obtained incrementally. At the initial position (0,ry) these two terms evaluate to 2 ry

2x = 0

2rx2 y =2rx

2 ry

11

Decision parameter is evaluated as

p10 = fellipse(1,ry -½ )

= ry2 + rx

2 (ry -½)2 - rx2 ry

2

Or p10 = ry

2 - rx2 ry + ¼ rx

2

12

REGION 2 For this region, the decision parameter is evaluated

as

p2k = fellipse(xk +½ ,yk - 1)

= ry2 (xk +½ )2 + rx

2 (yk - 1)2 - rx2 ry

2

P2k+1 = fellipse(xk+1 +½,yk+1 -1 )

= ry2(xk +½) 2 + rx

2 [(yk+1 -1) -1]2 - rx2 ry

2

or p2k+1 = p2k -2 rx

2(yk -1) + rx2 + ry

2 [(xk+1 +½)2 - (xk +½)2]

Decision parameter for region 2 p20 = fellipse(x0 +½ ,y0 - 1)

= ry2 (x0 +½ )2 + rx

2 (y0 - 1)2 - rx2 ry

2 13

ALGORITHM

14