elipsedrawing

8/6/2019 ElipseDrawing

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EllipseEllipse--Generating AlgorithmsGenerating Algorithms

•• EllipseEllipse – A modified circle whose radius varies

from a maximum value in one direction (majoraxis) to a minimum value in the perpendiculardirection (minor axis).

P=(x,y) F 1

F 2

d 1

d 2

• The sum of the two distances d 1 and d 2, between thefixed positions F 1 and F 2 (called the foci of the ellipse) toany point P on the ellipse, is the same value, i.e.

d 1 + d 2 = constant



Ellipse PropertiesEllipse Properties

• Expressing distances d 1 and d 2 in terms of the focal

coordinates F 1 = ( x 1, x 2) and F 2 = ( x 2, y 2), we have:

• Cartesian coordinates:

• Polar coordinates:

2 2 2 2

1 1 2 2( ) ( ) ( ) ( ) constant x x y y x x y y

r yr x

22

1c c

x y

x x y yr r

cos

sin

c x

c y

x x r

y y r



Ellipse AlgorithmsEllipse Algorithms

• Symmetry between quadrants

• Not symmetric between the two octants of aquadrant

• Thus, we must calculate pixel positions alongthe elliptical arc through one quadrant andthen we obtain positions in the remaining 3quadrants by symmetry

( x, y)(- x, y)

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

r x

r y




• Decision parameter:

2 2 2 2 2 2

( , )ellipse y x x y f x y r x r y r r

1

Slope = -1

r x

r y 2

0 if ( , ) is inside the ellipse

( , ) 0 if ( , ) is on the ellipse0 if ( , ) is outside the ellipse

ellipse

x y

f x y x y x y

2

2

2

2

y

x

r xdySlope

dx r y




• Starting at (0, r y ) we take unit steps in the x direction until we reach the boundary between

region 1 and region 2. Then we take unit steps inthe y direction over the remainder of the curve inthe first quadrant.

• At the boundary

• therefore, we move out of region 1 whenever

1

Slope = -1

r x

r y 2

2 21 2 2 y xdy r x r ydx

2 22 2 y x

r x r y



Midpoint Ellipse AlgorithmMidpoint Ellipse Algorithm

x i +2 x i +1 x i

y i -1

y i

Midpoint

Assuming that we have just plotted the pixels at ( x i , yi ).

The next position is determined by:

12

2 2 2 2 2 212

1 ( 1, )

( 1) ( )

i ellipse i i

i x i x y

p f x y

r x r y r r

If p1i < 0 the midpoint is inside the ellipse yi is closer

If p1i ≥ 0 the midpoint is outside the ellipse yi – 1 is closer



Decision Parameter (Region 1)Decision Parameter (Region 1)

At the next position [ x i +1 + 1 = x i + 2]

OR

where y i +1 = y i or y

i +1

= y i

– 1

11 1 1 2

2 2 2 2 2 211 2

1 ( 1, )

( 2) ( )

i ellipse i i

y i x i x y

p f x y

r x r y r r

2 2 2 2 2 21 1

1 1 2 21 1 2 ( 1) ( ) ( )i i y i y x i i p p r x r r y y




Decision parameters are incremented

by:

Use only addition and subtraction byobtaining

At initial position (0, r y )

2 2

1

2 2 2

1 1

2 if 1 02 2 if 1 0

y i y i

y i y x i i

r x r pincrement r x r r y p

2 22 and 2 y xr x r y

2

2 2

2 2 2 2 21 10 2 2

2 2 214

2 0

2 2

1 (1, ) ( )

y

x x y

ellipse y y x y x y

y x y x

r x

r y r r

p f r r r r r r

r r r r



Region 2Region 2

Over region 2, step in the negative y direction andmidpoint is taken between horizontal pixels at eachstep.

x i +2 x i +1 x i

y i

-1

y i

Midpoint

Decision parameter:

12

2 2 2 2 2 212

2 ( , 1)

( ) ( 1)

i ellipse i i

i x i x y

p f x y

r x r y r r

If p2i> 0 the midpoint is outside the ellipse x

i is closer

If p2i ≤ 0 the midpoint is inside the ellipse x i + 1 is closer



Decision Parameter (RegionDecision Parameter (Region2)2)

At the next position [y i +1 – 1 = y i – 2]

OR

where x i +1 = x i or x

i +1

= x i

+ 1

11 1 12

2 2 2 2 2 211 2

2 ( , 1)

( ) ( 2)

i ellipse i i

y i x i x y

p f x y

r x r y r r

2 2 2 2 21 1

1 1 2 22 2 2 ( 1) ( ) ( )i i x i x y i i p p r y r r x x




Decision parameters are incrementedby:

At initial position ( x 0, y 0) is taken at thelast position selected in region 1

2 2

1

2 2 2

1 1

2 if 2 0

2 2 if 2 0

x i x i

y i x i x i

r y r pincrement

r x r y r p

10 0 02

2 2 2 2 2 210 02

2 ( , 1)

( ) ( 1)

ellipse

x x y

p f x y

r x r y r r



Midpoint Ellipse AlgorithmMidpoint Ellipse Algorithm

1. Input r x , r y , and ellipse center ( x c , y c ), and obtain thefirst point on an ellipse centered on the origin as

( x 0, y 0) = (0, r y )

2. Calculate the initial parameter in region 1 as

3. At each x i position, starting at i = 0, if p1i < 0, thenext point along the ellipse centered on (0, 0) is ( x i +1, y i ) and

otherwise, the next point is ( x i + 1, y i – 1) and

and continue until

2 2 210 4

1 y x y x p r r r r

2 2

1 11 1 2i i y i y p p r x r

2 2 2

1 1 11 1 2 2i i y i x i y p p r x r y r

2 2

2 2 y xr x r y



Midpoint Ellipse AlgorithmMidpoint Ellipse Algorithm4. ( x 0, y 0) is the last position calculated in region 1. Calculate

the initial parameter in region 2 as

5. At each y i position, starting at i = 0, if p2i > 0, the nextpoint along the ellipse centered on (0, 0) is ( x i , y i – 1) and

otherwise, the next point is ( x i + 1, y i – 1) and

Use the same incremental calculations as in region 1.Continue until y = 0.

6. For both regions determine symmetry points in the other

three quadrants.7. Move each calculated pixel position (x, y) onto the elliptical

path centered on ( x c , y c ) and plot the coordinate values

x = x + x c , y = y + y c

2 2 2 2 2 210 0 022 ( ) ( 1) x x y p r x r y r r

2 2

1 12 2 2i i x i x p p r y r

2 2 2

1 1 12 2 2 2i i y i x i x p p r x r y r



ExampleExample

384504(7, 3)2446

512432(6, 4)2885

640360(5, 5)-108

4

640288(4, 5)2083768216(3, 6)-442

768144(2, 6)-224

1

76872(1, 6)-332

0

2r x 2y i

+1

2r y 2 x i

+1

x i +1,yi+1

pi i

r x = 8 , r y = 6

2r y2 x = 0 (with increment 2r y

2 = 72)

2r x 2 y = 2r x 2r y (with increment -2r x2 = -128)

Region 1Region 1

( x 0, y0) = (0, 6)

2 2 210 4

1 332 y x y x p r r r r

Move out of region 1 since

2r y2 x > 2r x

2 y



ExampleExample

8765432100

1

2

3

4

5

6

--(8, 0)7452

128576(8, 1)2331

256576(8, 2)-151

0

2r x 2y i

+1

2r y 2 x i

+1

x i +1,yi+1

pi i

Region 2Region 2

( x 0, y0) = (7, 3) (Last position in region 1)

10 2

2 (7 ,2) 151ellipse p f

Stop at y = 0



Midpoint Ellipse FunctionMidpoint Ellipse Function void ellipse(int Rx, int Ry){

int Rx2 = Rx * Rx, Ry2 = Ry * Ry;int twoRx2 = 2 * Rx2, twoRy2 = Ry2 * Ry2;

int p, x = 0, y = Ry;int px = 0, py = twoRx2 * y;

ellisePlotPoints(xcenter, ycenter, x, y);// Region 1 p = round(Ry2 – (Rx2 * Ry) + (0.25 * Rx2));while (px < py) {

x++; px += twoRy2;

if (p < 0) p += Ry2 + px;else {y--; py -= twoRx2; p += Ry2 + px – py;

}ellisePlotPoints(xcenter, ycenter, x, y);

}// Region 2 p = round(Ry2 * (x+0.5) * (x+0.5) + Rx2 * (y-1)*(y-1) – Rx2 * Ry2;while (y > 0) {

y--; py -= twoRx2;if (p > 0) p += Rx2 – py;else {

x++; px += twoRy2; p += Rx2 – py + px;

}

ellisePlotPoints(xcenter, ycenter, x, y);}}

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