Lecture03 MidPoiint Circle Algo

7/29/2019 Lecture03 MidPoiint Circle Algo

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Graphics Output Primitives

Drawing Line, Circle and Ellipse


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Objectives

Introduction to Primitives

Points & Lines

Line Drawing Algorithms Digital Differential Analyzer (DDA)

Bresenhams Algorithm

Mid-Point Algorithm

Circle Generating Algorithms Properties of Circles


Mid-Point Algorithm

Ellipse Generating Algorithms Properties of Ellipse


Mid-Point Algorithm

Other Curves Conic Sections

Polynomial Curves

Spline Curves


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Introduction

For a raster display, a picture is

completely specified by: intensity and position of pixels, or/and

set of complex objects

Shapes and contours can either be stored

in terms of pixel patterns (bitmaps) or as aset of basic geometric structures (for

example, line segments).


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Introduction

Output primitives are the basic geometric

structures which facilitate or describe a

scene/picture. Example of these include:

points, lines, curves (circles, conics etc),

surfaces, fill colour, character string etc.


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Points

A point is shown

by illuminating a

pixel on the

screen


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Lines

A line segment is completely defined in

terms of its two endpoints.

A line segment is thus defined as:Line_Seg = { (x1, y1), (x2, y2) }


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Lines

A line is

produced by

means of

illuminating aset of

intermediary

pixels between

the twoendpoints.

x

y

x1 x2

y1

y2


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Lines

Lines is digitized into a set of discrete

integer positions that approximate the

actual line path.

Example: A computed line position of

(10.48, 20.51) is converted to pixel

position (10, 21).


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Line

The rounding of coordinate values to

integer causes all but horizonatal and

vertical lines to be displayed with a stair

step appearance the jaggies.


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Line Drawing Algorithms

A straight line segment is defined by the

coordinate position for the end points of

the segment.

Given Points (x1, y1) and (x2, y2)


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Line

All line drawing algorithms make use ofthe fundamental equations:

Line Eqn. y= m.x+ b

Slope m = y2 y1 /x2x1= y/ x

y-intercept b = y1 m.x1 x-intervalx =y / m

y-interval y = mx


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DDA Algorithm (Digital Differential Analyzer)

A line algorithm Based on calculating

either yor x using the above equations.

There are two cases:

Positive slop

Negative slop


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DDA- Line with positive Slope

If m 1 then take x= 1

Compute successive yby

yk+1 = yk+ m (1)

Subscript ktakes integer values startingfrom 1, for the first point, and increases by1 until the final end point is reached.

Since 0.0 < m 1.0, the calculated yvalues must be rounded to the nearestinteger pixel position.


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DDA

If m > 1, reverse the role ofxand yand take y

= 1, calculate successivexfrom

xk+1 =xk+ 1/m (2)

In this case, each computed x value is rounded

to the nearest integer pixel position.

The above equations are based on the

assumption that lines are to be processed fromleft endpoint to right endpoint.


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DDA

In case the line is processed from Right

endpoint to Left endpoint, thenx= 1, yk+1 = yk m form 1 (3)

or

y= 1,xk+1 =xk1/m form > 1 (4)


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DDA- Line with negative Slope

If m < 1,

use(1) [provided line is calculated from left to

right] and

use(3) [provided line is calculated from right to

left].

If m 1

use (2) or (4).


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Merits + Demerits

Faster than the direct use of line Eqn.

It eliminates the multiplication in line Eqn.

For long line segments, the true line Pathmay be mislead due to round off.

Rounding operations and floating-pointarithmetic are still time consuming.

The algorithm can still be improved. Other algorithms, with better performance

also exist.


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Code for DDA AlgorithmProcedure lineDDA(xa,ya,xb,yb:integer);

Vardx,dy,steps,k:integer

xIncrement,yIncrement,x,y:real;

begin

dx:=xb-xa;

dy:=yb-ya;

if abs(dx)>abs(dy) then steps:=abs(dx)

else steps:=abs(dy);

xIncrement:=dx/steps;

yIncrement:=dy/steps;

x:=xa;

y:=ya;

setPixel(round(x),round(y),1);

for k:=1 to steps dobegin

x:=x+xIncrement;

y:=y+yIncrement;

setPixel(round(x),round(y),1)

end

end; {lineDDA}


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Bresenhams Line Algorithm

It is an efficient raster line generation

algorithm.

It can be adapted to display circles and

other curves.

The algorithm

After plotting a pixel position (xk, yk), what is

the next pixel to plot?

Consider lines with positive slope.


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Bresenhams Line

For a positive slope, 0 < m < 1 and line is

starting from left to right.

After plotting a pixel position (xk

, yk

) we

have two choices for next pixel:

(xk +1, yk)

(xk +1, yk+1)


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Bresenhams Line

At position xk +1, we

pay attention to the

intersection of the

vertical pixel and themathematical line

path.


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Bresenhams Line

At position xk +1,

we label vertical

pixel separations

from themathematical line

path as

dlower , dupper.


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Bresenhams Line

The y coordinate on the mathematical lineat xk+1 is calculated as

y = m(xk +1)+ b

then

dlower= y yk= m (xk+1) + b yk

anddupper=(yk+1) y

= yk+1 m(xk+1) b


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Bresenhams Line

To determine which of the two pixels is closest to the linepath, we set an efficient test based on the differencebetween the two pixel separations

dlower

- dupper

= 2m (xk+1) 2y

k+ 2b - 1

= 2 (y / x) (xk+1) 2yk + 2b - 1

Consider a decision parameter pk such that

pk = x (dlower- dupper)

= 2y.xk 2x.yk + c

where

c = 2y + x(2b 1)


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Bresenhams Line

Since x > 0, Comparing (dlower and dupper),

would tell which pixel is closer to the line path;

is it yk or yk + 1

If (dlower< dupper)

Then pk is negative

Hence plot lower pixel. Otherwise

Plot the upper pixel.


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Bresenhams Line

We can obtain the values of successivedecision parameter as follows:

pk = 2y.xk 2x.yk + c

pk+1=2y.xk+12x.yk+1+c Subtracting these two equations

pk+1 pk= 2y (xk+1 xk) 2x ( yk+1

yk) But xk+1 xk = 1, Therefore

pk+1 = pk+2y 2x (yk+1 yk)


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Bresenhams Line

( yk+1 yk) is either 0 or 1, depending on

the sign of pk (plotting lower or upper

pixel).

The recursive calculation of pk is

performed at integer x position, starting at

the left endpoint.

p0 can be evaluated as:

p0= 2y x


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Bresenhams Line-Drawing

Algorithm form < 11. Input the two line end points and store the left end point

in (x0 , y0 ).

2. Load (x0 , y0 ) into the frame buffer; that is, plot the firstpoint.

3. Calculate the constants x, y, 2y, and 2y 2x,and obtain the starting value for the decision parameteras

p0= 2y x

4. At eachxkalong the line, starting at k= 0 , perform thefollowing test: Ifpk< 0,the next point to plot is (xk+1, ykand

pk+1=pk+2y

Otherwise, the next point to plot is (xk+1, yk+1) andpk+1=pk+2y2x

5. Repeat step 4, x1 times.


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Summary

The constants 2yand 2y 2xare

calculated once for each line to be scan

converted.

Hence the arithmetic involves only integer

addition and subtraction of these two

constants.


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Example

To illustrate the algorithm, we digitize the linewith endpoints (20,10) and (30,18). This line hasslope of 0.8, with

x= 10

y=8

The initial decision parameter has the value

p0= 2y x = 6

and the increments for calculating successivedecision parameters are

2 y = 16

2 y - 2x = -4


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Example

We plot the initial point (x0 , y0)=(20,10) anddetermine successive pixel positions along the line

path from the decision parameter as

K pk (xk +1, yk +1) K pk (xk +1, yk +1)

0 6 (21,11) 5 6 (26,15)

1 2 (22,12) 6 2 (27,16)

2 -2 (23,12) 7 -2 (28,16)

3 14 (24,13) 8 14 (29,17)

4 10 (25,14) 9 10 (30,18)


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Example

C G


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Circle Generating Algorithms

A circle is defined as the set of points that are all at a

given distance rfrom a center point (xc, yc).

Circle function:

fcircle (x,y) = x2 + y2r2


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Circle Equations

Polar form

x= rCos

y= rSin (r= radius of circle)

P=(rCos, rSin)

rSin)

rCos)x

y

r


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Cartesian form

Use Pythagoras theorem

x2+ y2= r2

x

ry

y

x x

y

r

2 2,P x r x

Ci l l ith


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Circle algorithms Step throughx-axis to determine y-values

Disadvantages: Not all pixel filled in

Square root function is very slow

Ci l G ti Al ith


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Properties of Circle:

A circle is defined as the set of points that are all at a

given distance rfrom a center point (xc, yc).

For any circle point (x, y), this distance is expressed by the

Equation

(x xc)2

+ (y yc)2

= r2

We calculate the points by stepping along the x-axis in unit

steps from xc-rto xc+rand calculate y values as

Circle function:

fcircle (x,y) = x2 + y2r2


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There are some problems with this approach:

1. Involves Considerable computation at each step.

2. Spacing between plotted pixel positions is not unifom

as in this Figure.

Positive half of a circle

plotted with eqn and

with (xc, yc) = (0, 0)

Ci l G ti Al ith


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Problem (Spacing between plotted pixel

positions is not unifom ) can be removed

by adjusting the spacing by interchanging

x and y (stepping through y values and

calculating x values) whenever theabsolute value of the slope of the circle is

greater than 1.

But this increases the computation and

processing required by the algorithm.



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Circle Generating Algorithms Problem (Spacing between plotted pixel positions

is not uniform ) can be removed by calculatingpoints along the circular boundary using polar

coordinates r and .

x = xc + r cos

y = yc + r sin

using a fixed angular step size, a circle is plotted

with equally spaced points along the

circumference.


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Problem 1 can be overcome

by considering the symmetry

of circles as in Figure 3.

But it still requires a gooddeal of computation time.

Efficient Solutions

Midpoint Circle Algorithm

Mid i t Ci l D i


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To determine the closest pixel position tothe specified circle path at each step.

For given radius r and screen centerposition (xc, yc), calculate pixel positions

around a circle path centered at thecoodinate origin (0,0).

Then, move each calculated position (x, y)to its proper screen position by adding xc toxand

y

cto y.

Along the circle section from x=0 to x=y inthe first quadrant, the gradient varies from0 to -1.

Midpoint Circle Drawing

Algorithm


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Mid point Circle Algorithm

To apply the midpoint method, we define a

circle function:

Any point (x,y) on the boundary of thecircle with radius rsatisfies the equation

fcircle(x, y)= 0.


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If the points is in the interior of the circle,

the circle function is negative.

If the point is outside the circle, the circle

function is positive.

To summarize, the relative position of any

point (x,y) can be determined by checking

the sign of the circle function:


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The circle function tests in (3) are performed for the midpositions between pixels near the circle path at eachsampling step. Thus, the circle function is the decision

parameter in the midpoint algorithm, and we can set upincremental calculations for this function as we did in the

line algorithm.


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Figure 4 shows the midpoint between the two candidate

pixels at sampling positionxk+1. Assuming we have just

plotted the pixel at (xk, yk), we next need to determine

whether the pixel at position (xk+1, yk) or the one at

position (xk+1, yk1) is closer to the circle.


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Choosing the Next Pixel

M

(x, y) (x+1, y)

(x+1, y-1)

E

SE

0)2/1,1( yxF

0)2/1,1( yxF

choose E

choose SE

)2/1,1()( yxFMFd

decision variable d


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Our decision parameter is the circle

function (2) evaluated at the midpoint

between these two pixels:


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Ifpk< 0, this midpoint is inside the circle

and the pixel on scan line ykis closer to

the circle boundary.

Otherwise, the midpoint is outside or on

the circle boundary, and we select the

pixel on scan line yk1.

Successive decision parameters areobtained using incremental calculations.


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We obtain a recursive expression for the next decision parameter byevaluating the circle function at sampling position xk+1 +1 =xk+ 2

where yk+1 is eitherykoryk-1,depending on the sign ofpk.


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Increments for obtainingpk+1 are either

2xk+1 +1 (ifpkis negative) or

2xk+1+1 2yk+1 (ifpkis positive)

Evaluation of the terms 2xk+1 and 2yk+1can also be done incrementally as:


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At the start position (0, r), these two terms

(2x, 2y) have the values 0 and 2r,

respectively.

Each successive value is obtained byadding 2 to the previous value of2xand

subtracting 2 from the previous value of

2y.


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The initial decision parameter is obtained

by evaluating the circle function at the start

position (x0 , y0)=(0, r):


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If the radius r is specified as an integer, we

can simply roundp0 to

since all increments are integers.


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Summary of the Algorithm

As in Bresenhams line algorithm, the

midpoint method calculates pixel positions

along the circumference of a circle using

integer additions and subtractions,assuming that the circle parameters are

specified in screen coordinates. We can

summarize the steps in the midpoint circlealgorithm as follows.

Algorithm


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Example

Given a circle radius r= 10, we

demonstrate the midpoint circle algorithm

by determining positions along the circle

octant in the first quadrant fromx= 0 tox= y. The initial value of the decision

parameter is


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Example

For the circle centered on the coordinate origin,the initial point is (x0 , y0) =(0,10), and initialincrement terms for calculating the decisionparameters are

Successive decision parameter values andpositions along the circle path are calculatedusing the midpoint method as shown in the table.

Mid i t Ci l D i Al ith


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Midpoint Circle Drawing Algorithm

Initial (x0, y

0) = (1,10)

Decision parameters are: 2x0 = 2, 2y0 = 20

k Pk (xk+1, yk+1) 2xk+1 2yk+1

0 -9 (1, 10) 2 201 -9+2+1=-6 (2, 10) 4 20

2 -6+4+1=-1 (3, 10) 6 20

3 -1+6+1=6 (4, 9) 8 18

4 6+8+1-18=-3 (5, 9) 10 18

5 -3+10+1=8 (6, 8) 12 16

6 8+12+1-16=5 (7, 7) 14 14

Example


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Example

10

9

8

7

6

5

4

3

2

1

0

0 1 2 3 4 5 6 7 8 9 10

i pi xi+1, yi+1 2xi+1 2yi+1

0 -9 (1, 10) 2 20

1 -6 (2, 10) 4 20

2 -1 (3, 10) 6 20

3 6 (4, 9) 8 18

4 -3 (5, 9) 10 18

5 8 (6, 8) 12 16

6 5 (7, 7)

r= 10

p0 = 1r= -9 (ifris integer roundp0 = 5/4rto

integer)

Initial point (x0, y0) = (0, 10)


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Example

A plot of the generated pixel positions in

the first quadrant is shown in Figure 5.



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Midpoint Circle Drawing Algorithmvoid circleMidpoint (int xCenter, int yCenter, int radius){

int x = 0;Int y = radius;int f = 1 radius;

circlePlotPoints(xCenter, yCenter, x, y);while (x < y) {x++;if (f < 0)

f += 2*x+1;

else { y--;f += 2*(x-y)+1; }

}circlePlotPoints(xCenter, yCenter, x, y);

}



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Midpoint Circle Drawing Algorithm

void circlePlotPoints (int xCenter, int yCenter,int x, int y){

setPixel (xCenter + x, yCenter + y);setPixel (xCenter x, yCenter + y);

setPixel (xCenter + x, yCenter y);setPixel (xCenter x, yCenter y);setPixel (xCenter + y, yCenter + x);setPixel (xCentery, yCenter + x);setPixel (xCenter + y, yCenterx);

setPixel (xCentery, yCenterx);}

Lecture03 MidPoiint Circle Algo

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