University of Babylon
College of Computer Technology
Department of Information Network
Day: Wednesday
Date: 71-10-2012
Stage: Second.
Topic: Introduction to Computer Graphics
Lecture3: Circle Generating Algorithms .
General Objective:
The students will be able to use circle generating algorithms when they
need to build other algorithms.
Procedural Objective:
1- The students can use more than one algorithm to draw any circle.
2- The students will be able to alter the given circle generating
algorithm to draw other shapes.
3- The students can recognize the way that they could follow to build
an efficient circle generating algorithm.
Teaching aids: Lecture (presented by MS-power point program on the
laptop), discussion between lecturer and students, LCD monitor,
whiteboard, pens.
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CIRCLE-GENERATING ALGORITHM
3.1 Introduction
Since the circle is a frequently used component in pictures and graphs, a
procedure for generating either full circles or circular arcs is included in
most graphics packages.
3.2 Properties of Circles
A circle is defined as the set of points that are all at a given distance (r)
from a center position (xc, yc) Fig.(3-1). This distance relationship is
expressed by the Pythagorean theorem in Cartesian coordinates as:
(x-xc)2+(y-yc)2=r2
(3-1)
Figure (3-1) Circle with center coordinates
(xc,yc) and radius r.
We could use this equation to calculate the position of points on a circle
circumference by stepping along the x axis in unit steps from xc - r to xc
+ r and calculating the corresponding y values at each position as:
(3-2)
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But this is not the best method for generating a circle. One problem with
this approach is that it involves considerable computation at each step.
Moreover, the spacing between plotted pixel positions is not uniform, as
demonstrated in Fig.(3-2).
Figure(3-2) positive half of a circle
plotted with Eq. (3-2) and with
(xc,yc) = (0.0).
Another way to eliminate the unequal spacing shown in Fig.(3-2) is to
calculate points along the circular boundary using polar coordinates (r)
and (Ɵ) Fig.(3-1). Expressing the circle equation in parametric polar form
yields the pair of equations:
x=xc+r*cos Ɵ
y=yc+r*sin Ɵ
(3-3)
When a display is generated with these equations using a fixed angular
step size, a circle is plotted with equally spaced points along the
circumference. The step size chosen for (Ɵ) depends on the application
and the display device. Larger angular separations along the
circumference can be connected with straight line segments to
approximate the circular path. For a more continuous boundary on a
raster display, we can set the step size at (l/r). This plots pixel positions
that are approximately one unit apart.
Algorithm for generating a circle by using polar coordinates:
1-Get the center point (xc,yc) and radius (r).
2-Set the step size (dtheta) to (1/r).
3-Set the starting angle (theta) to zero.
4-While (theta<=6.28) do the following sub steps:
4-1 Calculate the value of x & y:
x=xc+r*cosƟ.
y=yc+r*sinƟ.
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4-2 Pset(x,y,color).
4-3 Adjust the value of theta:
theta=theta+dtheta.
5-End.
Computation can be reduced by considering the symmetry of circles. The
shape of the circle is similar in each quadrant. We can generate the circle
section in the second quadrant of the xy plane by noting that the two
circle sections are symmetric with respect to the y axis. And circle
sections in the third and fourth quadrants can be obtained from sections in
the first and second quadrants by considering symmetry about the x axis.
We can take this one step further and note that there is also symmetry
between octants. Circle sections in adjacent octants within one quadrant
are symmetric with respect to the 45° line dividing the two octants. These
symmetry conditions are illustrated in Fig. (3-3), where a point at position
(x, y) on a one-eighth circle sector is mapped into the seven circle points
in the other octants of the xy plane. Taking advantage of the circle
symmetry in this way we can generate all pixel positions around a circle
by calculating only the points within the sector from x = 0 (or x=r) to x =
y.
Figure (3-3) symmetry of a circle.
Calculation of a circle point (x,y) in
one octant yields the circle points
shown for the other seven octants.
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Algorithm for generating a circle by making use of the symmetry
between octants:
1- Get the center point (xc,yc) and radius (r).
2-Set the starting point as:
x=r.
y=0.
3-Set the step size (dtheta) to (1/r).
4- While (x>=y) do the following sub steps:
4-1 Call CirclePoints procedure* to display (energized) the
eight symmetrical points.
4-2 Adjust the value of x & y to get the new point:
x=x-y*dtheta.
y=y+x*dtheta.
5-End.
*Procedure CirclePoints (xc,yc,x,y)
{ PSet (xc + x, yc + y,color).
PSet (xc + x, yc - y,color).
PSet (xc + y, yc + x,color).
PSet (xc - y, yc + x,color).
PSet (xc - x, yc + y,color).
PSet (xc - x, yc - y,color).
PSet (xc - y, yc - x,color).
PSet (xc + y, yc - x,color).
}
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