1
Circle Drawing Algorithms
Pictures snagged from http://accad.osu.edu/~waynec/history/
2
Drawing Circles and Arcs
Similar to line drawing In that you have to determine what pixels to
activate But non-linear
Simple equation implementation Optimized
CS-321Dr. Mark L. Hornick
3
Cartesian form ofCircle Equations
(xc, yc)
2 2 2
22
c c
c c
x x y y r
or
y y r x x
r
Not a very good method. Why?
CS-321Dr. Mark L. Hornick
4
Be the algorithm!
CS-321Dr. Mark L. Hornick
5
What does it do?
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10 11
CS-321Dr. Mark L. Hornick
6
Polar Coordinate Form
(x, y)cos
sinc
c
x x r
y y r
r
Simple method: plot directly from parametric equations
θ
1
1
s r
s
r
CS-321Dr. Mark L. Hornick
7
Trig functions are expensive
( 1) 2 1(2 1)!
0
( 1) 2(2 )!
0
sin
cos
n
n
nn
n
nn
n
Plots of first 1, 3, 5, 7, 9, 11, and 13 terms in the Taylor’s series for sin
CS-321Dr. Mark L. Hornick
8
Polygon Approximation
(xi, yi)
Calculate polygon vertices from polar
equation; connect with line algorithm
CS-321Dr. Mark L. Hornick
9
The Bresenham Algorithms
Bresenham, J. E. Algorithm for computer control of a digital plotter, IBM Systems Journal, 4(1), 1965, pp. 25-30.
Bresenham, J. E. A linear algorithm for incremental digital display of circular arcs. Communications of the ACM, 20(2), 1977, pp. 100-106.
CS-321Dr. Mark L. Hornick
10
Bresenham’s Midpoint Circle Algorithm
0, if (x,y) inside circle
, 0, if (x,y) on circle
0, if (x,y) outside circlecirclef x y
2 2 2,circlef x y x y r
0<|m|<1 in this region
CS-321Dr. Mark L. Hornick
11
The Circle Decision Parameter
2
2 2
2 2 2
11,
2
11
2
12 1
4
k circle k k
k k
k k k k
p f x y
x y r
x x y y r
Calculate fcircle for point midway between candidate
pixels
xk
yk
Plot 1,k kx yIf pk < 0:
If pk >= 0: Plot 1, 1k kx y
CS-321Dr. Mark L. Hornick
12
Calculating pk+1
1 1
22 2
1
2
1
2 21 1
11,
2
11
2
14 4
4
1
k circle k
k
k k k k
k
k
p f y
y r
x
x
x y
x
y r
CS-321Dr. Mark L. Hornick
13
Recurrence Relation for pk
2 21 1 1
2 21 1 1
(2 2) 1
2 1
k k k k k k k
k k k k k k
p p x y y y y
p x y y y y
2 2 212 1
4k k k k kp x x y y r
2 2 21 1 1
14 4
4k k k k kp x x y y r
CS-321Dr. Mark L. Hornick
14
One Last Term …
2 21 1 ?k k k ky y y y
If yk+1 = yk, then:
2 2 0k k k ky y y y
If yk+1 = yk-1, then:
2 2
2 2
1 1
2 1 1
2 2 2 1
k k k k
k k k
k k
y y y y
y y y
y y
CS-321Dr. Mark L. Hornick
15
Initial Values 0 0
0
, 0,
10 1,
2circle
x y r
p f r
22
2 2
11
2
11
45
14
r r
r r r
r r
CS-321Dr. Mark L. Hornick
16
Bresenham Midpoint Circle Algorithm Summary
1
Plot 1,
2 1 1
k k
k k k
x y
p p x
At each point:
If pk < 0:
If pk >= 0:
1
Plot 1, 1
2 1 1 2 1
k k
k k k k
x y
p p x y
CS-321Dr. Mark L. Hornick
17
Circle Example 1
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10 11
r = 7
CS-321Dr. Mark L. Hornick
18
Symmetry Optimization
(a, b)
(b, a)
(a, -b)
(b, -a)
(-b, a)
(-b, -a)
(-a, b)
(-a, -b)
Calculate points for
one octant; replicate in other seven
CS-321Dr. Mark L. Hornick
19
General Ellipse Equation
In general, ellipse described by constant sum of distances from
foci
Difficult to solve equations and plot points
(use parametric polar form?)
CS-321Dr. Mark L. Hornick
20
Special Case Ellipses
(xc, yc)
22
1c c
x y
x x y y
r r
rx
ry
x x r
y y rc x
c y
cos
sin
Modified midpoint algorithm possible
CS-321Dr. Mark L. Hornick
21
Drawing Ellipses
General ellipses Polygon approximation Rotation (we’ll defer this until later, when we cover
coordinate transformations) Aligned axes
Constrained (no rotation) Midpoint algorithm
Recommended
