Bresenham’s Circle Drawing Algorithm

The Bresenham’s Circle Drawing Algorithm is used in computer graphics to draw a circle efficiently using only integer arithmetic.
It avoids floating-point calculations, making it faster and suitable for raster displays.

The algorithm determines the nearest pixel position to form a circle using a decision parameter.

Basic Idea of Circle Drawing

A circle with center (x_c, y_c) and radius r is represented by:

x² + y² = r²

Instead of calculating all points using trigonometric equations, Bresenham’s algorithm:

• Starts from the top point (0, r)
• Moves step-by-step in the first octant
• Uses symmetry to generate the remaining points

Why Only One Octant?

A circle is symmetric in 8 octants.

If we calculate one point (x, y) the remaining seven points are:

(x,y) (y,x) (−x,y) (−y,x) (−x,−y) (−y,−x) (x,−y) (y,−x)

Thus, computation becomes much faster.

Steps of Bresenham’s Circle Algorithm

Step 1: Initialize

x = 0 and y = r

Decision parameter:

p₀ = 3 − 2r

Step 2: Check Decision Parameter

Case 1: If p_k < 0

Choose East pixel (E)

x_k+1 = x_k + 1, and

y_k+1​ = y_k​

p_k+1 = p_k + 4x_k + 6

Case 2: If p_k ≥ 0

Choose South-East pixel (SE)

x_k+1 = x_k + 1

y_k+1​ = y_k​ − 1

p_k+1 = p_k + 4(x_k − y_k) + 10

Step 3: Stop Condition

Repeat until: x ≥ y

Numerical Example

Draw a Circle with:

• Center = (0, 0)
• Radius = 8

Initial Values

x₀ = 0 and y₀​ = 8

Decision parameter:

p₀ = 3 − 2r

p₀​ = 3 − 2(8) 

p₀ = 3 − 16

p₀​ = −13 

Iteration Table

Stop here because: x ≥ y

Detailed Calculations

Iteration 0

Current point: (0, 8) and p₀= −13

Since: p₀ < 0, Choose Case I.

Next point:

x_k+1 = 0 + 1 = 1

y_k+1 ​= 8

p_k+1 = p_k + 4 x 0 + 6

p_k+1 = −13 + 4(0) + 6

p1_k+1 = −7 

Iteration 1

Current point: (1, 8) and p₁ = −7

Again: p1 < 0 , Choose Case I.

Next point:

x_k+1 = 1 + 1 = 2

y_k+1 ​= 8

p_k+1 = −7 + 4(1) + 6

p_k+1 = 3

Iteration 2

Current point: (2, 8) and p2 = 3

Since, p2 ≥ 0 Choose Case II.

Next point: 

x_k+1 = 2 + 1 = 3

y_k+1 ​= 8 - 1 = 7

p_k+1 = p_k + 4(x_k − y_k) + 10

p_k+1 = 3 + 4(2 − 8) + 10

p_k+1 = 3 − 24 + 10

p_k+1 = −11

(Some books may show slightly different values depending on the variant used.)

Plotting Symmetric Points

For point: (3,7)

The 8 symmetric points are: 

(3,7) (7,3) (−3,7) (−7,3) (−3,−7) (−7,−3) (3,−7) (7,−3)

Advantages of Bresenham Circle Algorithm

1. Uses only integer arithmetic
2. Faster than direct circle equation
3. Efficient for raster graphics
4. Avoids floating-point calculations
5. Produces smooth circles

Disadvantages of Bresenham Circle Algorithm

1. Limited to raster displays
2. Slight approximation error may occur
3. Complex for beginners compared to DDA

Pseudocode

Input radius r and center (xc, yc)

x = 0
y = r
p = 3 - 2r

while (x <= y)

    plot 8 symmetric points

    if p < 0
        p = p + 4x + 6
    else
        p = p + 4(x - y) + 10
        y = y - 1

    x = x + 1

Conclusion

The Bresenham Circle Drawing Algorithm is one of the most important algorithms in computer graphics because it efficiently draws circles using integer calculations and symmetry. It is widely used in raster graphics systems for fast rendering of circles and curved shapes.