Bresenham’s Circle Generation Algorithm

1. Circle Drawing in Raster Graphics

Drawing a smooth circle on a raster display requires selecting pixels that best approximate the mathematical circle equation.

• Pixel-based approximation
• Requires efficient computation
• Uses symmetry to reduce work

2. What is Bresenham’s Circle Algorithm?

It is an incremental circle drawing algorithm that uses only integer arithmetic to decide the next pixel position while generating a circle.

• No floating-point operations
• Fast and accurate
• Hardware-friendly

3. Circle Symmetry

A circle is symmetric in all quadrants. The algorithm calculates points for one octant and mirrors them to draw the entire circle.

• 8-way symmetry
• Reduces computation by 1/8

4. Mathematical Basis

The circle equation used is:

x² + y² = r²

5. Decision Parameter

The decision parameter determines whether the next pixel should be chosen to the East or South-East direction.

Initial decision parameter:
p0 = 3 − 2r

6. Update Rules

If p < 0:
    p = p + 4x + 6
Else:
    p = p + 4(x − y) + 10
    y = y − 1
x = x + 1

7. Steps of Bresenham’s Circle Algorithm

• Input center (xc, yc) and radius r
• Initialize x = 0, y = r
• Initialize decision parameter
• Plot initial symmetric points
• Iterate until x ≤ y

8. Example

To draw a circle with radius r = 5 and center (0,0):

• p0 = 3 − 2×5 = −7
• Plot points using symmetry
• Update decision parameter iteratively

9. Advantages of Bresenham’s Circle Algorithm

• Uses integer arithmetic only
• Very fast execution
• High accuracy and smooth circles

10. Disadvantages

• Limited to raster displays
• More complex than naive methods

11. Applications

• Graphics primitives generation
• Game development
• GUI and CAD systems

Practice Questions

1. What is Bresenham’s circle algorithm?
2. Explain the role of decision parameter.
3. Why is symmetry used in circle drawing?
4. Write update rules of the algorithm.
5. List advantages of Bresenham’s circle algorithm.

Practice Task