Circle Drawing Techniques

📘 Circle Drawing Techniques — Exam Notes

---

🔹 1. Introduction

Circle Drawing Techniques are algorithms used in computer graphics to draw a circle efficiently on a pixel-based display.

👉 Since a circle is continuous but the screen has discrete pixels, we must approximate it using the best possible pixels.

---

🔹 2. Basic Concept of a Circle

✔️ Equation of a Circle

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

[ (x - x_c)^2 + (y - y_c)^2 = r^2 ]

---

✔️ Important Idea: Symmetry

A circle is highly symmetric, which helps reduce computation.

👉 If we find one point, we can generate 7 more points using symmetry.

---

✔️ 8-Way Symmetry

If ((x, y)) is a point on the circle, then:

[ (x, y), (-x, y), (x, -y), (-x, -y) ] [ (y, x), (-y, x), (y, -x), (-y, -x) ]

👉 This reduces calculations to only 1/8th of the circle.

---

🔹 3. Circle Drawing Algorithms

---

🔸 3.1 Polynomial Method (Direct Method)

✔️ Concept

Use circle equation directly:

[ y = \pm \sqrt{r^2 - x^2} ]

---

✔️ Steps

1. Input center ((x_c, y_c)) and radius (r)

2. For each x:

   • Compute y using formula

3. Plot points using symmetry

---

✔️ Disadvantages

• Uses square root (slow)
• Not efficient

---

🔸 3.2 Midpoint Circle Algorithm (Most Important)

✔️ Concept

• Uses decision parameter
• Chooses between two possible pixels
• Uses only integer calculations

---

✔️ Initial Conditions

• Start at point: [ (x, y) = (0, r) ]

• Initial decision parameter: [ p_0 = 1 - r ]

---

✔️ Steps

1. Input radius (r) and center ((x_c, y_c))

2. Set: [ x = 0,\quad y = r ]

3. Calculate: [ p = 1 - r ]

4. Repeat while (x < y):

   • If (p < 0): → Choose East pixel → (x = x + 1) → (p = p + 2x + 1)

   • Else: → Choose South-East pixel → (x = x + 1,\ y = y - 1) → (p = p + 2x + 1 - 2y)

5. Plot points using 8-way symmetry

---

✔️ Example (Simple)

For radius (r = 5):

• Start at (0, 5)
• p = 1 − 5 = −4

Then compute next points step-by-step using decision parameter.

---

✔️ Advantages

• Fast and efficient
• Uses integer arithmetic
• Most commonly used

---

✔️ Disadvantages

• Slightly complex logic

---

🔸 3.3 Bresenham’s Circle Algorithm

👉 Similar to midpoint algorithm (often considered a variation)

✔️ Concept

• Uses integer calculations
• Based on decision parameter

---

✔️ Key Idea

• Select between:

  • East pixel (E)
  • South-East pixel (SE)

---

✔️ Advantages

• Efficient
• No floating-point operations

---

🔹 4. Comparison of Methods

Method	Speed	Accuracy	Complexity
Polynomial	Slow	Good	Simple
Midpoint	Fast	High	Moderate
Bresenham	Very Fast	High	Moderate

---

🔹 5. Important Terms

• Radius (r): Distance from center to boundary
• Center (x_c, y_c): Middle point of circle
• Decision Parameter (p): Determines next pixel
• Symmetry: Repeating pattern to reduce computation

---

🔹 6. Diagram Descriptions

✔️ Circle Symmetry

• Draw a circle
• Divide into 8 equal parts
• Show mirrored points

---

✔️ Midpoint Algorithm

• Show two candidate pixels:

  • East (E)
  • South-East (SE)

• Highlight selected pixel

---

✔️ Pixel Approximation

• Show grid with circle approximated using pixels

---

🔹 7. Advantages of Using Algorithms

• Faster drawing
• Efficient memory usage
• Accurate approximation
• Suitable for real-time graphics

---

📝 Likely Exam Questions

1. Define circle drawing in computer graphics.
2. Write the equation of a circle.
3. Explain 8-way symmetry in circle drawing.
4. Describe the midpoint circle algorithm with steps.
5. Compare midpoint and Bresenham circle algorithms.
6. Why is the polynomial method inefficient?
7. Solve a numerical using midpoint algorithm.
8. What is a decision parameter?

---

⚡ Quick Revision Summary

• Circle equation: [ (x - x_c)^2 + (y - y_c)^2 = r^2 ]

• Use 8-way symmetry to reduce work

• Midpoint Algorithm is most important

• Decision parameter: [ p = 1 - r ]

• Choose between:

  • East pixel
  • South-East pixel

• Algorithms avoid floating-point operations for speed

---