---
title: "Symmetry property of a circle"
subject: "Computer Graphics"
chapter: 2
key: "semester_5/computer_graphics:circle-symmetry-property"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/circle-symmetry-property
site: BCT Engineering
---

# Symmetry property of a circle

Computer Graphics · Chapter 2: Scan-Conversion

- Asked in: 2069 Chaitra, 2 marks; 2075 Asoj, 3 marks; 2067 Mangsir (old course), 4 marks; 2072 Chaitra, 2 marks (asked 4 times)
- Video: [Lec: 06 | 8 Way Symmetry Properties of a Circle | Computer Graphics | Bangla Tutorial ](https://www.youtube.com/watch?v=6a7Cs9CT1cE) — Tech Tips With Ozifa
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/circle-symmetry-property
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-2

## Question

How is the symmetry property of a circle used to reduce computation (complexity) while drawing a complete circle?

## Short answer

A circle has 8-fold symmetry: points (±x,±y) and (±y,±x) all lie on it. Computing only one octant (e.g., from x=0 to x=y) and mirroring to the other seven reduces work by about eight times.

## Answer

A circle centred at the origin is symmetric about both axes and about the lines y = x and y = −x. So if (x, y) is a point on the circle, the following 8 points also lie on it:

```text
(x, y)  (y, x)  (−x, y)  (−y, x)
(x, −y) (y, −x) (−x, −y) (−y, −x)
```

For a circle centred at (xc, yc), the points are (xc ± x, yc ± y) and (xc ± y, yc ± x).

**Use in computation**
- Only the points of one octant (from x = 0 to x = y, i.e. a 45° arc) need to be calculated; the other 7 octants are obtained by changing signs and swapping x and y.
- The expensive computation (decision parameter, square root or trigonometric functions) is therefore done for only 1/8 of the circle, reducing the work by about 8 times.
- Each point calculated in this one octant gives 8 pixels.

Example: for r = 5 only the 4 octant points (0,5), (1,5), (2,5), (3,4) are calculated; symmetry then gives the whole circle of 28 distinct pixels (the 4 calculated points plus 24 more).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. One computed point (x,y) gives how many points?
   A. 2
   B. 4
   C. 8
   D. 16
   Answer: C. Symmetric about axes and y = ±x. (Key point: Symmetry points)

2. Calculation is needed for
   A. a quadrant
   B. half
   C. the whole
   D. one octant (x = 0 to x = y)
   Answer: D. Others by sign change/swap. (Key point: Octant)

3. The work is reduced by about
   A. 8 times
   B. 2 times
   C. 4 times
   D. 16 times
   Answer: A. Only 1/8 of the circle is computed. (Key point: Saving)

4. For centre (xc, yc) the points are
   A. (x ± xc, y)
   B. (xc ± x, yc ± y) and (xc ± y, yc ± x)
   C. (xc, yc ± x)
   D. (±xc, ±yc)
   Answer: B. Add the centre offset. (Key point: Centre shifted)

## Related in Computer Graphics

- [Ellipse symmetry and drawing algorithm](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aellipse-symmetry-algorithm) — Computer Graphics, chapter 2
- [Midpoint circle algorithm and decision parameter](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Amidpoint-circle-derive) — Computer Graphics, chapter 2, asked 8×
- [Midpoint circle: r=8, centre (10,10)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acircle-r8-center-10-10) — Computer Graphics, chapter 2, asked 2×
- [Midpoint circle: r=6, centre (20,10)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acircle-r6-center-20-10) — Computer Graphics, chapter 2, asked 2×
- [Midpoint circle: r=10, centre (20,30)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acircle-r10-center-20-30) — Computer Graphics, chapter 2

## Related in other subjects

- [Define computational complexity](https://www.bctengineer.com/question.md?key=semester_3%2Ftheory_of_computation%3Acomputational-complexity-define) — Theory of Computation, chapter 6, asked 2×
- [Harmonic exponentials; CTFS conjugate symmetry](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aharmonic-exponentials-conjugation) — Signals and Systems, chapter 2
- [NP-hard and NP-complete problems](https://www.bctengineer.com/question.md?key=semester_3%2Ftheory_of_computation%3Anp-hard-np-complete) — Theory of Computation, chapter 6, asked 2×
- [Class P, class NP and NP-complete](https://www.bctengineer.com/question.md?key=semester_3%2Ftheory_of_computation%3Aclass-p-np-relation) — Theory of Computation, chapter 6