Midpoint Circle Drawing

CSC-214 · Semester III · Computer Graphics

beginner20 minAvailable

Plot the first octant, then mirror it 8-fold.

cxcyr

Octant point

plot (19+0, 10+7)

Start with p = 1 - r. Each pixel x++, and y decrements when p ≥ 0. Because of 8-fold symmetry only 1/8 of the circle needs computing — each octant point paints 8 pixels. Decision parameter stays integer-only.

Lab brief

Objective

Generate a circle with the midpoint algorithm and watch 8-way symmetry fill all eight octants from one eighth of the arc.

Theory & concepts

The midpoint algorithm scans the arc from (0, r) until x = y using the implicit equation f(x, y) = x^2 + y^2 - r^2. Starting with p = 1 - r, the sign of p chooses the east or south-east pixel. Because of 8-way symmetry, each computed point is mirrored into the other seven octants.

Rendering diagram…

Midpoint circle algorithm and 8-way symmetry

x = 0;  y = r;  p = 1 - r
while x <= y:
  plot 8 symmetric points (x, y)
  x++
  if p < 0:  p += 2*x + 1
  else:      y--;  p += 2*x + 1 - 2*y

The arc is scanned from (0, r) down to the diagonal x = y
Each computed point generates 8 pixels through symmetry
Only integer additions and comparisons are used

Prerequisites

Lesson: Circle drawing: the midpoint algorithm
Lesson: Bresenham's line drawing algorithm

Instructions

1Choose a radius and center on the grid.
2Step through the algorithm and watch the arc advance from (0, r) toward the diagonal.
3Toggle the symmetry display to reveal the other seven octants being plotted from each computed point.
4Use Play to complete the circle and check for gaps or double-plotted pixels.
5Try radii of 5, 10 and 25 and compare the pixel density.
6Reflect on how the algorithm uses only integer arithmetic.

Tasks

Trace r = 10
Step the first octant and verify the table of (p, x, y) values.
Symmetry check
Confirm that the 8 mirrored points for (x, y) all satisfy x^2 + y^2 = r^2.
Gap-free circle
Draw several radii and check that octants meet without gaps.

Self-check

The first-octant pixel sequence matches the expected decision-parameter behavior.
The student can list the eight symmetric point positions.
The student explains why only an eighth of the circle is computed.

Environment

pixel grid canvasradius and center controlssymmetry togglestep/play animationdecision parameter readout