[rep_lodsb]

With the error initialized to zero, this algorithm will always step immediately after drawing the first pixel, which will cause that pixel to "stick out".

  y += 1;          // y == 1
  err += 2*y + 1;  // err == 3
  x -= 1;          // x == radius-1
  err -= 2*x + 1;  // err == 2-(2*(radius-1))

You could compare the absolute value of the new and old error, or start with err = -(radius-1) instead.

And you don't need calculus to come up with the algorithm, just simple high school algebra: (x+1)² - x² = 2x + 1.