Double-Step Circle Drawing Algorithm with and without Grey Scale

Some existing pixel-level algorithms for drawing circles are introduced first in this paper. We further pointed out that the Kuzmin's circle drawing algorithm, a neglected algorithm in fact has least calculations, but on the other hand the algorithm carries serious mistakes. This paper corrects the mistakes in the algorithm, and then presented a double-step circle drawing algorithm. The new algorithm chooses the best approximate pixels to the circle with only integer arithmetic. The comparison results show that its speed is higher than the existing circle drawing algorithms. Finally, based on this algorithm, an algorithm for drawing anti-aliased circles is proposed without increasing the calculations. It generates four more intermediate shades of grey scale than the double-step algorithm by Wu and Rokne and the greatest error of intensity is reduced by 40%. These algorithms are easy to be realized by hardware.