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midpoint circle algorithm
In computer graphics, the midpoint circle algorithm is an algorithm used to determine the points needed for drawing a circle. Bresenham's circle algorithm is derived from the midpoint circle algorithm. The algorithm can be generalized to conic sections.
The algorithm is related to work by Pitteway〔Pitteway, M.L.V., "Algorithm for Drawing Ellipses or Hyperbolae with a Digital Plotter", Computer J., 10(3) November 1967, pp 282-289〕 and Van Aken.〔Van Aken, J.R., "An Efficient Ellipse Drawing Algorithm", CG&A, 4(9), September 1984, pp 24-35〕
==Summary==
This algorithm draws all eight octants simultaneously, starting from each cardinal direction (0°, 90°, 180°, 270°) and extends both ways to reach the nearest multiple of 45° (45°, 135°, 225°, 315°). You can determine where to stop because when y = x, you have reached 45°. The reason for using these angles is shown in the above picture: As you increase y, you do not skip nor repeat any y value until you get to 45°. So during the while loop, y increments by 1 each iteration, and x decrements by 1 on occasion, never exceeding 1 in one iteration. This changes at 45° because that is the point where the tangent is rise=run. Whereas rise>run before and riseThe second part of the problem, the determinant, is far trickier. This determines when you decrement x. It usually comes after the drawing of the pixels in each iteration, because you never go below the radius on the first pixel. Because in a continuous function, the function for a sphere is the function for a circle with the radius dependent on z(or whatever the third variable is), it stands to reason that the algorithm for a discrete(voxel) sphere would also rely on this Midpoint_circle_algorithm. But if you look at a sphere, you will find that the integer radius of some adjacent circles is the same, but you would not expect to have the same exact circle adjacent to itself in the same hemisphere. Instead, you need to have a circle of the same radius with a different determinant, to allow the curve to come in slightly closer to the center or extend out farther. The circle charts seen relating to Minecraft, like the determinant listed below, only account for one possibility.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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