This paper describes a 2-D graphics algorithm that uses shifts and adds to precisely plot a series of points on an ellipse of any shape and orientation. The algorithm can also plot an elliptic arc that starts and ends at arbitrary angles. The ellipse algorithm described here is largely based on earlier papers by Van Aken and Simar [1,2], which extend Marvin Minsky's well-known circle algorithm [3,4,5] to ellipses, and show how to cancel out the sources of error in Minsky's original algorithm. A new flatness test is presented for automatically controlling the spacing between points plotted on an ellipse or elliptic arc. Most of the calculations performed by the ellipse algorithm and flatness test use fixed-point addition and shift operations, and thus are well-suited to run on less-powerful processors. C++ source code listings are included. Keywords: parametric ellipse algorithm, rotated ellipse, Minsky circle algorithm, flatness, elliptic arc, conjugate diameters, affine invariance

中文翻译：

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一种快速参数椭圆算法

本文介绍了一种二维图形算法，该算法使用移位和相加在任意形状和方向的椭圆上精确绘制一系列点。该算法还可以绘制以任意角度开始和结束的椭圆弧。这里描述的椭圆算法主要基于 Van Aken 和 Simar [1,2] 的早期论文，这些论文将 Marvin Minsky 著名的圆算法 [3,4,5] 扩展到椭圆，并展示了如何消除明斯基原始算法中的错误。提出了一种新的平整度测试，用于自动控制绘制在椭圆或椭圆弧上的点之间的间距。椭圆算法和平坦度测试执行的大多数计算都使用定点加法和移位运算，因此非常适合在功能较弱的处理器上运行。包括 C++ 源代码清单。