Editing Ellipse (section)

=== Computer graphics ===
Drawing an ellipse as a [[graphics primitive]] is common in standard display libraries, such as the MacIntosh [[QuickDraw]] API, and [[Direct2D]] on Windows. [[Jack Bresenham]] at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967.<ref>{{cite journal |author=Pitteway, M.L.V. |title=Algorithm for drawing ellipses or hyperbolae with a digital plotter |journal=The Computer Journal |volume=10 |issue=3 |pages=282–9 |year=1967 |doi=10.1093/comjnl/10.3.282 |doi-access=free }}</ref> Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.<ref>{{cite journal |author=Van Aken, J.R. |title=An Efficient Ellipse-Drawing Algorithm |journal=IEEE Computer Graphics and Applications |volume=4 |issue=9 |pages=24–35 |date=September 1984 |doi=10.1109/MCG.1984.275994 |s2cid=18995215 }}</ref>

In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties.<ref>{{cite journal |author=Smith, L.B. |title=Drawing ellipses, hyperbolae or parabolae with a fixed number of points |journal=The Computer Journal |volume=14 |issue=1 |pages=81–86 |year=1971 |doi=10.1093/comjnl/14.1.81 |doi-access=free }}</ref> These algorithms need only a few multiplications and additions to calculate each vector.

It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.

;Drawing with Bézier paths:
[[Composite Bézier curve]]s may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an [[affine transformation]] of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent [[Bézier curve]]s behave appropriately under such transformations.