Based on the ellipse, keep the polar diameter unchanged, and reduce the polar angle of the ellipse by an integer multiple to evolve a high-order ellipse, which is expressed as:
Where: R is the polar diameter; A is semi major axis; E is eccentricity; θ Is the polar angle; N is the order, n ≥ 1. When n = 1, it is a first-order ellipse (ellipse for short), when n = 2, it is a second-order ellipse (also known as oval), and when n ≥ 3, it is a high-order ellipse.
Several typical ellipses of a = 150, n = 1,…, 4 are shown in the figure. It can be seen that when n ≥ 2, the higher-order ellipse is a periodic curve of n. The gear with higher order ellipse as pitch curve can realize the periodic change of transmission ratio.