structural:reformatting, structural:scholar_edition, linguistic:translation
Ellipses

This section considers the family of ellipses defined by equations (x − x ∘) 2 a 2 + (y − y ∘) 2 b 2 = 1 {\displaystyle {\tfrac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\tfrac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1} with a fixed eccentricity e {\displaystyle e}. It is convenient to use the parameter: q = a 2 b 2 = 1 1 − e 2, {\displaystyle {\color {blue}q}={\frac {a^{2}}{b^{2}}}={\frac {1}{1-e^{2}}},}

and to write the ellipse equation as: (x − x ∘) 2 + q (y − y ∘) 2 = a 2, {\displaystyle \left(x-x_{\circ }\right)^{2}+{\color {blue}q}\,\left(y-y_{\circ }\right)^{2}=a^{2},}

where q is fixed and x ∘, y ∘, a {\displaystyle x_{\circ },\,y_{\circ },\,a} vary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if q < 1 {\displaystyle q<1}, the major axis is parallel to the x -axis; if q > 1 {\displaystyle q>1}, it is parallel to the y -axis.)

Like a circle, such an ellipse is determined by three points not on a line.

For this family of ellipses, one introduces the following q-analog angle measure, which is not a function of the usual angle measure θ: