Wikipedia:Reference desk/Archives/Mathematics/2021 March 19

= March 19 =

Ellipse eccentricity
An ellipse has two radii, a & b, with conventionally a >= b. When a=b eccentricity is zero and you have a circle. When b=0, you have a collapsed ellipse with eccentricity 1, forming two superimposed lines, each of length 2a. Why does the article say this is a parabola? -- SGBailey (talk) 14:53, 19 March 2021 (UTC)


 * It does not make sense, The eccentricity corresponds to the relative elongation, the ratio between the axes. If a family of ellipses is given by the equation $$\lambda^2x^2+y^2=1$$, then $$\textstyle\lim_{\lambda\to 0}e = 1$$, and the (non-uniform) limit of the locus of the equation is the pair of lines $$y=\pm 1$$. But we can also specify a family of ellipses with the parametric representation $$(x,y) = (\cos t,\lambda\sin t), 0\leq t\leq 2\pi$$, and then the limit (this time uniform) is the doubled line segment from $$(-1,0)$$ to $$(1,0)$$. Or we can use $$\lambda x^2+\lambda^{{-}1}y^2=1$$, a family of ellipses of constant area, and then the limiting figure, as $$\lambda\downarrow 0$$, is (again not uniform) a doubled x-axis. So the limiting case is not well defined. --Lambiam 19:30, 19 March 2021 (UTC)
 * Well, a parabola does have eccentricity 1, so it depends on which family of conics you're looking at. See File:Kegelschnitt-schar-ev.svg for a family of conics which includes all three non-degenerate types. When you're talking about the limiting case of a family of conics you have to be careful of the wording because I'm not sure if the limit on a familiy curves, especcially one with more than one parameter, being a another curve has ever been defined rigorously. But in the sense that an ellipses can have eccentricity up to but not including 1, and a parabola has eccentricity 1, the parabola is a limiting case. Not the limiting case (as stated in the article) since other limiting cases, given in your examples, are degenerate conics. --RDBury (talk) 13:07, 20 March 2021 (UTC)
 * The ellipse equation is
 * $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
 * where eccentricity $$e=1-b/a$$. Introducing new coordinate x': $$x=x'-a$$ we have
 * $$y=\sqrt{2\frac{b^2}{a}x'-\frac{b^2}{a^2}x'^2}=\sqrt{2px'-(1-e)^2x'^2}$$.
 * Suppose that $$p=b^2/a$$ is kept constant but both a and b go to infinity. Then e goes to unity and we get a parabola. Ruslik_ Zero 13:19, 20 March 2021 (UTC)
 * I think eccentricity is usually given as $$\sqrt{1-\frac{b^2}{a^2}}$$ where a>b, but the idea should still work. --RDBury (talk) 23:01, 20 March 2021 (UTC)
 * You are right. Ruslik_ Zero 20:54, 21 March 2021 (UTC)


 * The following is an attempt to define a notion of "limiting case" . Let $$S_p$$ be a parametrized family of subsets of a topological space, where the parameter $$p$$ ranges over the positive reals. We then define $$x$$ to be a limit point of $$S_p$$ for $$p\to\infty$$ if for every neighbourhood $$N$$ of $$x$$ the set $$S_p$$ is not disjoint from $$N$$ for all sufficiently large $$p$$. In the special case that each $$S_p$$ is a singleton set $$\{y_p\}$$, $$x$$ is a limit point if and only if $$\textstyle{\lim_{p\to\infty}y_p=x}$$. Using this, we define the fate of $$S_p$$ to be the set of its limit points. It is the largest set such that each of its points is eventually approached arbitrarily closely by the family members.
 * Unlike the concept of the limit of a function, which may be undefined, the fate of a parametrized family of subsets of a topological space is always defined, but it may be the empty set.
 * Let $$c$$ be any positive real number. The fate of the family of ellipses $$\frac{x-p}{p^2}+\frac{y^2}{cp}=1$$ can be determined by rewriting the equation in the form $$y^2=2cx-\frac{c}{p}x^2$$. As $$p\to\infty$$, for any fixed value of $$x$$ the values of $$y$$ tend to those for the parabola $$y^2=2cx$$. The value of $$c$$ can be varied to get any of these parabolas, showing once more that the fate of an ellipse family as the eccentricity goes to 1 depends on the specific family. The fate of the family $$\frac{x}{p^2}+\frac{y^2}{p}=1$$ is empty.
 * In the view of ellipses as conic sections, when the cutting plane is rotated around an axis orthogonal to a fixed generating line of the cone, as it gets closer and closer to being parallel to that line the eccentricity of an elliptic section approaches $$1$$; then, as the plane is parallel the section is for a flitting moment a parabola before becoming a hyperbola. This is, presumably, the origin of the statement that the parabola is "the limiting case" – but whether the fate of an ellipse family is a parabola depends critically on the way the family is produced. --Lambiam 09:52, 22 March 2021 (UTC)
 * Let $$c$$ be any positive real number. The fate of the family of ellipses $$\frac{x-p}{p^2}+\frac{y^2}{cp}=1$$ can be determined by rewriting the equation in the form $$y^2=2cx-\frac{c}{p}x^2$$. As $$p\to\infty$$, for any fixed value of $$x$$ the values of $$y$$ tend to those for the parabola $$y^2=2cx$$. The value of $$c$$ can be varied to get any of these parabolas, showing once more that the fate of an ellipse family as the eccentricity goes to 1 depends on the specific family. The fate of the family $$\frac{x}{p^2}+\frac{y^2}{p}=1$$ is empty.
 * In the view of ellipses as conic sections, when the cutting plane is rotated around an axis orthogonal to a fixed generating line of the cone, as it gets closer and closer to being parallel to that line the eccentricity of an elliptic section approaches $$1$$; then, as the plane is parallel the section is for a flitting moment a parabola before becoming a hyperbola. This is, presumably, the origin of the statement that the parabola is "the limiting case" – but whether the fate of an ellipse family is a parabola depends critically on the way the family is produced. --Lambiam 09:52, 22 March 2021 (UTC)
 * In the view of ellipses as conic sections, when the cutting plane is rotated around an axis orthogonal to a fixed generating line of the cone, as it gets closer and closer to being parallel to that line the eccentricity of an elliptic section approaches $$1$$; then, as the plane is parallel the section is for a flitting moment a parabola before becoming a hyperbola. This is, presumably, the origin of the statement that the parabola is "the limiting case" – but whether the fate of an ellipse family is a parabola depends critically on the way the family is produced. --Lambiam 09:52, 22 March 2021 (UTC)