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

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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 $\lambda ^{2}x^{2}+y^{2}=1$, 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

$y={\sqrt {2{\frac {b^{2}}{a}}x'-{\frac {b^{2}}{a^{2}}}x'^{2}}}={\sqrt {2px'-(1-e)^{2}x'^{2}}}$.

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)