What Is Ellipse?

The term ellipse has been coined by Apollonius of Perga, with a connotation of being "left out". The relation that suggested to him this term is rather obscure but nowadays could be justified, for example, by the fact that, ellipse is the only (non-degenerate) conic section that leaves out one of the halves of a cone. As a squashed circle, too, an ellipse leaves out a part of that circle.

There are many ways to define an ellipse. Some are purely geometrical and some are analytic. We cite several common definitions, prove that all are equivalent, and, based on these, derive additional properties of ellipse. Along the way, we shall introduce several relevant terms.

Definition 1

Ellipse is a bounded non-degenerate conic section.

Definition 2

Ellipse is a planar curve which in some Cartesian system of coordinates is described by the equation

(1)

$\displaystyle \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,$ $a, b \gt 0.$

Since in this definition the Cartesian system of coordinates is a matter of choice, it is always possible to request an additional condition, $a \gt b.$

When $a = b,$ ellipse "degenerates" into a circle which is considered as an exceptional kind of ellipse. When $a \gt b,$ a is called the major semi-axis while $b$ is the minor semi-axis. Correspondingly, the segment $[-a, a]$ on the $x$-axis is the major axis, while the segment $[-b, b]$ on the $y$-axis is the minor axis of the ellipse.

Definition 3

Ellipse is the locus of points the sum of whose distance to the two given points is constant.

Definition 4

Ellipse is the locus of points whose distances to a fixed point and to a fixed line are in a constant ratio less than $1.$

Definition 5

Ellipse is a squashed circle: given a circle $C,$ a straight line $L$ through the center of the circle and a coefficient $k.$ For point $P$ on $C$ find $P'$ on the perpendicular from $P$ to $L$ such that $\text{dist}(P', L) = k\,\text{dist}(P, L).$ The locus of point $P'$ is an ellipse.

Proofs

  1. Definition 1 implies Definition 3.

    This has a beautiful demonstration invented in 1822 by the Belgian mathematician G. P. Dandelin. The two points are known as the foci of the ellipse, and the constant sum is equal to the longest diameter of the ellipse - its major axis. Definition 3 is also known as the focal property of the ellipse.

  2. Definition 3 implies Definition 2.

    Let the $x$-axis run through the foci, with the origin midway between the two. The foci then will have coordinates $E(-c, 0)$ and $F(c, 0).$ Let $P(x, y)$ be a generic point on the ellipse, so that $|EP| + |FP| = 2a = const$ and $a \gt c.$ The segments $EP$ and $FP$ are known as the focal radii of $P,$ the left and the right radii, correspondingly. So we start with

    $\sqrt{(x + c)^{2} + y^{2}} + \sqrt{(x - c)^{2} + y^{2}} = 2a.$

    Move one of the square roots to the right and square both sides:

    $(x + c)^{2} + y^{2} = (x - c)^{2} + y^{2} - 4a\sqrt{(x - c)^{2} + y^{2}} + 4a^{2}$

    which simplifies to

    $a^{2} - xc = a\sqrt{(x - c)^{2} + y^{2}}.$

    With an additional squaring eliminate the radical:

    $a^{4} - 2a^{2}xc + x^{2}c^{2} = a^{2}(x^{2} - 2xc + c^{2} + y^{2}).$

    A simplification yields

    $a^{2}(a^{2} - c^{2}) = (a^{2} - c^{2}) x^{2} + a^{2}y^{2}.$

    Introducing $b^{2} = a^{2} - c^{2}$ we get equation (1).

    $a, b, c$ are called the major semiaxis, the minor semiaxis, and the linear eccentricity, respectively. $2c$ is the focal distance.

    Points $(\pm a, 0)$ and $(0, \pm b)$ lie on the ellipse and are known as its vertices. $e = c/a$ is called the eccentricity. For ellipse, $0 \le e \lt 1.$

  3. Definition 2 implies Definition 4.

    We may now find $|EP|$ and $|FP|$ separately. Say,

    $\begin{align} |EP|^{2} &= (x + c)^{2} + y^{2}\\ &= (x + c)^{2} + b^{2}(1 - x^{2} / a^{2})\\ &= (1 - b^{2} / a^{2}) x^{2} + 2xc + c^{2} + b^{2}\\ &= x^{2} c^{2} / a^{2} + 2xc + a^{2}\\ &= e^{2} x^{2} + 2xea + a^{2}\\ &= (a + ex)^{2}. \end{align}$

    Similarly, $|FP|^{2} = (a - ex)^{2}.$ Since $|x| \le a$ $|xe| \lt a,$

    $\begin{align} |EP| &= a + ex,\\ |FP| &= a - ex. \end{align}$

    Now, consider the straight line $d$: $x = -a/e$ perpendicular to the major axis. The distance between $P(x, y)$ and that line is

    $x + a/e = |EP| / e$

    making the ratio of the distances from $P$ to $E$ and to $d$ equal $e \lt 1.$ The line $d$ is called the (left) directrix of the ellipse. There is the right directrix, of course, whose equation is $x = a/e.$ It plays the role similar to the left one but with respect to the right focus $F.$

  4. Definition 4 implies Definition 2.

    Let $e$ be the ratio of distances and choose the system of coordinates so that the focus and the related directrix be $E(-ae, 0)$ and $x = -a/e.$ Let $P(x, y)$ be a generic point on the locus in question. Then, squaring the ratio of the distances to the point and to the line gives

    (2)

    $\displaystyle\frac{(x + ae)^{2} + y^{2}}{(x + a/e)^{2}} = e^{2}.$

    Multiply out and simplify to obtain

    (3)

    $x^{2}(1 - e^{2}) + y^{2} = a^{2}(1 - e^{2})$

    which, upon setting $b^{2} = a^{2} (1 - e^{2}),$ leads to the equation (1) in Definition 2.

  5. Definition 4 implies Definition 3.

    We may change the sign of $x$ in equation (2) to obtain

    (2')

    $\displaystyle\frac{(x - ae)^{2} + y^{2}}{(x - a/e)^{2}} = e^{2}.$

    which tells us that in Definition 4 there is a second pair of point-line with equations $F(ae, 0)$ and $x = a/e.$ (Naturally (2') leads to the same equation (3).)

    From (2) and (2'),

    (4)

    $\begin{align} |EP| + |FP| &= |e(x - a/e)| + |e(x + a/e)|\\ &= |a - xe| + |a + xe|.\\ \end{align}$

    Now observe that for $e$ to be less than $1,$ the whole locus has to be to the left of the directrix $x = a/e$ (and to the right of $x = -a/e.)$ For, otherwise, $P$ would be father away from $F$ than from the line $x = a/e$. This remark allows us to drop the symbols of absolute value in (4):

    $\begin{align} |EP| + |FP|&= |a - xea| + |a + xea|\\ &= a - xea + a + xea\\ &= 2a.\\ \end{align}$

  6. Definition 2 implies Definition 5.

    Assuming $a \gt b,$ let $L$ be the $x$-axis and $k = b/a.$

  7. Definition 5 implies Definition 2.

    Assuming $1 \gt k \gt 0,$ take $L$ as the $x$-axis and let $b = kr,$ where $r$ is the radius of circle $C.$ The locus of point $P'$ will then have equation (1). If $k$ is negative, the curve also reflects around $L.$ For $k \gt 1,$ the circle is rather stretched than squashed; it is squashed in the direction perpendicular to $L.$

  8. Definition 2 implies Definition 1.

    There is a point not in the plane of the ellipse that the lines through that point and the points on the ellipse form a right circular cone. For such a cone, the ellipse is a cross-section as in Dandelin's proof. The foci found in the proof must coincide with the original foci of the ellipse due to the uniqueness of the foci. There is actually the whole locus of these points - a straight line perpendicular to any of the circular cross-section. What is this line?

    There is an intuitive construction that could be seen as a guiding concept but which I shall not - at this point - justify in any way rigorously. In the plane perpendicular to the plane of the ellipse and through its major axis find a direction $\alpha$ for which $\tan\alpha = a/c.$ This direction will define a circular cylinder with the ellipse as a cross-section. The axis of the cylinder is the sought line, the cylinder being just the limiting case of a cone, with the apex moved to a point at infinity.

References

  1. H. Eves, A Survey of Geometry, Allyn and Bacon, 1972
  2. V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook, Birkhauser; 1 edition (July3, 2004)
  3. K. Kendig, Conics, MAA, 2005
  4. D. Pedoe, Geometry: a Comprehensive Course, Dover, 1970
  5. M. M. Postnikov, Lectures in Geometry, Semester 1: Analytic Geometry, Mir, Moscow, 1983
  6. G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960
  7. S. Schwartzman, The Words of Mathematics, MAA, 1994
  8. R. C. Yates, Curves and Their Properties , NCTM, 1974
  9. C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, 2005
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