Concept
Ellipse
Overview
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of both distances to the two focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity
e
{\displaystyle e}
, a number ranging from
e
=
0
{\displaystyle e=0}
(the limiting case of a circle) to
e
=
1
{\displaystyle e=1}
(the limiting case of infinite elongation, no longer an ellipse but a parabola).
An ellipse has a simple algebraic formula for its area, but for its perimeter (also known as circumference), integration is required to obtain the exact solution.
The largest and smallest diameters of an ellipse, also known as its width and height, are typically denoted 2a and 2b. An ellipse has four extreme points: two vertices at the endpoints of the major axis and two co-vertices at the endpoints of the minor axis.
Analytically, the equation of a standard ellipse centered at the origin is:
x
2
a
2
+
y
2
b
2
=
1.
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}
Assuming
a
≥
b
{\displaystyle a\geq b}
, the foci are
(
±
c
,
0
)
{\displaystyle (\pm c,0)}
, where
c
=
a
2
−
b
2
{\textstyle c={\sqrt {a^{2}-b^{2}}}}
(the linear eccentricity) is the distance from the center to a focus. The standard parametrization is:
(
x
,
y
)
=
(
a
cos
(
t
)
,
b
sin
(
t
)
)
for
0
≤
t
≤
2
π
.
{\displaystyle (x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .}
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a right circular cylinder is also an ellipse.
An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant, called the eccentricity:
e
=
c
a
=
1
−
b
2
a
2
.
{\displaystyle e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.}
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.
The name, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his Conics.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Ellipse is discussed.
If the velocity of the solar system, that is to say if the constant part existed alone, the observed direction would be invariable. This position that one would thus observe is called the _mean_ apparent position of the star. Taking account now at the same time of the two parts of the velocity of the earth, we shall have the actual apparent position, which describes a little ellipse around the mean apparent…
Henri Poincaré, The Foundations of Science
names the idea; states what it is
Such ellipses are generally narrow and vertically directed, for the amount of lateral movement is small. That there is some lateral movement is shown by the ascending and descending lines not coinciding, and occasionally, as with Desmodium gyrans and Thalia dealbata, it was strongly marked. In the case of Melilotus the ellipses described by the terminal leaflet during the day are laterally extended, instead of…
Charles Darwin, The Power of Movement in Plants
names the idea; states what it is; opens on it
The case would then exactly resemble that of the navigator who discovers the land to be an island by sailing round it. If the path was visible, no one I think would dispute that to identify it with an ellipse is to describe it: and I cannot see why any difference should be made by its not being directly an object of sense, when every point in it is as exactly ascertained as if it were so.
John Stuart Mill, A System of Logic
states what it is; names the idea; opens on it
Yet it is a fact surely, that the planet does describe an ellipse; and a fact which we could see, if we had adequate visual organs and a suitable position. Not having these advantages, but possessing the conception of an ellipse, or (to express the meaning in less technical language) knowing what an ellipse was, Kepler tried whether the observed places of the planet were consistent with such a path. He found they…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
names the idea; opens on it
Decreasing the angle minute by minute, the ellipse becomes first perceptibly eccentric, then manifestly so, and by and by acquires so immensely elongated a form, as to bear no recognizable resemblance to a circle. By continuing this process, the ellipse passes insensibly into a parabola; and, ultimately, by still further diminishing the angle, into an hyperbola. Now here we have four different species of…
Herbert Spencer, Essays
names the idea
Related ideas
- Springer Science+Business Media
- Isaac Newton
- Johannes Kepler
- Kepler's laws of planetary motion
- Leonardo da Vinci
- Algebra
- Mathematics
- Real number
- Optics
- Physics
- Newton's law of universal gravitation
- Proclus
- Quantum mechanics
- Statistics
- Solar System
- Bijection
- Astronomy
- Circle
- Chicago
- Diameter
Place in the tradition
Part of Direct and indirect realism.