Difference between revisions of "Ellipse" - New World Encyclopedia

Revision as of 05:21, 27 November 2007

"Elliptical" redirects here.

An ellipse and some of its mathematical parameters.

An ellipse (shaded green) is a type of conic section.

In mathematics, an ellipse (from the Greek word ἔλλειψις, which literally means "absence") is a closed curve on a plane, such that the sum of the distances from any point on the curve to two fixed points is a constant. The two fixed points are called foci (plural of focus). A circle is a special case of an ellipse, in which the two foci coincide.

An ellipse is a type of conic section. If a conical surface is cut with a plane that does not intersect the base of the cone, the intersection of the cone and the plane is an ellipse.

The orbits of planets and many comets have been found to be elliptical.

Parameters of an ellipse

The point that lies halfway between the two foci is called the center of the ellipse. The line segment (or chord) AB (in the figure), which passes through the foci (F1, F2) and terminates on the ellipse, is called the major axis. This axis is the longest segment that can be obtained by joining two points on the ellipse. The two points at which the major axis intersects the curve are called the vertices. The line segment (chord) CD, which passes through the center, is perpendicular to the major axis, and terminates on the ellipse, is called the minor axis.

The semimajor axis (denoted by a in the figure) is one-half of the major axis and corresponds to the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis (denoted by b in the figure) is one-half the minor axis.

A chord drawn through a focus perpendicular to the major axis is called the latus rectum.

The area enclosed by an ellipse is πab, where a and b are the semimajor and semiminor axes.

Drawing an ellipse

An ellipse can be drawn on a sheet of paper using two pins, a pencil, and a loop of string. The pins are placed vertically at two separate points (which will become the foci of the ellipse), and the loop of string is placed around the pins and the pencil. The pencil is held vertically such that the string is taut and forms a triangle. If the pencil is moved around so that the string stays taut, it will trace out an ellipse, because the sum of the distances from the pencil to the pins will remain a constant.

Eccentricity

The shape of an ellipse can be expressed by a number called the eccentricity of the ellipse, conventionally denoted $\,\varepsilon$ . It is the ratio obtained when the distance from the center to a focus is divided by the distance from the center to one of the vertices. Thus, eccentricity is a non-negative number less than 1 and greater than or equal to 0. An eccentricity of 0 implies that the two foci occupy the same point (corresponding to the center), and in such a case, the ellipse is a circle.

For an ellipse with semimajor axis a and semiminor axis b, the eccentricity is

\varepsilon ={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}

.

The greater the eccentricity is, the larger the ratio of a to b, and therefore the more elongated the ellipse.

If c equals the distance from the center to either focus, then

\varepsilon ={\frac {c}{a}}

.

The distance c is known as the linear eccentricity of the ellipse. The distance between the foci is 2aε.

Ellipses in physics

In the seventeenth century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.

More generally, in the gravitational two-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus.

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse.

Stretching and projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

An ellipse can be produced by multiplying the x coordinates of all points on a circle by a constant, without changing the y coordinates. This is equivalent to stretching the circle out in the x-direction.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse. In a circle, all light would be reflected back to the center since all tangents are orthogonal to the radius.

Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at another focus remarkably well. Such a room is called a whisper chamber. Examples are the National Statuary Hall Collection at the U.S. Capitol (where John Quincy Adams is said to have used this property for eavesdropping on political matters), at an exhibit on sound at the Museum of Science and Industry in Chicago, in front of the University of Illinois at Urbana-Champaign Foellinger Auditorium, and also at a side chamber of the Palace of Charles V, in the Alhambra.

Equations

An ellipse with a semimajor axis a and semiminor axis b, centered at the point $(h,k)$ and having its major axis parallel to the x-axis may be specified by the equation

{\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1

.

This ellipse can be expressed parametrically as

x=h+a\,\cos t,\,\!

y=k+b\,\sin t\,\!

where $t$ may be restricted to the interval $-\pi \leq t\leq \pi \,\!$ .

If $h$ = 0 and $k$ = 0 (i.e., if the center is the origin (0,0)), then we can express this ellipse in polar coordinates by the equation

r={\frac {ab}{\sqrt {a^{2}\sin ^{2}\theta +b^{2}\cos ^{2}\theta }}}={\frac {b}{\sqrt {1-\varepsilon ^{2}\cos ^{2}\theta }}}

where $\varepsilon$ is the eccentricity of the ellipse.

With one focus at the origin, the ellipse's polar equation is

r={\frac {a\cdot (1-\varepsilon ^{2})}{1+\varepsilon \cdot \cos \theta }}

.

Circumference

The circumference of an ellipse is $4aE(\varepsilon )$ , where the function E is the complete elliptic integral of the second kind.

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted $l\,\!$ (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to $a\,\!$ and $b\,\!$ (the ellipse's semi-axes) by the formula $al=b^{2}\,\!$ or, if using the eccentricity, $l=a\cdot (1-\varepsilon ^{2})\,\!$ .

In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation

r\cdot (1+\varepsilon \cdot \cos \theta )=l\,\!

An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

References
ISBN links support NWE through referral fees

Arnone, Wendy. 2001. Geometry for Dummies. Hoboken, NJ: For Dummies (Wiley). ISBN 0764553240.

Research and Education Association. 1999. Math Made Nice-n-Easy Books #7: Trigonometric Identities & Equations, Straight Lines, Conic Sections. Piscataway, N.J.: Research & Education Association. ISBN 0878912061.

Smith, Karen E. 2000. An Invitation to Algebraic Geometry. New York: Springer. ISBN 0387989803.

Stillwell, John. 1998. Numbers and Geometry. Undergraduate Texts in Mathematics. New York: Springer. ISBN 0387982892.

External links

Apollonius' Derivation of the Ellipse at Convergence. Retrieved November 3, 2007.
Ellipse & Hyperbola Construction - An interactive sketch showing how to trace the curves of the ellipse and hyperbola. (Requires Java). Retrieved November 3, 2007.
Ellipse Construction - Another interactive sketch, this time showing a different method of tracing the ellipse. (Requires Java). Retrieved November 3, 2007.
Ellipse on MathWorld - More on Ellipse. Retrieved November 3, 2007.
Kimberling, Clark. The Shape and History of The Ellipse in Washington, D.C. Retrieved November 3, 2007.
Collection of animated ellipse demonstrations. Ellipse, axes, semi-axes, area, perimeter, tangent, foci. Retrieved November 3, 2007.
Woodworking videos showing how to work with ellipses in wood. Retrieved November 3, 2007.

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@@ Line 15: / Line 15: @@
 The '''[[semimajor axis]]''' (denoted by ''a'' in the figure) is one-half of the major axis and corresponds to the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the '''[[Semi-minor axis|semiminor axis]]''' (denoted by ''b'' in the figure) is one-half the minor axis.
+A chord drawn through a focus perpendicular to the major axis is called the ''latus rectum''.
 The [[Area of an ellipse|area]] enclosed by an ellipse is ''πab'', where ''a'' and ''b'' are the semimajor and semiminor axes.
@@ Line 34: / Line 36: @@
 :<math>\varepsilon = \frac{c}{a}</math>.
 The distance ''c'' is known as the '''linear eccentricity''' of the ellipse.  The distance between the foci is 2''a''&epsilon;.
+== Ellipses in physics ==
+In the seventeenth century, [[Johannes Kepler]] explained that the orbits along which the planets travel around the Sun are ellipses in his [[Kepler's laws of planetary motion|first law of planetary motion]]. Later, [[Isaac Newton]] explained this as a corollary of his [[Newton's law of universal gravitation|law of universal gravitation]].
+More generally, in the gravitational [[two-body problem]], if the two bodies are bound to each other (that is, the total energy is negative), their orbits are [[similar]] ellipses with the common [[barycenter]] being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus.
+The general solution for a [[harmonic oscillator]] in two or more [[dimension]]s is also an ellipse, but this time with the origin of the force located at the center of the ellipse.
+== Stretching and projection ==
+An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line).  Similarly, any [[oblique projection]] onto a plane results in a conic section.  If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.
+An ellipse can be produced by multiplying the ''x'' coordinates of all points on a circle by a constant, without changing the ''y'' coordinates. This is equivalent to ''stretching'' the circle out in the x-direction.
+== Reflection property ==
+Assume an elliptic [[mirror]] with a light source at one of the foci. Then all rays are [[Reflection (physics)|reflected]] to a single point &mdash; the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse. In a circle, all light would be reflected back to the center since all tangents are [[orthogonal]] to the radius.
+Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at another focus remarkably well.  Such a room is called a ''whisper chamber''.  Examples are the [[National Statuary Hall Collection]] at the [[U.S. Capitol]] (where John Quincy Adams is said to have used this property for eavesdropping on political matters), at an exhibit on sound at the [[Museum of Science and Industry (Chicago)|Museum of Science and Industry]] in [[Chicago]], in front of the [[University of Illinois at Urbana-Champaign]] Foellinger Auditorium, and also at a side chamber of the Palace of Charles V, in the [[Alhambra]].
 ==Equations==
@@ Line 54: / Line 76: @@
 :<math>r = \frac{ a\cdot(1-\varepsilon^{2})}{1 + \varepsilon\cdot\cos\theta} </math>.
-==Semi-latus rectum and polar coordinates==
+===Circumference===
+The [[circumference]] of an ellipse is <math>4 a E(\varepsilon)</math>,
+where the function ''E'' is the complete [[elliptic integral]] of the [[Elliptic integral#Complete elliptic integral of the second kind|second kind]].
+===Semi-latus rectum and polar coordinates===
 The ''[[semi-latus rectum]]'' of an ellipse, usually denoted <math>l\,\!</math> ([[lowercase]] L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line [[perpendicular]] to the major axis. It is related to <math>a\,\!</math> and <math>b\,\!</math> (the ellipse's semi-axes) by the formula <math>al=b^2\,\!</math> or, if using the eccentricity, <math>l=a\cdot(1-\varepsilon^2)\,\!</math>.
@@ Line 64: / Line 91: @@
 An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle &phi; to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin&nbsp;&phi;, provided &phi; is not 90°.
-==Circumference==
-The [[circumference]] of an ellipse is <math>4 a E(\varepsilon)</math>,
-where the function ''E'' is the complete [[elliptic integral]] of the [[Elliptic integral#Complete elliptic integral of the second kind|second kind]].
-The exact [[infinite series]] is:
-:<math>C = 2\pi a \left[{1 - \left({1\over 2}\right)^2\varepsilon^2 - \left({1\cdot 3\over 2\cdot 4}\right)^2{\varepsilon^4\over 3} - \left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2{\varepsilon^6\over5} - \dots}\right]\!\,</math>
-Or:
-:<math>C = 2\pi a \sum_{n=0}^\infty {\left\lbrace - \left[\prod_{m=1}^n \left({ 2m-1 \over 2m}\right)\right]^2 {\varepsilon^{2n}\over 2n - 1}\right\rbrace}</math>
-A good [[approximation]] is [[Srinivasa_Ramanujan|Ramanujan]]'s:
-:<math>C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]\!\,</math>
-which can also be written as:
-:<math>C \approx \pi a \left[ 3 (1+\sqrt{1-\varepsilon^2}) - \sqrt{(3+ \sqrt{1-\varepsilon^2})(1+3 \sqrt{1-\varepsilon^2})} \right] \!\,</math>
-For the special case where the minor axis is half the major axis, we get:
-:<math>C \approx \pi a (9 - \sqrt{35})/2 \!\,</math>
-or
-<math>C \approx \frac{a}{2} \sqrt{93 + \frac{1}{2} \sqrt{3}} \!\,</math> (better approximation).
-More generally, the [[arc length]] of a portion of the circumference, as a function of the angle subtended, is given by an incomplete [[elliptic integral]].  The [[inverse function]], the angle subtended as a function of the arc length, is given by the [[elliptic functions]].
-== Stretching and projection ==
-An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse.  The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line).  Similarly, any [[oblique projection]] onto a plane results in a conic section.  If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.
-An ellipse can be produced by multiplying the ''x'' coordinates of all points on a circle by a constant, without changing the ''y'' coordinates. This is equivalent to ''stretching'' the circle out in the x-direction.
-== Reflection property ==
-Assume an elliptic [[mirror]] with a light source at one of the foci. Then all rays are [[Reflection (physics)|reflected]] to a single point &mdash; the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse. In a circle, all light would be reflected back to the center since all tangents are [[orthogonal]] to the radius.
-Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at another focus remarkably well.  Such a room is called a ''whisper chamber''.  Examples are the [[National Statuary Hall Collection]] at the [[U.S. Capitol]] (where John Quincy Adams is said to have used this property for eavesdropping on political matters), at an exhibit on sound at the [[Museum of Science and Industry (Chicago)|Museum of Science and Industry]] in [[Chicago]], in front of the [[University of Illinois at Urbana-Champaign]] Foellinger Auditorium, and also at a side chamber of the Palace of Charles V, in the [[Alhambra]].
-== Ellipses in physics ==
-In the seventeenth century, [[Johannes Kepler]] explained that the orbits along which the planets travel around the Sun are ellipses in his [[Kepler's laws of planetary motion|first law of planetary motion]]. Later, [[Isaac Newton]] explained this as a corollary of his [[Newton's law of universal gravitation|law of universal gravitation]].
-More generally, in the gravitational [[two-body problem]], if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are [[similar]] ellipses with the common [[barycenter]] being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus.
-The general solution for a [[harmonic oscillator]] in two or more [[dimension]]s is also an ellipse, but this time with the origin of the force located at the center of the ellipse.
-==Ellipses in computer graphics==
-Drawing an ellipse is a common graphics primitive in standard display libraries, such as the Macintosh [[QuickDraw]] API and the Windows [[Graphics Device Interface]] (GDI).  Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. [[Jack Bresenham]] at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit.  An efficient generalization to draw ellipses was invented in 1984 by [[Jerry Van Aken]] (IEEE CG&A, Sept. 1984).
-Sample JavaScript code to calculate the points of an ellipse.
-<source lang="javascript">
-/**
-* This functions returns an array containing 36 points to draw an
-* ellipse.
-*
-* @param x {double} X coordinate
-* @param y {double} Y coordinate
-* @param a {double} Semimajor axis
-* @param b {double} Semiminor axis
-* @param angle {double} Angle of the ellipse
-*/
-function calculateEllipse(x, y, a, b, angle, steps) {
-  if (steps == null)
-    steps = 36;
-  var points = [];
-  var beta = -angle / 180 * Math.PI;
-  var sinbeta = Math.sin(beta);
-  var cosbeta = Math.cos(beta);
-  for (var i = 0; i < 360; i += 360 / steps) {
-    var alpha = i / 180 * Math.PI;
-    var sinalpha = Math.sin(alpha);
-    var cosalpha = Math.cos(alpha);
-    var X = x + (a * cosalpha * cosbeta - b * sinalpha * sinbeta);
-    var Y = y + (a * cosalpha * sinbeta + b * sinalpha * cosbeta);
-    points.push(new OpenLayers.Geometry.Point(X, Y));
-  }
-  return points;
-}
-</source>
 == See also ==
@@ Line 163: / Line 100: @@
 * [[Oval]]
 * [[Parabola]]
-== Notes ==
-<references/>
 ==References==