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ellipse

ellipse, closed plane curve consisting of all points for which the sum of the distances between a point on the curve and two fixed points (foci) is the same. It is the conic section formed by a plane cutting all the elements of the cone in the same nappe. The center of an ellipse is the point halfway between its foci. The major axis is the chord that passes through the foci. The minor axis is the chord that passes through the center perpendicular to the major axis. The latus rectum is the chord through either focus perpendicular to the major axis. The vertices are the two points of intersection of the major axis with the curve. The eccentricity of an ellipse, a ratio of two lengths, is a measure of its flatness; it is the distance from the center to either focus divided by the distance from the center to either vertex. The circle may be considered an ellipse of eccentricity zero, i.e., one in which the center and the two foci all coincide.

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ellipse

ellipse Conic section formed by cutting a right circular cone with a plane inclined at such an angle that the plane does not intersect the base of the cone. When the intersecting plane is parallel to the base, the conic section is a circle. In rectangular Cartesian coordinates its standard equation is x2/a2 + y2/b2 = 1. Most planetary orbits are ellipses.

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ellipse

el·lipse / iˈlips/ • n. a regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points (the foci) is constant, or resulting when a cone is cut by an oblique plane that does not intersect the base.

ellipse

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ellipse

ellipse. Figure formed by section made by a plane passing obliquely through the axis of a regular cone. Unlike an oval, it is identical at each end, i.e. on both sides of its dividing axes. See also arch.

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ellipse

ellipse XVIII. — F. — L. ell¯psis (see next).
Hence ellipsoid XVIII.

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ellipse

ellipseapse, collapse, craps, elapse, lapse, perhaps, schnapps •prolapse • synapse • Lesseps •quadriceps •biceps, triceps •forceps •traipse, trapes •jackanapes • Pepys •Chips, eclipse, ellipse, thrips •Phillips • apocalypse •amidships, midships •cripes, Stars and Stripes •copse • Cheops • Pelops • Cyclops •triceratops • corpse • Stopes •oops, whoops •turps • mumps • goosebumps

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Ellipse

Ellipse

Other definitions of an ellipse

Features

Drawing ellipses

Uses

Resources

An ellipse is a two-dimensional shape similar to a stretched circle, only it has a bit longer and flatter sides. It is often classified as an oval. In fact, an ellipse is the oval formed by the intersection of a plane and a right circular coneone of the four types of conic sections. The other three are the circle, hyperbola, and parabola. The ellipse is symmetrical along two lines, called axes. The major axis runs through the longest part of the ellipse and its center, and the minor axis is perpendicular to the major axis through the ellipses center.

Other definitions of an ellipse

Ellipses are described in several ways, each way having its own advantages and limitations:

  1. An ellipse is a set of points, the sum of whose distances from two fixed points (the foci, which lie on the major axis) is constant. That is, P: PF1 + PF2 = constant.
  2. An ellipse is a set of points whose distances from a fixed point (the focus) and fixed line (the directrix) are in a constant ratio less than 1. That is, P: PF/PD = e, where 0 < e < 1. The constant, e, is the eccentricity of the ellipse.
  3. An ellipse is a set of points (x, y) in a Cartesian plane satisfying an equation of the form x2/a2 + y2/b2 = 1, where a is the distance from the origin to the end of the x axis on the ellipse (semimajor axis), b is the distance from the origin to the end of the y axis on the ellipse (semiminor axis), and a > b. The equation of an ellipse can have other forms, but this one, with the center at the origin and the major axis coinciding with one of the coordinate axes, is the simplest.
  4. An ellipse is a set of points (x, y) in a Cartesian plane satisfying the parametric equations x = a cos t and y = b sin t, where a and bare constants and t is a variable that is 0 < t < 2π. Other parametric equations are possible, but these are the simplest.

Features

In working with ellipses it is useful to identify several special points, chords, measurements, and properties:

The major axis: The longest chord in an ellipse that passes through the foci. It is equal in length to the constant sum in Definition 1 above.

The center: The midpoint, C, of the major axis.

The vertices: The end points of the major axis.

The minor axis: The chord which is perpendicular to the major axis at the center. It is the shortest chord that passes through the center.

The foci: The fixes points in Definitions 1 and 2. In any ellipse, these points lie on the major axis and are at a distance c on either side of the center. When a and b are the semimajor and semiminor axes respectively, then a2 = b2 + c2.

The eccentricity: A measure of the relative elongation of an ellipse. It is the ratio e in Definition 2, or the ratio FC/VC (center-to-focus divided by center-to-vertex). These two definitions are mathematically equivalent. When the eccentricity is close to zero, the ellipse is almost circular; when it is close to 1, the ellipse is almost a parabola. All ellipses having the same eccentricity are geometrically similar figures.

The angle measure of eccentricity: Another measure of eccentricity. It is the acute angle formed by the major axis and a line passing through one focus and an end point of the minor axis. This angle is the arc cosine of the eccentricity.

The area: The area of an ellipse is given by the simple formula πab, where a and b are the semimajor and semiminor axes.

The perimeter: There is no simple formula for the perimeter of an ellipse. The formula is an elliptic integral that can be evaluated only by approximation.

The reflective property of an ellipse: If an ellipse is thought of as a mirror, any ray that passes through one focus and strikes the ellipse will be reflected through the other focus. This is the principle behind rooms designed so that a small sound made at one location can be easily heard at another, but not elsewhere in the room. The two locations are the foci of an ellipse.

Drawing ellipses

There are mechanical devices, called ellipsographs, based on Definition 4 for drawing ellipses precisely, but lacking such a device, one can use simple equipment and the definitions above to draw ellipses that are accurate enough for most practical purposes.

To draw large ellipses one can use the pin-and-string method based on Definition 1: Stick pins into

KEY TERMS

Axes Chords through the center of an ellipse. The major axis is the longest chord and the minor axis the shortest. They are perpendicular to each other.

Eccentricity A measure of the relative lengths the major and minor axes of an ellipse.

Foci Two points in the interior of an ellipse. Their spacing, along with one other dimension, determines the size and shape of an ellipse.

the drawing board at the two foci and at one end of the minor axis. Tie a loop of string snugly around the three pins. Replace the pin at the end of the minor axis with a pencil. While keeping the loop taut, draw the ellipse. If string is used that does not stretch, the resulting ellipse will be quite accurate.

To draw small and medium sized ellipses a technique based on Definition 4 can be used: Draw two concentric circles whose radii (make up values for a and b) are equal to the semimajor axis and the semi-minor axis respectively. Draw a ray from the center, intersecting the inner circle at y and outer circle at x. From y, draw a short horizontal line, and from x, draw a short vertical line. Where these lines intersect is a point on the ellipse. Continue this procedure with many different rays until points all around the ellipse have been located. Connect these points with a smooth curve. If this is done carefully, using ordinary drafting equipment, the resulting ellipse will be quite accurate.

Uses

Ellipses are found in both natural and artificial objects. The paths of the planets and some comets around the Sun are approximately elliptical, with the Sun at one of the foci. The seam where two cylindrical pipes are joined is an ellipse. Artists drawing circular objects such as the tops of vases use ellipses to render them in proper perspective. In Salt Lake City, the roof of the Mormon Tabernacle has the shape of an ellipse rotated around its major axis, and its reflective properties give the auditorium its unusual acoustical properties. For instance, a person standing at one focus can hear a pin dropped at the other focus clearly. An ellipsoidal reflector in a lamp such as those dentists use will, if the light source is placed at its focus, concentrate the light at the other focus.

Because the ellipse is a particularly graceful sort of oval, it is widely used for esthetic purposes, in the design of formal gardens, in table tops, in mirrors, in picture frames, and in other decorative uses.

Resources

BOOKS

Finney, Ross L, et al. Calculus: Graphical, Numerical,Algebraic. Glenview, IL: Prentice Hall, 2003.

Jeffrey, Alan. Mathematics for Engineers and Scientists. Boca Raton, FL: Chapman & Hall/CRC, 2005.

Larson, Ron. Calculus: An Applied Approach. Boston, MA:Houghton Mifflin, 2003.

Lyublinskava, Irina E. Connecting Mathematics with Science: Experiments for Precalculus. Emeryville, CA: Key Curriculum Press, 2003.

J. Paul Moulton

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Ellipse

Ellipse

An ellipse is a kind of oval. It is the oval formed by the intersection of a plane and a right circular cone-one of the four types of conic sections . The other three are the circle , the hyperbola , and the parabola . The ellipse is symmetrical along two lines, called axes. The major axis runs through the longest part of the ellipse and its center, and the minor axis is perpendicular to the major axis through the ellipse's center.


Other definitions of an ellipse

Ellipses are described in several ways, each way having its own advantages and limitations:

  1. The set of points, the sum of whose distances from two fixed points (the foci, which lie on the major axis) is constant. That is, P: PF1 + PF2 = constant.
  2. The set of points whose distances from a fixed point (the focus) and fixed line (the directrix) are in a constant ratio less than 1. That is, P: PF/PD = e, where 0 < e < 1. The constant, e, is the eccentricity of the ellipse.
  3. The set of points (x,y) in a Cartesian plane satisfying an equation of the form x2/25 + y2/16 = 1. The equation of an ellipse can have other forms, but this one, with the center at the origin and the major axis coinciding with one of the coordinate axes, is the simplest.
  4. The set of points (x,y) in a Cartesian plane satisfying the parametric equations x = a cos t and y = b sin t, where a and b are constants and t is a variable . Other parametric equations are possible, but these are the simplest.

Features

In working with ellipses it is useful to identify several special points, chords, measurements, and properties:

The major axis: The longest chord in an ellipse that passes through the foci. It is equal in length to the constant sum in Definition 1 above. In Definitions 3 and 4 the larger of the constants a or b is equal to the semimajor axis.

The center: The midpoint, C, of the major axis.

The vertices: The end points of the major axis.

The minor axis: The chord which is perpendicular to the major axis at the center. It is the shortest chord which passes through the center. In Definitions 3 and 4 the smaller of a or b is the semiminor axis.

The foci: The fixes points in Definitions 1 and 2. In any ellipse, these points lie on the major axis and are at a distance c on either side of the center. If a and b are the semimajor and semiminor axes respectively, then a2 = b2 + c2.. In the examples in Definitions 3 and 4, the foci are 3 units from the center.

The eccentricity: A measure of the relative elongation of an ellipse. It is the ratio e in Definition 2, or the ratio FC/VC (center-to-focus divided by center-to-vertex). These two definitions are mathematically equivalent. When the eccentricity is close to zero , the ellipse is almost circular; when it is close to 1, the ellipse is almost a parabola. All ellipses having the same eccentricity are geometrically similar figures.

The angle measure of eccentricity: Another measure of eccentricity. It is the acute angle formed by the major axis and a line passing through one focus and an end point of the minor axis. This angle is the arc cosine of the eccentricity.

The area: The area of an ellipse is given by the simple formula PIab, where a and b are the semimajor and semiminor axes.

The perimeter: There is no simple formula for the perimeter of an ellipse. The formula is an elliptic integral which can be evaluated only by approximation .

The reflective property of an ellipse: If an ellipse is thought of as a mirror, any ray which passes through one focus and strikes the ellipse will be reflected through the other focus. This is the principle behind rooms designed so that a small sound made at one location can be easily heard at another, but not elsewhere in the room. The two locations are the foci of an ellipse.


Drawing ellipses

There are mechanical devices, called ellipsographs, based on Definition 4 for drawing ellipses precisely, but lacking such a device, one can use simple equipment and the definitions above to draw ellipses which are accurate enough for most practical purposes.

To draw large ellipses one can use the pin-and-string method based on Definition 1: Stick pins into the drawing board at the two foci and at one end of the minor axis. Tie a loop of string snugly around the three pins. Replace the pin at the end of the minor axis with a pencil and, keeping the loop taut, draw the ellipse. If string is used that does not stretch, the resulting ellipse will be quite accurate.

To draw small and medium sized ellipses a technique based on Definition 4 can be used: Draw two concentric circles whose radii are equal to the semimajor axis and the semiminor axis respectively. Draw a ray from the center, intersecting the inner circle at y and outer circle at x. From y draw a short horizontal line and from x a short vertical line. Where these lines intersect is a point on the ellipse. Continue this procedure with many different rays until points all around the ellipse have been located. Connect these points with a smoothcurve . If this is done carefully, using ordinary drafting equipment, the resulting ellipse will be quite accurate.


Uses

Ellipses are found in both natural and artificial objects. The paths of the planets and some comets around the Sun are approximately elliptical, with the sun at one of the foci. The seam where two cylindrical pipes are joined is an ellipse. Artists drawing circular objects such as the tops of vases use ellipses to render them in proper perspective. In Salt Lake City the roof of the Mormon Tabernacle has the shape of an ellipse rotated around its major axis, and its reflective properties give the auditorium its unusual acoustical properties. (A pin dropped at one focus can be heard clearly by a person standing at the other focus.) An ellipsoidal reflector in a lamp such as those dentists use will, if the light source is placed at its focus, concentrate the light at the other focus.

Because the ellipse is a particularly graceful sort of oval, it is widely used for esthetic purposes, in the design of formal gardens, in table tops, in mirrors , in picture frames, and in other decorative uses.


Resources

books

Finney, Thomas, Demana, and Waits. Calculus: Graphical, Numerical, Algebraic. Reading, MA: Addison Wesley Publishing Co., 1994.


J. Paul Moulton

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http://www.chicagomanualofstyle.org/tools_citationguide.html

American Psychological Association

http://apastyle.apa.org/

Notes:
  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.