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# definitions - ELLIPSES

## ellipsis(n.)

1.omission or suppression of parts of words or sentences

## ellipse(n.)

1.a closed plane curve resulting from the intersection of a circular cone and a plane cutting completely through it"the sums of the distances from the foci to any point on an ellipse is constant"

# Merriam Webster

EllipseEl*lipse" (ĕl*lĭps"), n. [Gr. 'elleipsis, prop., a defect, the inclination of the ellipse to the base of the cone being in defect when compared with that of the side to the base: cf. F. ellipse. See Ellipsis.]
1. (Geom.) An oval or oblong figure, bounded by a regular curve, which corresponds to an oblique projection of a circle, or an oblique section of a cone through its opposite sides. The greatest diameter of the ellipse is the major axis, and the least diameter is the minor axis. See Conic section, under Conic, and cf. Focus.

2. (Gram.) Omission. See Ellipsis.

3. The elliptical orbit of a planet.

The Sun flies forward to his brother Sun;
The dark Earth follows wheeled in her ellipse.
Tennyson.

# definition (more)

definition of Wikipedia

eclipsis

# analogical dictionary

MESH root[Thème]

ellipsis [MeSH]

MESH root[Thème]

ellipsis [MeSH]

ellipsis; eclipsis[ClasseHyper.]

(allusion)[termes liés]

linguistics[Domaine]

Removing[Domaine]

ellipsis (n.)

ligne courbe (fr)[Classe]

ligne (géométrie) (fr)[Classe]

line[Classe]

chose courbe (fr)[ClasseParExt.]

figure géométrique (fr)[Classe]

ligne courbe (géométrie) (fr)[Classe]

en forme d'œuf (fr)[Classe]

ligne courbe (fr)[Thème]

ovale (fr)[Thème]

cône (fr)[termes liés]

ovale (fr)[termes liés]

geometry[Domaine]

ShapeAttribute[Domaine]

factotum[Domaine]

GeometricFigure[Domaine]

visage (fr)[DomaineDescription]

geometry[Domaine]

rounded[Similaire]

ellipse (n.)

# Ellipsis

‌…
Ellipsis
…  . . .
Precomposed ellipsis  Spaced 3 periods  Mid-line ellipsis
Punctuation Word dividers General typography Intellectual property Currency apostrophe ( ’ ' ) brackets ( [ ], ( ), { }, ⟨ ⟩ ) colon ( : ) comma ( , ، 、 ) dash ( ‒, –, —, ― ) ellipsis ( …, ..., . . . ) exclamation mark ( ! ) full stop/period ( . ) guillemets ( « » ) hyphen ( ‐ ) hyphen-minus ( - ) question mark ( ? ) quotation marks ( ‘ ’, “ ”, ' ', " " ) semicolon ( ; ) slash‌/stroke‌/solidus ( /,  ⁄  ) space ( ) ( ) ( ) interpunct ( · ) ampersand ( & ) at sign ( @ ) asterisk ( * ) backslash ( \ ) bullet ( • ) caret ( ^ ) dagger ( †, ‡ ) degree ( ° ) ditto mark ( 〃 ) inverted exclamation mark ( ¡ ) inverted question mark ( ¿ ) number sign‌/pound‌/hash ( # ) numero sign ( № ) obelus ( ÷ ) ordinal indicator ( º, ª ) percent, per mil ( %, ‰ ) basis point ( ‱ ) pilcrow ( ¶ ) prime ( ′, ″, ‴ ) section sign ( § ) tilde ( ~ ) underscore‌/understrike ( _ ) vertical bar‌/broken bar‌/pipe ( ¦, | ) copyright symbol ( © ) registered trademark ( ® ) service mark ( ℠ ) sound recording copyright ( ℗ ) trademark ( ™ ) currency (generic) ( ¤ ) currency (specific) ( ) asterism ( ⁂ ) tee ( ⊤ ) up tack ( ⊥ ) index/fist ( ☞ ) therefore sign ( ∴ ) because sign ( ∵ ) interrobang ( ‽ ) irony punctuation ( ؟ ) lozenge ( ◊ ) reference mark ( ※ ) tie ( ⁀ ) diacritical marks whitespace characters non-English quotation style ( « », „ ” ) Chinese punctuation Hebrew punctuation Japanese punctuation Korean punctuation
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Ellipsis (plural ellipses; from the Ancient Greek: ἔλλειψις, élleipsis, "omission" or "falling short") is a series of marks that usually indicate an intentional omission of a word, sentence or whole section from the original text being quoted. An ellipsis can also be used to indicate an unfinished thought or, at the end of a sentence, a trailing off into silence (aposiopesis), example: "But I thought he was . . ." When placed at the beginning or end of a sentence, the ellipsis can also inspire a feeling of melancholy or longing. The ellipsis calls for a slight pause in speech or any other form of text, but it is incorrect to use ellipses solely to indicate a pause in speech.

The most common form of an ellipsis is a row of three periods or full stops (. . .) or a pre-composed triple-dot glyph (). The usage of the em dash (—) can overlap the usage of the ellipsis. The Chicago Manual of Style recommends that an ellipsis be formed by typing three periods, each with a space on both sides.

The triple-dot punctuation mark is also called a suspension point, points of ellipsis, periods of ellipsis, or colloquially, dot-dot-dot.

## In writing

In the 19th and early 20th centuries, an ellipsis was often used when a writer intentionally omitted a specific proper noun, such as a location: "Jan was born on . . . Street in Warsaw."

As commonly used, this juxtaposition of characters is referred to as "dots of ellipsis" in the English language.

Occasionally, it would be used in pulp fiction and other works of early 20th C. fiction to denote expletives that would otherwise have been censored.[1]

An ellipsis may also imply an unstated alternative indicated by context. For example, when Count Dracula says "I never drink . . . wine", the implication is that he does drink something else.

In reported speech, the ellipsis is sometimes used to represent an intentional silence, perhaps indicating irritation, dismay, shock or disgust.[citation needed]

In poetry, this is used to highlight sarcasm or make the reader think about the last points in the poem.

In news reporting, it is used to indicate that a quotation has been condensed for space, brevity or relevance.

Herb Caen, the decades-long, Pulitzer-prize-winning columnist for the San Francisco Chronicle, was famously known for his "Three-dot journalism".

## Across different languages

### In English

The Chicago Manual of Style suggests the use of an ellipsis for any omitted word, phrase, line, or paragraph from within a quoted passage. There are two commonly used methods of using ellipses: one uses three dots for any omission, while the second one makes a distinction between omissions within a sentence (using three dots: . . .) and omissions between sentences (using a period and a space followed by three dots: . ...). An ellipsis at the end of a sentence with no sentence following should be preceded by a period (for a total of four dots).

The Modern Language Association (MLA), however, used to indicate that an ellipsis must include spaces before and after each dot in all uses. If an ellipsis is meant to represent an omission, square brackets must surround the ellipsis to make it clear that there was no pause in the original quote: [ . . . ]. Currently, the MLA has removed the requirement of brackets in its style handbooks. However, some maintain that the use of brackets is still correct because it clears confusion.[2]

The MLA now indicates that a three-dot, spaced ellipsis ( … ) should be used for removing material from within one sentence within a quote. When crossing sentences (when the omitted text contains a period, so omitting the end of a sentence counts), a four-dot, spaced (except for before the first dot) ellipsis (. . . . ) should be used. When ellipsis points are in the original text, the ellipsis points should be enclosed in square brackets. (text . . . text would be quoted as "text […] text") [3]

According to the Associated Press, the ellipsis should be used to condense quotations. It is less commonly used to indicate a pause in speech or an unfinished thought or to separate items in material such as show business gossip. The stylebook indicates that if the shortened sentence before the mark can stand as a sentence, it should do so, with an ellipsis placed after the period or other ending punctuation. When material is omitted at the end of a paragraph and also immediately following it, an ellipsis goes both at the end of that paragraph and in front of the beginning of the next, according to this style.[4]

According to Robert Bringhurst's Elements of Typographic Style, the details of typesetting ellipsis depend on the character and size of the font being set and the typographer's preference. Bringhurst writes that a full space between each dot is "another Victorian eccentricity. In most contexts, the Chicago ellipsis is much too wide"—he recommends using flush dots, or thin-spaced dots (up to one-fifth of an em), or the prefabricated ellipsis character (Unicode U+2026, Latin entity &hellip;). Bringhurst suggests that normally, an ellipsis should be spaced fore-and-aft to separate it from the text, but when it combines with other punctuation, the leading space disappears and the other punctuation follows. This is the usual practice in typesetting. He provides the following examples:

 i … j k…. l…, l l, … l m…? n…!

In legal writing in the United States, Rule 5.3 in the Bluebook citation guide governs the use of ellipsis and requires a space before the first dot and between the two subsequent dots. If an ellipsis ends the sentence, then there are three dots, each separated by a space, followed by the final punctuation, unless the final mark of punctuation is also a period.

### In Polish

When applied in Polish language syntax, the ellipsis is called wielokropek, which means "multidot". The word wielokropek distinguishes the ellipsis of Polish syntax from that of mathematical notation, in which it is known as an elipsa.

When an ellipsis replaces a fragment omitted from a quotation, the ellipsis is enclosed in parentheses or square brackets. An unbracketed ellipsis indicates an interruption or pause in speech.

The syntactical rules for ellipses are standardized by the 1983 Polska Norma document PN-83/P-55366, Zasady składania tekstów w języku polskim ("Rules for setting texts in the Polish Language").

### In Japanese

The most common character corresponding to an ellipsis is called 3-ten rīdā ("3-dot leaders", ). 2-ten rīdā exists as a character, but it is used less commonly. In writing, the ellipsis consists usually of six dots (two 3-ten rīdā characters, ……). Three dots (one 3-ten rīdā character) may be used where space is limited, such as in a header. However, variations in the number of dots exist. In horizontally written text the dots are commonly vertically centered within the text height (between the baseline and the ascent line), as in the standard Japanese Windows fonts; in vertically written text the dots are always centered horizontally. As the Japanese word for dot is pronounced "ten", the dots are colloquially called "ten-ten-ten" (てんてんてん, akin to the English "dot dot dot").

In Japanese manga, the ellipsis by itself represents speechlessness, or a "pregnant pause". Given the context, this could be anything from an admission of guilt to an expression of being dumbfounded at another person's words or actions. As a device, the ten-ten-ten is intended to focus the reader on a character while allowing the character to not speak any dialogue. This conveys to the reader a focus of the narrative "camera" on the silent subject, implying an expectation of some motion or action. It is not unheard of to see inanimate objects "speaking" the ellipsis.

### In Chinese

In Chinese, the ellipsis is six dots (in two groups of three dots, occupying the same horizontal space as two characters) (i.e. ……). The dots are always centered within the baseline and the ascender when horizontal (on the baseline has become acceptable)[citation needed] and centered horizontally when vertical.

## In mathematical notation

An ellipsis is also often used in mathematics to mean "and so forth". In a list, between commas, or following a comma, a normal ellipsis is used, as in:

$1,2,3,\ldots,100\,.$

To indicate the omission of values in a repeated operation, an ellipsis raised to the center of the line is used between two operation symbols or following the last operation symbol, as in:

$1+2+3+\cdots+100\,$

(though sometimes, for example, in Russian mathematical texts, normal, non-raised, ellipses are used even in repeated summations[5]).

The latter formula means the sum of all natural numbers from 1 to 100. However, it is not a formally defined mathematical symbol. Repeated summations or products may similarly be denoted using capital sigma and capital pi notation, respectively:

$1+2+3+\cdots+100\ = \sum_{n=1}^{100} n$
$1 \times 2 \times 3 \times \cdots \times 100\ = \prod_{n=1}^{100} n = 100!$ (see factorial)

Normally dots should be used only where the pattern to be followed is clear, the exception being to show the indefinite continuation of an irrational number such as:

$\pi=3.14159265\ldots$

Sometimes, it is useful to display a formula compactly, for example:

$1+4+9+\cdots+n^2+\cdots+400\,.$

Another example is the set of zeros of the cosine function.

$\left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}, \ldots \right\}\,.$

There are many related uses of the ellipsis in set notation.

The diagonal and vertical forms of the ellipsis are particularly useful for showing missing terms in matrices, such as the size-n identity matrix

$I_n = \begin{bmatrix}1 & 0 & \cdots & 0 \\0 & 1 & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 1 \end{bmatrix}.$

The use of ellipses in mathematical proofs is often discouraged because of the potential for ambiguity. For this reason, and because the ellipsis supports no systematic rules for symbolic calculation, in recent years some authors have recommended avoiding its use in mathematics altogether.[6]

## Computer interfaces and programming

Ellipses are often used in an operating system's taskbars or web browser tabs to indicate longer titles than will fit. Hovering the cursor over the tab often displays a tooltip of the full title. When many programs are open, or during a "tab explosion" in web browsing, the tabs may be reduced in size so much that no characters from the actual titles show, and ellipses take up all the space besides the program icon or favicon.

In many user interface guidelines, a "…" after the name of a command implies that the user will need to provide further information, for example in a subsequent dialog box, before the action can be completed. A typical example is the Save As… command, which after being clicked will usually require the user to enter a filename, as opposed to Save where the file will usually be saved under its existing name.

The ellipsis is used as an operator in some programming languages. The precise meaning varies by language, but it generally involves something dealing with multiple items. See Ellipsis (programming operator).

## On the Internet and in text messaging

The ellipsis is[citation needed] one of the favorite constructions of Internet chat rooms, and it has[citation needed] evolved over the past ten years into a staple of text-messaging. Although an ellipsis is technically complete with three periods (...), its rise in popularity as a "trailing-off" or "silence" indicator, particularly in mid-20th century comic strip and comic book prose writing, has led to expanded uses online. It has been used in new ways online, sometimes at the end of a message "to signal that the rest of the message is forthcoming."[7]

Today, extended ellipsis of two, seven, ten, or even dozens of periods have become common constructions in Internet chat rooms and text messages.[8] Often, the extended ellipses indicate an awkward silence or a "no comment" response to the previous statement made by the other party. They are sometimes used jokingly or for emphatic confusion about what the other person has said.[citation needed]

The incorrect use of "elliptical commas", or commas used in plurality for the effect of an ellipsis or multiple ellipses, has also grown in popularity online—although no style journal or manual has yet embraced them.[citation needed]

## Computer representations

In computing, several ellipsis characters have been codified, depending on the system used.

In the Unicode standard, there are the following characters:

Name Character Unicode HTML Entity Name Use
Horizontal ellipsis U+2026 &hellip; General
Laotian ellipsis U+0EAF General
Mongolian ellipsis U+1801 General
Thai ellipsis U+0E2F General
Vertical ellipsis U+22EE Mathematics
Midline horizontal ellipsis U+22EF Mathematics
Up-right diagonal ellipsis U+22F0 Mathematics
Down-right diagonal ellipsis U+22F1 Mathematics

In Windows, it can be inserted with Alt+0133.

In MacOS, it can be inserted with Opt+; (on an English language keyboard).

In Chinese and sometimes in Japanese, ellipsis characters are done by entering[clarification needed] two consecutive horizontal ellipsis (U+2026). In vertical texts, the application should rotate the symbol accordingly.

Unicode recognizes a series of three period characters (U+002E) as compatibility equivalent (though not canonical) to the horizontal ellipsis character.[9]

In HTML, the horizontal ellipsis character may be represented by the entity reference &hellip; (since HTML 4.0). Alternatively, in HTML, XML, and SGML, a numeric character reference such as &#x2026; or &#8230; can be used.

In the TeX typesetting system, the following types of ellipsis are available:

Character name Character TeX markup
Lower ellipsis $\ldots\,\!$ \ldots
Centred ellipsis $\cdots\,\!$ \cdots
Diagonal ellipsis $\ddots\,\!$ \ddots
Vertical ellipsis $\vdots\,\!$ \vdots
Up-right diagonal ellipsis \reflectbox{\ddots}

The horizontal ellipsis character also appears in the following older character maps:

Note that ISO/IEC 8859 encoding series provides no code point for ellipsis.

As with all characters, especially those outside of the ASCII range, the author, sender and receiver of an encoded ellipsis must be in agreement upon what bytes are being used to represent the character. Naive text processing software may improperly assume that a particular encoding is being used, resulting in mojibake.

The Chicago Style Q&A recommends to avoid the use of  (U+2026) character in manuscripts and to place three periods plus two nonbreaking spaces (. . .) instead, so that an editor, publisher, or designer can replace them later.[10]

In Abstract Syntax Notation One (ASN.1), the ellipsis is used as an extension marker to indicate the possibility of type extensions in future revisions of a protocol specification. In a type constraint expression like A ::= INTEGER (0..127, ..., 256..511) an ellipsis is used to separate the extension root from extension additions. The definition of type A in version 1 system of the form A ::= INTEGER (0..127, ...) and the definition of type A in version 2 system of the form A ::= INTEGER (0..127, ..., 256..511) constitute an extension series of the same type A in different versions of the same specification. The ellipsis can also be used in compound type definitions to separate the set of fields belonging to the extension root from the set of fields constituting extension additions. Here is an example: B ::= SEQUENCE { a INTEGER, b INTEGER, ..., c INTEGER }

Cohesion (linguistics)

## References

1. ^ Raymond Chandler, Frank MacShane. Raymond Chandler: Stories and Early Novels. First Edition. New York: Library of America. 1995. Note on the Texts.
2. ^ Fowler, H. Ramsey, Jane E. Aaron, Murray McArthur. The Little, Brown Handbook. Fourth Canadian Edition. Toronto: Pearson Longman. 2005. p. 440.
3. ^ http://www.naropa.edu/nwc/documents/citationcomparisonsp11.pdf
4. ^ Godlstein, Norm, editor. "Associated Press Stylebook and Briefing on Media Law". 2005. pp.328–329.
5. ^ Мильчин А. Э. Издательский словарь-справочник.— Изд. 3-е, испр. и доп., Электронное — М.: ОЛМА-Пресс, 2006. (in Russian)
6. ^ Roland Backhouse, Program Construction: Calculating Implementations from Specifications. Wiley (2003), page 138
7. ^ Judith C. Lapadat (July 2002). "Written Interaction: A Key Component in Online Learning". Journal of Computer-Mediated Communication. Retrieved 2009-06-24.
8. ^ Maness, Jack M. (2007). "The Power of Dots: Using Nonverbal Compensators in Chat Reference" (PDF). Proceedings of the 2007 Annual Meeting of ASIS&T. Annual Meeting of ASIS&T. University Libraries − University of Colorado at Boulder. Retrieved 24 October 2011.
9. ^ UnicodeData.txt: 2026;HORIZONTAL ELLIPSIS;Po;0;ON;<compat> 002E 002E 002E;;;;N;;;;;`
10. ^

# Ellipses

Ellipses is the plural form of two different English words:

# Ellipse

An ellipse obtained as the intersection of a cone with a plane.
The rings of Saturn are circular, but when seen partially edge on, as in this image, they appear to be ellipses. Picture by ESO

In mathematics, an ellipse (from Greek ἔλλειψις elleipsis, a "falling short") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas. Ellipses arise from the intersection of a right circular cylinder with a plane that is not parallel to the cylinder's main axis of symmetry. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.

## Elements of an ellipse

The ellipse and some of its mathematical properties.

An ellipse is a smooth closed curve which is symmetric about its horizontal and vertical axes. The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter.[1]

The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes,[2][3] the major and minor semiaxes,[4][5] or major radius and minor radius.[6][7][8][9]

The foci of the ellipse are two special points F1 and F2 on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF1 + PF2 = 2a ). Each of these two points is called a focus of the ellipse.

Refer to the lower Directrix section of this article for a second equivalent construction of an ellipse.

The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0<e<1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape. It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away.
The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse (f = ae).

## Drawing ellipses

### Pins-and-string method

Drawing an ellipse with two pins, a loop, and a pen.

The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil.[10] In this method, pins are pushed into the paper at two points which will become the ellipse's foci. A string tied at each end to the two pins and the tip of a pen is used to pull the loop taut so as to form a triangle. The tip of the pen will then trace an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed; thus it is called the gardener's ellipse.[11]

### Other methods

Trammel of Archimedes (ellipsograph) animation

An ellipse can also be drawn using a ruler, a set square, and a pencil:

Draw two perpendicular lines M,N on the paper; these will be the major and minor axes of the ellipse. Mark three points A, B, C on the ruler. A->C being the length of the major axis and B->C the length of the minor axis. With one hand, move the ruler on the paper, turning and sliding it so as to keep point A always on line N, and B on line M. With the other hand, keep the pencil's tip on the paper, following point C of the ruler. The tip will trace out an ellipse.

The trammel of Archimedes or ellipsograph is a mechanical device that implements this principle. The ruler is replaced by a rod with a pencil holder (point C) at one end, and two adjustable side pins (points A and B) that slide into two perpendicular slots cut into a metal plate.[12] The mechanism can be used with a router to cut ellipses from board material. The mechanism is also used in a toy called the "nothing grinder".

### Approximations to ellipses

An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. To draw the orbit with a pair of compasses the centre of the circle should be offset from the focus by an amount equal to the eccentricity multiplied by the radius.

## Mathematical definitions and properties

### In Euclidean geometry

#### Definition

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the directrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).

The equivalence of these definitions can be proved using the Dandelin spheres.

#### Equations

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is $\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1.$

#### Focus

The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

$f = \sqrt{a^2-b^2}.$

#### Eccentricity

The eccentricity of the ellipse (commonly denoted as either e or $\epsilon$) is

$e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}} =\sqrt{1-\left(\frac{b}{a}\right)^2} =f/a$

(where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening factor $g=1-\frac {b}{a}=1-\sqrt{1-e^2},$

$e=\sqrt{g(2-g)}.$

When the ellipse is expressed in general quadratic form, its eccentricity is given by this expression. Other expressions for the eccentricity are given here.

#### Directrix

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix resulting in the equality, e=PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse.
Besides the well–known ratio e=f/a, it is also true that e=a/d.

#### Circular directrix

The ellipse can also be defined as the set of points that are equidistant from one focus and a particular circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle, as is the entire ellipse.

#### Ellipse as hypotrochoid

An ellipse (in red) as a special case of the hypotrochoid with R = 2r.

The ellipse is a special case of the hypotrochoid when R = 2r.

#### Area

The area enclosed by an ellipse is πab, where a and b are one-half of the ellipse's major and minor axes respectively.

If the ellipse is given by the implicit equation $A x^2+ B x y + C y^2 = 1$, then the area is $\frac{2\pi}{\sqrt{ 4 A C - B^2 }}$.

#### Circumference

The circumference $C$ of an ellipse is:

$C = 4 a E(e)$

where again e is the eccentricity and where the function $E$ is the complete elliptic integral of the second kind.

The exact infinite series is:

$C = 2\pi a \left[{1 - \left({1\over 2}\right)^2e^2 - \left({1\cdot 3\over 2\cdot 4}\right)^2{e^4\over 3} - \left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2{e^6\over5} - \cdots}\right]\,\!$

or

$C = 2\pi a \left[1 - \sum_{n=1}^\infty {e^{2n}\over 2n - 1} \prod_{m=1}^n \left({ 2m-1 \over 2m}\right)^2\right] \,\!$

For computational purposes a much faster series where the denominators vanish at a rate $\tfrac{27}{1024} \left (\tfrac{a-b}{a+b} \right )^{8}$ is given by:[13]

$C = \frac{8\pi}{Q^{5/4}}\sum_{n=0}^\infty \frac{(\tfrac{1}{12})_{n}(\tfrac{5}{12})_{n}(v_{1}+nv_{2})r^{n}}{(n!)^{2}}$
$r = \frac{432(a^{2}-b^{2})^{2}(a-b)^{6}ba}{Q^3}$
$Q = b^{4}+60ab^{3}+134a^{2}b^{2}+60a^{3}b+a^{4}\,$
$v_{1} = ba(15b^{4}+68ab^{3}+90a^{2}b^{2}+68a^{3}b+15a^{4})\,$
$v_{2} = -a^{6}-b^{6}+126ab^{5}+1041a^{2}b^{4}+1764a^{3}b^{3}+1041a^{4}b^{2}+126a^{5}b\,$

A good approximation is Ramanujan's:

$C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi \left[3(a+b)-\sqrt{10ab+3(a^2+b^2)}\right]$

and a better approximation is

$C\approx\pi\left(a+b\right)\left(1+\frac{3\left(\frac{a-b}{a+b}\right)^2}{10+\sqrt{4-3\left(\frac{a-b}{a+b}\right)^2}}\right).\!\,$

For the special case where the minor axis is half the major axis, these become:

$C \approx \frac{\pi a (9 - \sqrt{35})}{2}$

or, as an estimate of the better approximation,

$C \approx \frac{a}{2} \sqrt{93 + \frac{1}{2} \sqrt{3}}$

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.[citation needed]

#### Chords

The midpoints of a set of parallel chords of an ellipse are collinear.[14]:p.147

### In projective geometry

In projective geometry, an ellipse can be defined as the set of all points of intersection between corresponding lines of two pencils of lines which are related by a projective map. By projective duality, an ellipse can be defined also as the envelope of all lines that connect corresponding points of two lines which are related by a projective map.

This definition also generates hyperbolae and parabolae. However, in projective geometry every conic section is equivalent to an ellipse. A parabola is an ellipse that is tangent to the line at infinity Ω, and the hyperbola is an ellipse that crosses Ω.

An ellipse is also the result of projecting a circle, sphere, or ellipse in three dimensions onto a plane, by parallel lines. It is also the result of conical (perspective) projection of any of those geometric objects from a point O onto a plane P, provided that the plane Q that goes through O and is parallel to P does not cut the object. The image of an ellipse by any affine map is an ellipse, and so is the image of an ellipse by any projective map M such that the line M−1(Ω) does not touch or cross the ellipse.

### In analytic geometry

#### General ellipse

In analytic geometry, the ellipse is defined as the set of points $(X,Y)$ of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation[15][16]

$~A X^2 + B X Y + C Y^2 + D X + E Y + F = 0$

provided $B^2 - 4AC < 0.$

To distinguish the degenerate cases from the non-degenerate case, let be the determinant of the 3×3 matrix [A, B/2, D/2 ; B/2, C, E/2 ; D/2, E/2, F ]: that is, = (AC - B2/4)F + BED/4 - CD2/4 - AE2/4. Then the ellipse is a non-degenerate real ellipse if and only if C∆<0. If C∆>0 we have an imaginary ellipse, and if =0 we have a point ellipse.[17]:p.63

#### Canonical form

Let $a>b$. By a proper choice of coordinate system, the ellipse can be described by the canonical implicit equation

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Here $(x,y)$ are the point coordinates in the canonical system, whose origin is the center $(X_c,Y_c)$ of the ellipse, whose $x$-axis is the unit vector $(X_a,Y_a)$ coinciding with the major axis, and whose $y$-axis is the perpendicular vector $(-Y_a,X_a)$ coinciding with the minor axis. That is, $x = X_a(X - X_c) + Y_a(Y - Y_c)$ and $y = -Y_a(X - X_c) + X_a(Y - Y_c)$.

In this system, the center is the origin $(0,0)$ and the foci are $(-e a, 0)$ and $(+e a, 0)$.

Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters. Translation of an ellipse centered at $(X_c,Y_c)$ is expressed as

$\frac{(x - X_c)^2}{a^2}+\frac{(y - Y_c)^2}{b^2}=1$

Moreover, any canonical ellipse can be obtained by scaling the unit circle of $\reals^2$, defined by the equation

$X^2+Y^2=1\,$

by factors a and b along the two axes.

For an ellipse in canonical form, we have

$Y = \pm b\sqrt{1 - (X/a)^2} = \pm \sqrt{(a^2-X^2)(1 - e^2)}$

The distances from a point $(X,Y)$ on the ellipse to the left and right foci are $a + e X$ and $a - e X$, respectively.

### In trigonometry

#### General parametric form

An ellipse in general position can be expressed parametrically as the path of a point $(X(t),Y(t))$, where

$X(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi$
$Y(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi$

as the parameter t varies from 0 to 2π. Here $(X_c,Y_c)$ is the center of the ellipse, and $\varphi$ is the angle between the $X$-axis and the major axis of the ellipse.

#### Parametric form in canonical position

Parametric equation for the ellipse (red) in canonical position. The eccentric anomaly t is the angle of the blue line with the X-axis. Click on image to see animation.

For an ellipse in canonical position (center at origin, major axis along the X-axis), the equation simplifies to

$X(t)=a\,\cos t$
$Y(t)=b\,\sin t$

Note that the parameter t (called the eccentric anomaly in astronomy) is not the angle of $(X(t),Y(t))$ with the X-axis.

Formulae connecting a tangential angle $\phi$, the angle anchored at the ellipse's center $\phi^\prime$ (called also the polar angle from the ellipse center), and the parametric angle t[18] are[19][20][21]:

$\tan \phi=\frac {a}{b} \tan t=\frac {\tan \phi'}{(1-g)^2}=\frac {\tan \phi'}{1-e^2}$
$\tan \phi^\prime=(1-f) \tan t$
$\tan t=\frac {b}{a} \tan \phi=\sqrt{(1-e^2)} \tan \phi=(1-g) \tan \phi=\frac {\tan \phi'}{\sqrt{(1-e^2)}}=\frac {a}{b} \tan \phi'$

#### Polar form relative to center

Polar coordinates centered at the center.

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate $\theta$ measured from the major axis, the ellipse's equation is

$r(\theta)=\frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}$

#### Polar form relative to focus

Polar coordinates centered at focus.

If instead we use polar coordinates with the origin at one focus, with the angular coordinate $\theta = 0$ still measured from the major axis, the ellipse's equation is

$r(\theta)=\frac{a (1-e^{2})}{1 \pm e\cos\theta}$

where the sign in the denominator is negative if the reference direction $\theta = 0$ points towards the center (as illustrated on the right), and positive if that direction points away from the center.

In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate $\phi$, the polar form is

$r=\frac{a (1-e^{2})}{1 - e\cos(\theta - \phi)}.$

The angle $\theta$ in these formulas is called the true anomaly of the point. The numerator $a (1-e^{2})$ of these formulas is the semi-latus rectum of the ellipse, usually denoted $l$. It is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis.

Semi-latus rectum.

#### General polar form

The following equation on the polar coordinates (rθ) describes a general ellipse with semidiameters a and b, centered at a point (r0θ0), with the a axis rotated by φ relative to the polar axis:

$r(\theta )=\frac{P(\theta )+Q(\theta )}{R(\theta )}$

where

$P(\theta )=r_0 \left[\left(b^2-a^2\right) \cos \left(\theta +\theta _0-2 \varphi \right)+\left(a^2+b^2\right) \cos \left(\theta -\theta_0\right)\right]$
$Q(\theta )=\sqrt{2} a b \sqrt{R(\theta )-2 r_0^2 \sin ^2\left(\theta -\theta_0\right)}$
$R(\theta )=\left(b^2-a^2\right) \cos (2 \theta -2 \varphi )+a^2+b^2$

#### Angular eccentricity

The angular eccentricity $\alpha$ is the angle whose sine is the eccentricity e; that is,

$\alpha=\sin^{-1}(e)=\cos^{-1}\left(\frac{b}{a}\right)=2\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!$

### Degrees of freedom

An ellipse in the plane has five degrees of freedom (the same as a general conic section), defining its position, orientation, shape, and scale. In comparison, circles have only three degrees of freedom (position and scale), while parabolae have four. Said another way, the set of all ellipses in the plane, with any natural metric (such as the Hausdorff distance) is a five-dimensional manifold. These degrees can be identified with, for example, the coefficients A,B,C,D,E of the implicit equation, or with the coefficients Xc, Yc, φ, a, b of the general parametric form.

## Ellipses in physics

### Elliptical reflectors and acoustics

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves created by that disturbance, after being reflected by the walls, will converge simultaneously to a single point — the second focus. This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.

Similarly, if a light source is placed at one focus of an elliptic mirror, all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolate spheroid), this property will hold for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear fluorescent lamp along a line of the paper; such mirrors are used in some document scanners.

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 the other focus remarkably well. The effect is even more evident under a vaulted roof shaped as a section of a prolate spheroid. Such a room is called a whisper chamber. The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the United States 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.

### Planetary orbits

In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun at one focus, 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 (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus.

Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation and quantum effects which become significant when the particles are moving at high speed.)

For elliptical orbits, useful relations involving the eccentricity, $e$ are:

$e={{r_\mathrm{ap}-r_\mathrm{per}}\over{r_\mathrm{ap}+r_\mathrm{per}}}={{r_\mathrm{ap}-r_\mathrm{per}}\over{2a}}$

$r_\mathrm{ap}=(1+e)a\!\,$             $r_\mathrm{per}=(1-e)a\!\,$

where:

• $r_\mathrm{ap}\,\!$ is radius at apoapsis (i.e., the farthest distance).
• $r_\mathrm{per}\,\!$ is radius at periapsis (the closest distance).
• $a\,\!$ is the length of the semi-major axis.

In addition, the semimajor axis $a\,\!$ is the arithmetic mean of $r_\text{ap}\,\!$ and $r_\mathrm{per}\,\!$, or $a=\frac{{r_\text{ap}} + {r_\text{per}}}{2}$, and the semiminor axis $b\,\!$ is the geometric mean of $r_\text{ap}\,\!$ and $r_\mathrm{per}\,\!$, or $b=\, ^2 \! \! \! \! \sqrt{r_\mathrm{ap} \times r_\mathrm{per}}$ .
Also the semi-latus rectum (the distance from a focus to a point on the ellipse along a line parallel to the minor axis) is the harmonic mean of $r_\text{ap}\,\!$ and $r_\mathrm{per}\,\!$, or $\text {semi-latus rectum} = \frac{2}{\frac{1}{r_\text{ap}} + \frac{1}{r_\text{per}}}$ .

### Harmonic oscillators

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.

### Phase visualization

In electronics, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope. If the display is an ellipse, rather than a straight line, the two signals are out of phase.

### Elliptical gears

Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, will turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage.

Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.[22]

An example gear application would be a device that winds thread onto a conical bobbin on a spinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.[23]

### Optics

In a material that is optically anisotropic (birefringent), the refractive index depends on the direction of the light. The dependency can be described by an index ellipsoid. (If the material is optically isotropic, this ellipsoid is a sphere.)

## Ellipses in statistics and finance

In statistics, a bivariate random vector (X, Y) is jointly elliptically distributed if its iso-density contours — loci of equal values of the density function — are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance — that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.[24][25]

## Ellipses in computer graphics

Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw API, and Direct2D on Windows. 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. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967.[26] Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.[27]

In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties.[28] These algorithms need only a few multiplications and additions to calculate each vector.

It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.

### Drawing with Bezier spline paths

Multiple Bezier splines may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an affine transformation of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bezier curves will behave appropriately under such transformations.

## Line segment as a type of degenerate ellipse

A line segment is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1, and with the focal points at the ends.[29] Although the eccentricity is 1 this is not a parabola. A radial elliptic trajectory is a non-trivial special case of an elliptic orbit, where the ellipse is a line segment.

## Ellipses in optimization theory

It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for attacking this problem.

## Notes

1. ^
2. ^ Herschel, Sir John Frederick William (1842). A treatise on astronomy. Lea & Blanchard. p. 256.
3. ^ Lankford, John (1997). History of Astronomy: An Encyclopedia. Taylor & Francis. pp. 194. ISBN 978-0-8153-0322-0.
4. ^ Prasolov, Viktor Vasilʹevich; Tikhomirov, Vladimir Mikhaĭlovich (2001). Geometry. American Mathematical Society. p. 80. ISBN 978-0-8218-2038-4.
5. ^ Fenna, Donald (2007). Cartographic Science: A Compendium of Map Projections, With Derivations. CRC Press. p. 24. ISBN 978-0-8493-8169-0.
6. ^ AutoCAD release 13 command reference. Autodesk, Inc.. 1994. p. 216.
7. ^ Salomon, David (2006). Curves And Surfaces for Computer Graphics. Birkhäuser. pp. 365. ISBN 978-0-387-24196-8.
8. ^ Kreith, Frank; Goswami, D. Yogi (2005). The CRC Handbook Of Mechanical Engineering. CRC Press. pp. 11-8. ISBN 978-0-8493-0866-6. "Circles and Ellipses (11.3.2)"
9. ^
10. ^ Besant 1907, p. 57
11. ^ Armengaud, Aîné (1853). "Ovals, Ellipses, Parabolas, Volutes, etc. §53". The Practical Draughtsman's Book of Industrial Design. Longman, Brown, Green, and Longmans. p. 16.
12. ^
13. ^ Cetin Hakimoglu-Brown iamned.com math page
14. ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.
15. ^ Larson, Ron; Hostetler, Robert P.; Falvo, David C. (2006). "Chapter 10". Precalculus with Limits. Cengage Learning. p. 767. ISBN 0-618-66089-5.
16. ^ Young, Cynthia Y. (2010). "Chapter 9". Precalculus. John Wiley and Sons. p. 831. ISBN 0-471-75684-9.
17. ^ Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.
18. ^ If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, the angle $\phi'$ is the geocentric latitude, and parametric angle t is a parametric (or reduced) latitude of auxiliary circle
19. ^ Ellipse at MathWorld, derived from formula (58)
20. ^ Auxiliary circle and various ellipse formulas
21. ^ Meeus, J. (1991). "Ch. 10: The Earth's Globe". Astronomical Algorithms. Willmann-Bell. p. 78. ISBN 0-943396-35-2.
22. ^ David Drew. "Elliptical Gears". [1]
23. ^ Grant, George B. (1906). A treatise on gear wheels. Philadelphia Gear Works. p. 72.
24. ^ Chamberlain, G. (February 1983). "A characterization of the distributions that imply mean—Variance utility functions". Journal of Economic Theory 29 (1): 185–201. DOI:10.1016/0022-0531(83)90129-1.
25. ^ Owen, J.; Rabinovitch, R. (June 1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice". Journal of Finance 38: 745–752. JSTOR 2328079.
26. ^ Pitteway, M.L.V. (1967). "Algorithm for drawing ellipses or hyperbolae with a digital plotter". The Computer Journal 10 (3): 282–9. DOI:10.1093/comjnl/10.3.282.
27. ^ Van Aken, J.R. (September 1984). "An Efficient Ellipse-Drawing Algorithm". IEEE Computer Graphics and Applications 4 (9): 24–35. DOI:10.1109/MCG.1984.275994.
28. ^ Smith, L.B. (1971). "Drawing ellipses, hyperbolae or parabolae with a fixed number of points". The Computer Journal 14 (1): 81–86. DOI:10.1093/comjnl/14.1.81.
29. ^ Seligman, Courtney (1993–2010). "Orbital Motions Ellipses and Other Conic Sections". Online Astronomy eText.

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