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Locus (mathematics)

Locus (mathematics)

In mathematics, a locus (Latin for "place", plural loci) is a collection of points which share a common property. A locus of points usually forms a continuous figure or figures. For example, the conic sections are defined in terms of loci: # A circle is the locus of points from which the distance to the center is a given value, the radius. # An ellipse is the locus of points, the sum of the distances from which to the foci is a given value. # A parabola is the locus of points, the distances from which to the focus and to the directrix are equal. # A hyperbola is the locus of points, the difference of the distances from which to the foci is a given value. Other geometrical shapes are defined in terms of loci: # A line is the locus of points equidistant from two fixed points Category:Elementary geometry

Latin

Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. It gained great importance as the formal language of the Roman Empire. All Romance languages, those being most notably Spanish, French, Portuguese, Italian, and Romanian, are descended from Latin, and many words based on Latin are found in other modern languages such as English. The Latin alphabet, derived from the Greek, remains the most widely-used alphabet in the world. It is said that 80 percent of scholarly English words are derived from Latin (in a large number of cases by way of French). Moreover, in the Western world, Latin was a lingua franca, the learned language for scientific and political affairs, for more than a thousand years, being eventually replaced by French in the 18th century and English in the late 19th. Ecclesiastical Latin remains the formal language of the Roman Catholic Church to this day, and thus the official national language of the Vatican. The Church used Latin as its primary liturgical language until the Second Vatican Council in the 1960s. Latin is also still used (drawing heavily on Greek roots) to furnish the names used in the scientific classification of living things. The modern study of Latin, along with Greek, is known as Classics.

Main features

Latin is a synthetic inflectional language: affixes (which usually encode more than one grammatical category) are attached to fixed stems to express gender, number, and case in adjectives, nouns, and pronouns, which is called declension; and person, number, tense, voice, mood, and aspect in verbs, which is called conjugation. There are five declensions (declinationes) of nouns and four conjugations of verbs. There are six noun cases: #nominative (used as the subject of the verb or the predicate nominative), #genitive (used to indicate relation or possession, often represented by the English of or the addition of s to a noun), #dative (used of the indirect object of the verb, often represented by the English to or for), #accusative (used of the direct object of the verb, or object of the preposition in some cases), #ablative (separation, source, cause, or instrument, often represented by the English by, with, from), #vocative (used of the person or thing being addressed). In addition, some nouns have a locative case used to express location (otherwise expressed by the ablative with a preposition such as in), but this survival from Proto-Indo-European is found only in the names of lakes, cities, towns, small islands, and a few other words related to locations, such as "house", "ground", and "countryside". Latin itself, being a very old language, is far closer to Proto-Indo-European than are most modern Western European languages; it has, in fact, about the same relationship with PIE as modern Italian or French has to Latin. There are six general tenses in Latin (technically they are tense/aspect/mood complexes). The indicative mood can be used with all of them. The subjunctive mood, however, has only present, imperfect, perfect, and pluperfect tenses. These tenses in the subjunctive mood do not completely correlate in meaning to the tenses in the indicative. The following examples are of the first conjugation verb "laudare" ("to praise") in the indicative mood and the active voice:
Primary sequence tenses
# present (laudo, "I praise") # imperfect (laudabam, "I was praising") # future (laudabo, "I shall praise," "I will praise")
Secondary sequence tenses
# perfect (laudavi, "I praised", "I have praised") # pluperfect (laudaveram, "I had praised") # future perfect (laudavero, "I shall have praised," "I will have praised") The future perfect tense can also imply a normal future idea (like in "When I will have run...") and so may also sometimes be included in the primary sequence.
Latin and Romance
After the collapse of the Roman Empire, Latin evolved into the various Romance languages. These were for many centuries only spoken languages, Latin still being used for writing. For example, Latin was the official language of Portugal until 1296 when it was replaced by Portuguese. The Romance languages evolved from Vulgar Latin, the spoken language of common usage, which in turn evolved from an older speech which also produced the formal classical standard. Latin and Romance differ (for example) in that Romance had distinctive stress, whereas Latin had distinctive length of vowels. In Italian and Sardo logudorese, there is distinctive length of consonants and stress, in Spanish only distinctive stress, and in French even stress is no longer distinctive. Another major distinction between Romance and Latin is that all Romance languages, excluding Romanian, have lost their case endings in most words except for some pronouns. Romanian retains a direct case (nominative/accusative), an indirect case (dative/genitive), and vocative. In Italy, Latin is still compulsory in secondary schools as Liceo Classico and Liceo Scientifico which are usually attended by people who aim to the highest level of education. In Liceo Classico Ancient Greek is a compulsory subject.
Latin and English
See Latin influence in English for a more complete exposition. English grammar is independent of Latin grammar, though prescriptive grammarians in English have been heavily influenced by Latin. Attempts to make English grammar follow Latin rules — such as the prohibition against the split infinitive — have not worked successfully in regular usage. However, as many as half the words in English were derived from Latin, including many words of Greek origin first adopted by the Romans, not to mention the thousands of French, hundreds of Spanish, Portuguese and Italian words of Latin origin that have also enriched English. During the 16th and on through the 18th century English writers created huge numbers of new words from Latin and Greek roots. These words were dubbed "inkhorn" or "inkpot" words (as if they had spilled from a pot of ink). Many of these words were used once by the author and then forgotten, but some remain. Imbibe, extrapolate, dormant and inebriation are all inkhorn terms carved from Latin words. In fact, the word etymology is derived from the Greek word etymologia, meaning "true sense of the word." Latin was once taught in many of the schools in Britain with academic leanings - perhaps 25% of the total [http://www.channel4.com/history/microsites/T/teachem2/thennow/]. However, the requirement for it was gradually abandoned in the professions such as the law and medicine, and then, from around the late 1960s, for admission to university. After the introduction of the Modern Language GCSE in the 1980s, it was gradually replaced by other languages, although it is now being taught by more schools along with other classical languages.
Latin education
The linguistic element of Latin courses offered in high schools or secondary schools, and in universities, is primarily geared toward an ability to translate Latin texts into modern languages, rather than using it in oral communication. As such, the skill of reading is heavily emphasized, whereas speaking and listening skills are barely touched upon. However, there is a growing movement, sometimes known as the Living Latin movement, whose supporters believe that Latin can, or should, be taught in the same way that modern "living" languages are taught, that is, as a means of both spoken and written communication. One of the most interesting aspects of such an approach is that it assists speculative insight into how many of the ancient authors spoke and incorporated sounds of the language stylistically; without understanding how the language is meant to be heard it is very difficult to identify patterns in Latin poetry. Institutions offering Living Latin instruction include the Vatican and the University of Kentucky. In Britain the Classical Association encourages this approach, and there has been something of a vogue for books describing the adventures of a mouse called Minimus. In the United States there is a thriving competitive organization for high school Latin students, the National Junior Classical League (the second-largest youth organization in the world after the Boy Scouts), backed up by the Senior Classical League for college students. Many would-be international auxiliary languages have been heavily influenced by Latin, and the moderately successful Interlingua considers itself to be the modernized and simplified version of the language (le latino moderne international e simplificate). Latin translations of modern literature such as Paddington Bear, Winnie the Pooh, Harry Potter and the Philosopher's Stone, Le Petit Prince, Max und Moritz, and The Cat in the Hat have also helped boost interest in the language.
See also

About the Latin language

- Latin grammar
- Latin spelling and pronunciation
- Latin declension
- Latin conjugation
- Latin alphabet
- List of Latin words with English derivatives
- Latin verbs with English derivatives
- Latin nouns with English derivatives
- ablative absolute
- Word order in Latin
About the Latin literary heritage

- Latin literature
- Romance languages
- Loeb Classical Library
- List of Latin phrases
- List of Latin proverbs
- Brocard
- List of Latin and Greek words commonly used in systematic names
- List of Latin place names in Europe
- Carmen Possum
Other related topics

- Roman Empire
- Internationalism
References

- Bennett, Charles E. Latin Grammar (Allyn and Bacon, Chicago, 1908)
- N. Vincent: "Latin", in The Romance Languages, M. Harris and N. Vincent, eds., (Oxford Univ. Press. 1990), ISBN 0195208293
- Waquet, Françoise, Latin, or the Empire of a Sign: From the Sixteenth to the Twentieth Centuries (Verso, 2003) ISBN 1859844022; translated from the French by John Howe.
- Wheelock, Frederic. Latin: An Introduction (Collins, 6th ed., 2005) ISBN 0060784237
External links

- [http://www.jambell.com/latin.html Latin Phrases for after dinner conversation (Thanks to Elaine Poole)]
- [http://www.ethnologue.com/show_language.asp?code=lat Ethnologue report for Latin]
- [http://forumromanum.org/literature/index.html Corpus Scriptorum Latinorum] is a comprehensive webography of Latin texts and their translations.
- [http://www.perseus.tufts.edu/ The Perseus Project] has many useful pages for the study of classical languages and literatures, including [http://www.perseus.tufts.edu/cgi-bin/resolveform?lang=Latin an interactive Latin dictionary].
- [http://lysy2.archives.nd.edu/cgi-bin/words.exe words by William whitaker] is a dictionary program online capable of looking up various word forms.
- [http://retiarius.org/ Retiarius.Org] includes a Latin text search engine.
- [http://www.nd.edu/~archives/latgramm.htm Latin-English dictionary and Latin grammar from U of Notre Dame]
- [http://latin-language.co.uk/ Latin language] History of Latin language, Latin texts with English translation and a collection of dictionaries.
- [http://augustinus.eresmas.net/scl/ Societas Circulorum Latinorum] gathers together Latin Circles all over the world.
- [http://www.learnlatin.tk LearnLatin.tk] - Free online course in Latin
- [http://www.latintests.net/ LatinTests.net] - Lets Latin learners test their grammar and vocabulary with self-checking quizzes.
- [http://thelatinlibrary.com/ The Latin Library] contains many Latin etexts
- [http://www.textkit.com/ Textkit] has Latin textbooks and etexts.
- [http://www.websters-online-dictionary.org/definition/Latin-english/ Latin–English Dictionary]: from Webster's Rosetta Edition.
- [http://www.language-reference.com/ Language reference] Cross-foreign-language lexicon powered by its own search engine. All cross combinations between Latin and French, German, Italian, Spanish.
- [http://comp.uark.edu/~mreynold/rhetor.html Rhetor by Gabriel Harvey] was originally published in 1577 and never again reprinted.
- [http://freewebs.com/omniamundamundis omniamundamundis] Latin hypertexts from fourteen ancient Roman authors.
- [http://www.saltspring.com/capewest/pron.htm Pronunciation of Biological Latin, Including Taxonomic Names of Plants and Animals]
- [http://www.yleradio1.fi/nuntii Nuntii Latini (News in Latin)], written and spoken (RealAudio) news in latin. Weekly review of world news in Classical Latin, the only international broadcast of its kind in the world, produced by YLE, the Finnish Broadcasting Company.
- [http://www.tranexp.com:2000/InterTran?url=http%3A%2F%2F&type=text&text=Replace%20Me&from=eng&to=ltt InterTran Latin], Translate from Latin to ENGLISH or vice versa.
- [http://www.latinvulgate.com Latin Vulgate] The Latin and English of the Old & New Testaments in parallel, along with the Complete Sayings of Jesus in parallel Latin and English. Category:Classical languages Category:Ancient languages Category:Fusional languages Category:Languages of Italy Category:Languages of Vatican City als:Latein zh-min-nan:Latin-gí ko:라틴어 ja:ラテン語 simple:Latin language th:ภาษาละติน

Conic sections

In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

Types of conics

Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.) The degenerate cases, where the plane passes through the apex of the cone, resulting in an intersection figure of a point, a straight line, or a pair of intersecting lines, are often excluded from the list of conic sections. In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. If the equation is of the form :

ax^2 + 2hxy + by^2 +2gx + 2fy + c = 0\;

quadratic equation then:
- if h² = ab, the equation represents a parabola;
- if h² < ab and a

\ne

b and/or h

\ne

0 , the equation represents an ellipse;
- if h² > ab, the equation represents a hyperbola;
- if h² < ab and a = b and h = 0, the equation represents a circle;
- if a + b = 0, the equation represents a rectangular hyperbola.

Eccentricity

An alternative definition of conic sections starts with a point F (the focus), a line L (the directrix) not containing F and a positive number e (the eccentricity). The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is

\over

, where

a

is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is

ae

. In the case of a circle e = 0 and one imagines the directrix to be infinitely far removed from the center. However, the statement that the circle consists of all points whose distance is e times the distance to L is not useful, because we get zero times infinity. The eccentricity of a conic section is thus a measure of how far it deviates from being circular. For a given

a

, the closer

e

is to 1, the smaller is the semi-minor axis.

Semi-latus rectum and polar coordinates

semi-minor axis The semi-latus rectum of a conic section, usually denoted l, is the distance from the single focus, or one of the two foci, to the conic section itself, measured along a line perpendicular to the major axis. It is related to a and b by the formula

al=b^2\,\!

, or

l=a(1-e^2)\,\!

. In polar coordinates, a conic section with one focus at the origin and, if any, the other on the positive x-axis, is given by the equation :

r (1 - e \cos \theta) = l\,\!

Properties

Conic sections are always "smooth". More precisely, they never contain any inflection points. This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow/prevent turbulence.

Applications

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem. In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola.

Dandelin spheres

See Dandelin spheres for a short elementary argument showing that the characterization of these curves as intersections of a plane with a cone is equivalent to the characterization in terms of foci, or of a focus and a directrix.

Derivation

Let there be a cone whose axis is the z-axis. Let its vertex be the origin. The equation for the cone is :

x^2 + y^2 - a^2 z^2 = 0 \qquad \qquad (1)

where :

a = \tan \theta > 0 \;

and

\theta

is the angle which the generators of the cone make with respect to the axis. Notice that this cone is actually a pair of cones: one cone standing upside down on the vertex of the other cone—or, as mathematicians say, this cone consists of two "nappes." Let there be a plane with a slope running along the x direction but which is level along the y direction. Its equation is :

z = mx + b \qquad \qquad (2)

where :

m = \tan \phi > 0 \;

and

\phi

is the angle of the plane with respect to the x-y plane. We are interested in finding the intersection of the cone and the plane, which means that equations (1) and (2) shall be combined. Both equations can be solved for z and then equate the two values of z. Solving equation (1) for z yields :

z = \sqrt

therefore :

\sqrt = m x + b.

Square both sides and expand the squared binomial on the right side, :

= m^2 x^2 + 2 m b x + b^2. \;

Grouping by variables yields :

x^2 \left(  - m^2 \right) +  - 2 m b x - b^2 = 0. \qquad \qquad (3)

Note that this is the equation of the projection of the conic section on the xy-plane, hence contracted in the x-direction compared with the shape of the conic section itself.

Derivation of the parabola

The parabola is obtained when the slope of the plane is equal to the slope of the generators of the cone. When these two slopes are equal, then the angles

\theta

and

\phi

become complementary. This implies that :

\tan \theta = \cot \phi \;

therefore :

m = . \qquad \qquad (4)

Substituting equation (4) into equation (3) makes the first term in equation (3) vanish, and the remaining equation is :

-  b x - b^2 = 0.

Multiply both sides by a², :

y^2 - 2 a b x - a^2 b^2 = 0 \;

then solve for x, :

x =  y^2 - . \qquad \qquad (5)

Equation (5) describes a parabola whose axis is parallel to the x-axis. Other versions of equation (5) can be obtained by rotating the plane around the z-axis.

Derivation of the ellipse

An ellipse happens when the angles

\theta

and

\phi

, when added together, do not measure up to a right angle: :

\theta + \phi <  \qquad \qquad \mbox

which implies that the tangent of the sum of these two angles is positive. :

\tan (\theta + \phi) > 0. \;

But a trigonometric identity states that :

\tan (\theta + \phi) =

therefore :

\tan (\theta + \phi) =  > 0 \qquad \qquad (6)

but m + a is positive, since the summands are given to be positive, so inequality (6) is positive if the denominator is also positive: :

1 - m a > 0. \qquad \qquad (7)

From inequality (7) we can deduce :

m a < 1, \;

m^2 a^2 < 1, \;

1 - m^2 a^2 > 0, \;

> 1,

- 1 > 0,

- m^2 > 0 \qquad \qquad \mbox.

Let us start out again from equation (3), :

x^2 \left(  - m^2 \right) +  - 2 m b x - b^2 = 0, \qquad \qquad (3)

but this time the coefficient of the x² term does not vanish but is instead positive. Solve for y, :

y = a \sqrt. \qquad \qquad (8)

This would clearly describe an ellipse were it not for the second term under the radical, the 2 m b x: it would be the equation of a circle which has been stretched proportionally along the directions of the x-axis and the y-axis. Equation (8) is an ellipse but it is not obvious, so it will be rearranged further until it is obvious. Complete the square under the radical, :

y = a \sqrt.

Group together the b² terms, :

y = a \sqrt.

Divide by a then square both sides, :

+ \left( x \sqrt -  \right)^2 = b^2 \left( 1 +  \right).

The x has a coefficient. It is desired to pull this coefficient out by factoring it out of the second term which is a square, :

+ \left(  - m^2 \right) \left( x -  \right)^2 = b^2 \left( 1 +  \right).

Further rearrangements of constants finally leads to :

+ \left( x -  \right)^2 = .

The coefficient of the y term is positive (for an ellipse). Renaming of coefficients and constants leads to :

+ (x - C)^2 = R^2 \qquad \qquad (9)

which is clearly the equation of an ellipse. That is, equation (9) describes a circle of radius R and center (C,0) which is then stretched vertically by a factor of

\sqrt

. The second term on the left side (the x term) has no coefficient but is a square, so that it must be positive. The radius is a product of squares, so it must also be positive. The first term on the left side(the y term) has a coefficient which is positive (one of the inequalities derived earlier), so the equation describes an ellipse.

Derivation of the hyperbola

The hyperbola happens when the angles

\theta

and

\phi

add up to an obtuse angle, which is greater than a right angle. The tangent of an obtuse angle is negative. All the inequalities which were valid for the ellipse become reversed. Therefore :

1 - a^2 m^2 < 0 \qquad \qquad \mbox.

Otherwise the equation for the hyperbola is the same as equation (9) for the ellipse, except that the coefficient A of the y term is negative. The sign change is enough to convert an ellipse into a hyperbola. This is because the equation of a real ellipse contains an imaginary hyperbola, and the equation of a real hyperbola contains an imaginary ellipse (see imaginary number). The sign change of coefficient A causes real and imaginary values of the function y=f(x) equivalent to equation (9) to swap.

External links

- [http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html Special plane curves: Conic sections]
- http://mathworld.wolfram.com/Focus.html
- [http://ccins.camosun.bc.ca/~jbritton/jbconics.htm Occurrence of the conics] in nature and elsewhere
- [http://fishrock.com/conics/default.htm Conic structures] in architecture
- Category:Euclidean solid geometry ja:円錐曲線

Ellipse

Elliptical redirects here, for the exercise machine, see Elliptical trainer. Elliptical trainer In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form :

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 < 4 AC

, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. triangle The line segment which passes through the foci and terminates on the ellipse is called the major axis. The major axis is along the longest segment that passes through the ellipse. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix

A = PDP^T

, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues.

Parametrisation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. length An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation :

\frac + \frac = 1

The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). center The same ellipse is also represented by the parametric equations: :

x = a\,\cos t

y = b\,\sin t

0 \leq t < 2\pi

which use the trigonometric functions sine and cosine. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation :

\frac + \frac = 1

where (h,k) is the center. A Gauss-mapped form: :

\left(\frac,\frac\right)

has normal

(\cos\phi,\sin\phi)

Eccentricity

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement :

e = \sqrt

or where

c

(the linear eccentricity of the ellipse) equals the distance from the center to either focus :

e = \frac

The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The distance between the foci is 2ae.

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(1-e^2)\,\!

. perpendicular 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 (1 + e \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°.

Area

The area enclosed by an ellipse is

\pi ab\,\!

, where

\pi

is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :

c = 2\pi a \left[\right]\!\,

A good approximation is Ramanujan's: :

c \approx \pi \left[3(a+b) - \sqrt\right]\!\,

which can also be written as: :

c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\,

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 point). 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.

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.

Ellipses in physics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. 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. 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. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

Ellipses in computer graphics

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. 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). A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example.

Focus (geometry)

In geometry, the focus (pl. foci) is a special point used in describing conic sections. A conic section can be defined as the set of points whose distance to its focus is equal to the eccentricity times the distance to the corresponding directrix. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section. Note that (non-circular) ellipses and hyperbolas each have a pair of foci. An ellipse can be described as the set of points for which the sum of the distances to the foci is constant, while a hyperbola is the set of points for which the absolute value of the difference of the distances to the foci is constant. In the gravitational two-body problem, the orbits of the two bodies are described by conic sections with foci at the center of mass. Category:Conic sections ja:焦点

Hyperbola

:For hyperbole, the figure of speech, see hyperbole. hyperbole In mathematics, a hyperbola is a type of conic section (literally: 'exaggeration' from the Greek ) defined as the intersection between a cone and a plane which cuts through both halves of the cone. It may also be defined as the set of all points for which the difference in the distance to two fixed points (called the foci) is constant. For a simple geometric proof that the two characterizations above are equivalent to each other, see Dandelin spheres. Algebraically, a hyperbola is a curve in the Cartesian plane defined by an equation of the form :

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 > 4 AC

, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the hyperbola, exists.

Definitions

- It can also be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola. These foci lie on the transverse axis and their midpoint is called the center. A hyperbola comprises two disconnected curves called its arms which separate the foci. At large distances from the foci the hyperbola begins to approximate two lines, known as asymptotes. A hyperbola has the property that a ray originating at one of the foci is reflected in such a way as to appear to have originated at the other focus. reflectedA special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angles. The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy=c, where c is a constant. Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola. If on the hyperbola equation one switches x and y, the conjugate hyperbola is obtained. A hyperbola and its conjugate have the same asymptotes.

Equations

Cartesian

(center (h, k) ) :

\frac - \frac = 1

\frac - \frac = 1

In both formulas a is called the semi-major axis; it is half the distance between the two branches; b is called the semi-minor axis. Note that b can be larger than a! The reason for this is because the values of a and b do not dictate which way the hyperbola opens. It is the order in which y and x are subtracted. If y is positive, then the hyperbola opens up and down. If x is positive, then the hyperbola opens left and right. The eccentricity is given by :

e = \sqrt

For rectangular hyperbolas with the coordinate axes parallel to their asymptotes: :

(x-h)(y-k) = c \,

Polar

r^2 =\ \ \,  a\,\sec 2t

r^2 =    -a\,\sec 2t

r^2 =\ \ \, a\,\csc 2t

r^2 =    -a\,\csc 2t

Parametric

x = a\,\cosh \theta;\; y = b\,\sinh \theta

x = a\,\tan \theta;\ \ y = b\,\sec \theta

External links

-
-
-
- [http://mathworld.wolfram.com/Hyperbola.html Mathworld - Hyperbola] Category:Conic sections ja:双曲線

Line (mathematics)

A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve is not always a line. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points. Three or more points that lie on the same line are called collinear. Two different lines can intersect in at most one point; two different planes can intersect in at most one line. This intuitive concept of a line can be formalized in various ways. If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties. "Everything that satisfies the axioms for a line is a line." While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development. In Euclidean space Rⁿ (and analogously in all other vector spaces), we define a line L as a subset of the form :

L = \

where a and b are given vectors in Rⁿ with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line. In a two-dimensional space, such as the plane, two different lines must either be parallel lines or must intersect at one point. In higher-dimensional spaces however, two lines may do neither, and two such lines are called skew lines. In R², every line L is described by a linear equation of the form :

L=\

with fixed real coefficients a, b and c such that a and b are not both zero (see Linear equation for other forms). Important properties of these lines are their slope, x-intercept and y-intercept. The eccentricity of a straight line is infinity. More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology. The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.

Line segment

In mathematics, a line segment is a part of a line that is bounded by two end points. See also interval (mathematics). When the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. The midpoint of a line segment is its 'middle' point: the unique point at an equal distance from the two end points.

Ray

In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----
- ---> A B C In geometric optics a ray or a (light) beam is a line or curve that describes the direction in which light or other electromagnetic radiation is propagated. The ray is perpendicular to the wavefront in wave optics. In most media, light rays are straight lines. Light passing from one medium to another undergoes refraction or total internal reflection following Snell's law.

External links

- [http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml Equations of the Straight Line] at cut-the-knot
- [http://mathworld.wolfram.com/Line.html Rigorous definition of a line] Category:Elementary geometry ja:線分 ja:直線 simple:Line ja:半直線

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