lens

en-noun

lens (plural lenses or (obsolete) lens or (rare) lentes)

lenses (plural)
lens (obsolete plural)
lentes (plural rare)

→ Bengali: লেন্স (lenś)

→ Turkish: lens

Borrowed from Latin lēns (“lentil”), with Medieval Latin later taking on the sense of “lens”, a semantic loan from Arabic عَدْسَة (ʕadsa, “lentil; optic lens”).

semantic loan

(geometry) A convex shape bounded by two circular arcs, joined at their endpoints, the corresponding concave shape being a lune.

lens

geometry: convex shape bounded by two circular arcs


pi

en-noun

pi (countable and uncountable, plural pis)

From Ancient Greek πεῖ (peî), ultimately from Proto-Semitic *pay- (“mouth”). Doublet of pe. Its mathematical use apparently stems from its use as the first letter in περιφέρεια (periphéreia, “periphery; circumference”) and was first cited in 1706 in the Synopsis Palmariorum Matheseos by William Jones.

(mathematics) An irrational and transcendental constant representing the ratio of the circumference of a Euclidean circle to its diameter; approximately 3.14159265358979323846264338327950; usually written π.


angle

From Middle English angle, angul, angule, borrowed from Middle French angle, from Latin angulus, anglus (“corner, remote area”). Cognate with Old High German ancha (“nape of the neck”), Middle High German anke (“joint of the foot, nape of neck”). Doublet of angulus and ankle.

cog

The angle between lines A and B is π/4 radians, or 45 degrees.

The single-imaging optic of the mammalian eye offers some distinct visual advantages. Such lenses can take in photons from a wide range of angles, increasing light sensitivity. They also have high spatial resolution, resolving incoming images in minute detail.

2013 July-August, Fenella Saunders, “Tiny Lenses See the Big Picture”, in American Scientist:

(geometry) The measure of such a figure. In the case of a plane angle, this is the ratio (or proportional to the ratio) of the arc length to the radius of a section of a circle cut by the two rays, centered at their common point. In the case of a solid angle, this is the ratio of the surface area to the square of the radius of the section of a sphere.


circle

Pitcairn-Norfolk: sirkil

From Middle English circle, cercle, from Old French cercle and Latin circulus, diminutive of Latin circus (“circle, circus”), from Ancient Greek κίρκος (kírkos, “circle, ring”), related to Old English hring (“ring”). Compare also Old English ċircul (“circle, zodiac”), which came from the same Latin source.

دَائِرَة

geometry: set of all points in a plane within a radius

The set of all points (x, y) such that (x − 1)² + y² = r² is a circle of radius r around the point (1, 0).

(geometry) A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from a given point (center).

аикәагьежь

geometry: set of points that are equally distant from a center


lune

What he worried about was any eventual convexity, a shrinking, it might be, of the planet itself to some palpable curvature of whatever he would be standing on, so that he would be left sticking out like a projected radius, unsheltered and reeling across the empty lunes of his tiny sphere.

(geometry) A concave figure formed by the intersection of the arcs of two circles on a plane, or on a sphere the intersection between two great semicircles.


cosine

From co- + sine, short for complementi sinus in Latin, meaning “sine of the complement”. Doublet of cosinus.

(trigonometry) In a right triangle, the ratio of the length of the side adjacent to an acute angle to the length of the hypotenuse. Symbol: cos


tangent

Borrowed from Latin tangentem, the accusative of tangēns (“touching”) (in the phrase līnea tangēns (“a touching line”)), the present participle of the verb tangō (“touch”, verb), from Proto-Italic *tangō, from Proto-Indo-European *teh₂g- (“to touch”).
Cognate with Old English þaccian (“to touch lightly, pat, stroke”). More at thack, thwack.

cog

The collision occurred on a two-mile stretch of tangent track.

(rail transport, of track) Straight; not horizontally curved.


degree

From Middle English degre, borrowed from Old French degré (French: degré), itself from Latin gradus, with the prefix de-.

Appendix:Roget MICRA thesaurus/Class I § 26. Degree

Appendix:Roget MICRA thesaurus/Class I § 31. Greatness

Appendix:Roget MICRA thesaurus/Class I § 32. Smallness

A right angle is a ninety-degree angle.

Most humans have a field of vision of almost 180 degrees.

(geometry) A unit of measurement of angle equal to ¹⁄₃₆₀ of a circle's circumference.


sine

Borrowed from Latin sinus (“curve, bend; bosom”), a translation of Arabic جَيْب (jayb, “bosom”), a misidentification of the notation جيب (j-y-b), written without vowel diacritics, standing for Arabic جِيبَ (jība, “sine”), in turn from Sanskrit ज्या (jyā, “sine, chord, bowstring”) through the similar Sanskrit जीव (jīva, “sine, chord, life, existence”). Doublet of sinus.

der

(trigonometry, mathematics) In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse.


radian

International System of Units
International System of Units

(geometry) In the International System of Units, the derived unit of plane angle: the angle subtended at the centre of a circle by an arc of its circumference equal in length to the radius of the circle. Symbol: rad.


radius

Borrowed from Latin radius (“ray”). Doublet of ray.

Fatima claims to have visited all the bars within a five-mile radius of her Manhattan apartment.

We start with spherical geometries. The two geometries on spheres of radiuses R₁ and R₂ are obviously identical if R₁ = R₂; moreover, the converse also holds.

1994, Viacheslav V. Nikulin, Igor R. Shafarevich, translated by Miles Reid, Geometries and Groups, Springer, page 194:

The velocity dispersions of early- and late-type galaxies in compact groups change little with groupcentric radius; the radii sampled are less than 100~h#x7B;-1#x7D; kpc, smaller than the radii typically sampled by members of massive clusters of galaxies.

2016, Jubee Sohn, Ho Seong Hwang, Margaret J. Geller, Antonaldo Diaferio, Kenneth J. Rines, Myung Gyoon Lee, Gwang-Ho Lee, “Compact Groups of Galaxies with Complete Spectroscopic Redshifts in the Local Universe”, in JKAS, volume 2015:

(geometry) A line segment between any point of a circle or sphere and its center.

This contribution reduces with increasing distance p from the emitting surface element dA, due to conservation of energy, as the wave energy distributed across the spherical wave front remains constant, while the radius increases during ...

1998, Dieter Schuocker, Handbook of the Eurolaser Academy, Springer Science & Business Media, →ISBN, page 51:

(geometry) The length of this line segment.

полупре́чник m (polupréčnik, literally “semidiameter”)

(graph theory) The minimum eccentricity of any vertex, for a given graph.


chord

Variant of cord, with spelling alteration due to Latin chorda (“cord”), ultimately from Ancient Greek χορδή (khordḗ, “string of gut, the string of a lyre”)

The simplest method of calculating the radius of a curve in situ is to measure the versine; in railway practice this is done by extending a tape 66 ft. (1 ch.) long in a straight line or chord between two points on the periphery of a curve, and then measuring the maximum distance of the rail from the chord at the centre of the 66 ft. The radius in chains is found by dividing the versine in inches into 99. Thus if a versine measures 3 in., the radius will be 33 ch.

1941 September, “The Why and the Wherefore: Curves”, in Railway Magazine, page 430:

(geometry) A line segment between two points of a curve.

Ctrl-K is the default first key for chords, but you can create chords using any keys that you want.

2005, James Avery, Visual Studio hacks, page 99:

(computing) A keyboard shortcut that involves two or more distinct keypresses, such as Ctrl+M followed by P.

(graph theory) An edge that is not part of a cycle but connects two vertices of the cycle.


circumference

From Latin circumferentia, from circum (“around”) + ferō (“I carry”). Displaced native Old English ymbgang.

(geometry) The line that bounds a circle or other two-dimensional figure.

مُحِيط

line that bounds a circle or other two-dimensional object


diameter

From Old French diametre (French diamètre), from Latin diametrus, from Ancient Greek διάμετρος (diámetros) (γραμμή (grammḗ)) (diametros grammē, “line measuring across”), from διά (diá, “across”) + μέτρον (métron, “measure”). By surface analysis, dia- + -meter.

(geometry) Any straight line between two points on the circumference of a circle that passes through the centre/center of the circle; a chord that passes through the center of the circle.


disk

→ Japanese: ディスク (disuku)

→ Korean: 디스크 (diseukeu)

→ Norwegian Bokmål: disk

→ Thai: ดิสก์ (dìt)

→ Turkish: disk

→ Welsh: disg

From Ancient Greek δίσκος (dískos, “a circular plate suited for hurling”), from δικέω (dikéō, “to hurl, to launch”). Doublet of dais, desk, disc, discus, dish, and diskos.

doublet

She burned some disks yesterday to back up her computer.

(computer hardware, nonstandard) An optical disc, such as a LaserDisc, CD, DVD, Blu-ray, or similar removable optical storage medium.


circular

en-adj

circular (comparative more circular, superlative most circular)

more circular (comparative)
most circular (superlative)

From Middle English circuler, circuleer, circulere, from Old French circulier, from Late Latin circularis, from Latin circulus, diminutive of circus (“ring”).

Here, in the transept and choir, where the service was being held, one was conscious every moment of an increasing brightness; colours glowing vividly beneath the circular chandeliers, and the rows of small lights on the choristers' desks flashed and sparkled in front of the boys' faces, deep linen collars, and red neckbands.

1918, W[illiam] B[abington] Maxwell, chapter V, in The Mirror and the Lamp, Indianapolis, Ind.: The Bobbs-Merrill Company, →OCLC:


oblique circle

(projection) a circle whose plane is oblique to the axis of the primitive plane


secant

LL-Q1860 (eng)-Flame, not lame-secant.wav

From Latin secāns, present participle of secō (“to cut”).

(trigonometry) In a right triangle, the reciprocal of the cosine of an angle. Symbol: sec


equator

From Late Latin (circulus) aequator (diei et noctis).

bor

Baishui is about as close to the equator as Tampa, Florida.

2018, VOA Learning English > China's Melting Glacier Brings Visitors, Adds to Climate Concerns:

(geography, often “the Equator”) An imaginary great circle around Earth, equidistant from the two poles, and dividing earth's surface into the northern and southern hemisphere.

(astronomy) A similar great circle on any sphere, especially on a celestial body, or on other reasonably symmetrical three-dimensional body.


sine wave

(trigonometry, signal processing) A periodically oscillating waveform as described by the sine function, possibly multiplied by some amplitude, shifted by some phase, and translated by an additive constant.


arccosine

(trigonometry) Any of several single-valued or multivalued functions that are inverses of the cosine function.


arcsine

/ˈɑːɹkˌsaɪn/

(trigonometry) Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin⁻¹


arc-

From Latin arcus (“bow, arc”), as they represent an angle or arc. Compare etymology of arch.

arc-sine (British), arcsine (US)

arctan

inverse trigonometric function
inverse trigonometric function#English

(mathematics) Used to form the names of inverse trigonometric functions, and the symbols for these functions.


great circle route

en-noun

great circle route (plural great circle routes)

A route, taken by intercontinental aircraft for instance, which is the shortest distance between two destinations on the surface of the Earth; an arc of a great circle.


great circle

(geometry) A circle defined as the intersection of the surface of a sphere and a plane which passes through the centre of the sphere.


simple harmonic motion

simple harmonic motion (plural simple harmonic motions)

(mathematics, mechanics) oscillating motion (as of a pendulum) in which the acceleration of the oscillator has an equal magnitude but opposite direction to the displacement of it from the equilibrium position.


incircle

LL-Q1860 (eng)-Flame, not lame-incircle.wav

To construct the incircle of ΔABC, we need the center of the circle and a pass-through point.

2003, Clark Kimberling, Geometry in Action: A Discovery Approach Using the Geometer's Sketchpad, page 16:

Having constructed K, the incircle of the quadrilateral can be easily constructed since it is tangent to BA and BC.

2005, István Reiman, International Mathematical Olympiad: 1959-1975, page 51:

For example, in Figure 4.2a only two edges of the polygon are tangent to the incircle. The other four edges do not even touch the incircle, but their extensions, shown by dotted lines, are tangent to the incircle.

2012, Tom M. Apostol, Mamikon A. Mnatsakanian, New Horizons in Geometry, Mathematical Association of America, page 106:

(geometry) A circle within a polygon, especially a triangle, that is tangent to each side.

вписана окръжност

the circle within a polygon that is tangent to each side


waveform

LL-Q1860 (eng)-Flame, not lame-waveform.wav

independent variable
independent variable

(mathematics) The shape of a wave function represented by a graph showing some dependent variable as function of an independent variable.


semicircular

LL-Q1860 (eng)-Flame, not lame-semicircular.wav

In the shape of half of a circle or a semicircle.


geodesic

LL-Q1860 (eng)-Flame, not lame-geodesic.wav

(spherical geometry) A segment of a great circle.


trigonometric function

trigonometric function (plural trigonometric functions)

(trigonometry) Any function of an angle expressible as the ratio of two of the sides of a right triangle that has that angle (e.g., sine), or that subtracts 1 from such a ratio or subtracts such a ratio from 1 (e.g., coversed sine).


cosecant

LL-Q1860 (eng)-Flame, not lame-cosecant.wav

(trigonometry) In a right triangle, the reciprocal of the sine of an angle. Symbols: cosec, csc


cotangent

LL-Q1860 (eng)-Flame, not lame-cotangent.wav

(trigonometry) In a right triangle, the reciprocal of the tangent of an angle. Symbols: cot, ctg, or ctn


arctangent

(trigonometry) Any of several single-valued or multivalued functions that are inverses of the tangent function. Symbol: arctan, tan⁻¹


circular arc

A continuous section of the circumference of a circle.


sinus

Learned borrowing from Latin sinus (“a bent surface, curve, hollow”). Doublet of sine.

lbor

(anatomy, zootomy) A pouch or cavity in a bone or other tissue, especially one in the bones of the face or skull connecting with the nasal cavities (the paranasal sinus).


epicycle

LL-Q1860 (eng)-Flame, not lame-epicycle.wav

From Latin epicyclus, from Ancient Greek ἐπίκυκλος (epíkuklos), from ἐπί (epí, “upon”) + κύκλος (kúklos, “circle”).

(geometry) Any circle whose circumference rolls around that of another circle, thus creating a hypocycloid or epicycloid.

episykli

any circle whose circumference rolls around that of another circle


quadrature of the circle

(geometry) The problem, proposed by ancient Greek geometers, of using a finite ruler-and-compass construction to make a square with the same area as a given circle.


small circle

a circle formed by a plane that cuts a sphere not through its centre


arccos

arccos (1/2ϕ)=2π/5

[…]with the appropriate constraint angles measured in terms of the arccoses and arctans.

1973, The Soviet Economy in Regional Perspective, Praeger Publishers, page 3:


arctan

tan (arctan 2/2)=1/ϕ


cline

Let C₁ and C₂ be two nonintersecting clines. Prove that there is a unique pair of points that are simultaneously symmetric to both C₁ and C₂.

2001, Michael Henle, Modern Geometries: Non-Euclidean, Projective, and Discrete, page 77:

To visualize Möbius transformations, it is helpful to focus on fixed points and, in the case of two fixed points, on two families of clines with respect to these points.

2009, Michael P. Hitchman, Geometry with an Introduction to Cosmic Topology, page 64:

2011, Dominique Michelucci, What is a Line?, Pascal Schreck, Julien Narboux, Jürgen Richter-Gebert (editors), Automated Deduction in Geometry, 8th International Workshop, ADG 2010, Revised Selected Papers, LNAI 6877, page 139,
Let Ω be a fixed, arbitrary, point. Then circles (in the classical sense) through Ω can be considered as lines. For convenience, such circles are called clines in this section. Two distinct clines cut in one point (ignoring Ω and the two cyclic points); it can happen that Ω is a double intersection point; in this case, one may say that the two clines are parallel, and that they meet at a point at infinity, which is Ω.

(geometry, inversive geometry) A generalized circle.


hacoversed sine

en-noun

hacoversed sine (plural hacoversed sines)

trigonometric function
trigonometric function

(trigonometry) The trigonometric function equal to half the coversed sine. Abbreviation: hacoversin


coversed sine

From contraction of complement + versed sine.

The cosine of an arc is the sine of its complement.† In like manner, the coversed sine, cotangent, and cosecant of an arc, are respectively the versed sine, tangent, and secant of its complement.

1825, Jas Thomson, Elements of Plane and Spherical Trigonometry, page 2:

At the top of the page on the left-hand side is placed the number of degrees, and in the left-hand column each minute of the degree, opposite to which are arranged the numerical values of the sine, coversed sine, &c, of the corresponding angle in those columns, at the top of which those terms are placed.

1830, W. Hopkins, “Book II: Elements of Trigonometry”, in Geometry, Plane, Solid, and Spherical, page 46:

For the coversed sine; subtract the sine of the angle from 1. Thus, for the same angle we have the coversed sine,
1 — ·36650 = ·6335.

1838, William Grier, The Mechanic′s Pocket Dictionary, page 21:

Hence the cosines and versed sines of zenith distances are respectively equal to the sines and coversed sines of the corresponding angular elevations.

1871, Homersham Cox, The Law and Science of Ancient Lights, 2nd edition, page 109:

(trigonometry) The trigonometric function 1 − sin(x).


haversed sine

(trigonometry, rare) The trigonometric function defined as half the versed sine. Abbreviation: haversin


versed sine

From New Latin sinus versus (“flipped sine”); the sine function being sometimes called sinus rectus.

For the divisions, to be laid on this line, are the differences between the logarithm versed sine of 180°, and the logarithm versed sines of the successive arcs.

1753, John Robertson, “The Construction of the Logarithmic Lines on the Gunter′s Scale”, in Philosophical Transactions: 1750-1755, Great Britain: Royal Society, page 340:

But if the given arch exceed 90°, find the natural co-sine of its supplement, which add to 1000000, and the sum will be the natural versed sine required.

1805, John William Norie, A New and Complete Epitome of Practical Navigation, page xxii:

The Natural Sines in this table are fractions (radius being unit); and are particularly adapted to Dunthorne′s and other methods of clearing the moon′s distance from the effects of parallax and refraction, where natural sines or versed sines are used ;[…].

1806, John Garnett, Tables Requisite To Be Used with The Nautical Ephemeris, page 41:

(trigonometry) The trigonometric function 1 − cos(x).


unit circle

(trigonometry, the unit circle) The circle of radius 1 with centre at the origin, used in defining trigonometric functions.

jednotková kružnice

circle of radius 1 with centre at the origin, used in trigonometry


havercosine

(trigonometry) the trigonometric function (1 + cos(x)) / 2, equal to half the vercosine


excosecant

(trigonometry) The trigonometric function csc(x) − 1. Abbreviation: excsc


exsecant

(trigonometry) The trigonometric function sec(x) − 1. Abbreviation: exsec


semidiameter

From Middle English semidiametre, semydiametre, from Medieval Latin sēmidiameter, equivalent to semi- + diameter.

(astronomy) The apparent radius of a star etc, when viewed from Earth.

[B]etweene the ſphere of Saturne and the Firmament, there is ſuch an incredible and vaſt ſpace or diſtance (7000000. ſemidiameters of the earth, as Tycho [Brahe] calculates) void of ſtarres: [...]

1638, Democritus Junior [pseudonym; Robert Burton], “Ayre Rectified. With a Digression of the Ayre.”, in The Anatomy of Melancholy. […], 5th edition, Oxford, Oxfordshire: […] [Robert Young, Miles Flesher, and Leonard Lichfield and William Turner] for Henry Cripps, →OCLC, partition 2, section 2, member 3, page 254:


cofunction

Cosine and sine are each other's cofunctions.

(mathematics) The trigonometric function of the complement of the supplied angle.

(programming) A costatement that takes the form of a function and can accept parameters and return a value.


inverse trigonometric function

inverse trigonometric function (plural inverse trigonometric functions)

inverse trigonometric functions (plural)

(mathematics) A function that is the inverse of an appropriate restriction of a trigonometric function.


trigonometric cofunction

trigonometric cofunction (plural trigonometric cofunctions)

(mathematics) Either of a pair of trigonometric functions that have equal values when their arguments are complementary angles, e.g. sine and cosine.


plane angle

plane angle (countable and uncountable, plural plane angles)

(geometry) An angle formed by two intersecting straight lines.

rovinný úhel

(geometry) An angle formed by two intersecting planes, measured by the angle between two intersecting straight lines that are in respective planes and perpendicular to the intersection of the planes.


celestial equator

celestial equator (plural celestial equators)

(astronomy) A great circle on the celestial sphere, coincident with the plane of Earth's Equator.


circular sector

circular sector (plural circular sectors)

(geometry) A portion of a circle enclosed by two radii and an arc.


cosinus

Learned borrowing from New Latin cosinus, abbreviation of complementi sinus. Doublet of cosine.

lbor

When I came to these very buildings to pass my examination I knew far better the names of all the plants in this garden than the theory of the cubic roots or the long formulæ of the sum of two cosinus.

1884, A[lbert] A[ugustin] Fauvel, Chinese Plants in Normandy, Hong Kong: […], page 4, column 1:

So, in the helicopteron, as the helix is at the same time a sustaining plane, it should be likened to a surface moving horizontally, and in which, consequenty, the resistance to motion will be to the lifting power as the sinus is to the cosinus of the angle formed by such plane with the horizon.

1884 November 29, “Aerial Navigation”, in Scientific American: A Weekly Journal of Practical Information, Art, Science, Mechanics, Chemistry, and Manufactures, volume LI, number 22, New York, N.Y.: Munn & Co., translation of original by Victor Tatin in La Nature, page 342, column 1:

And according to our choice of a symmetrical conjunction or opposition, all the cosinuses are reduced to 1, namely to coefficients build up solely by scalar Keplerian elements a, e.

1949, Contributions from the Astronomical Institute of the Charles University Prague, page 38:

The variations are described in terms of cycles of sinuses and cosinuses.

1996, Pentti Zetterberg, Matti Eronen, Markus Lindholm, “Construction of a 7500-Year Tree-Ring Record for Scots Pine (Pinus sylvestris, L.) in Northern Fennoscandia and its Application to Growth Variation and Palaeoclimatic Studies”, in Heinrich Spiecker, Kari Mielikäinen, Michael Köhl, Jens Peter Skovsgaard, editors, Growth Trends in European Forests (European Forest Institute Research Report; No. 5), Springer-Verlag Berlin Heidelberg, →ISBN, page 15:

Basically, the rotation of the matrix of the factor loadings L represents its post-multiplication, i.e. L* = LO by the rotation matrix O, which itself resembles one of the matrices included in the classical rotational Lie groups SO(m) (containing the specific m–fold combination of sinuses and cosinuses.

2007, Vladimir G. Ivancevic, Tijana T. Ivancevic, “Introduction: Human and Computational Mind”, in Computational Mind: A Complex Dynamics Perspective (Studies in Computational Intelligence; 60), Springer-Verlag Berlin Heidelberg, →ISBN, →LCCN, section 1 (Natural Intelligence and Human Mind), pages 60–61:


sinusoid

From the French sinusoïde, equivalent to sinus + -oid.

Temperature during the week can be represented with a sinusoid.

(mathematics) A curve having the shape of a sine wave.


sinusoid

en-adj

sinusoid (comparative more sinusoid, superlative most sinusoid)

more sinusoid (comparative)
most sinusoid (superlative)


osculating circle

en-noun

osculating circle (plural osculating circles)

(geometry, for any point on a curve) The circle that has the same tangent, and the same curvature at the point on the curve.


circular segment

circular segment (plural circular segments)

(geometry) A region of a circle bounded by an arc and the chord connecting the endpoints of the arc.


Fourier series

Named after French mathematician and physicist Joseph Fourier (1768-1830), who pioneered research into such series and their application to problems of heat transfer and vibration.

1951 [Blackie & Son], R. C. H. Young (translator), Konrad Knopp, Theory and Application of Infinite Series, [1947, Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, 4th Edition], 1990, Dover, Unabridged unaltered reprint of 2nd English Edition, page 493,
As we have seen (pp. 369—370), the question of the necessary and sufficient conditions under which the Fourier series of an integrable function converges and represents the given function is one which presents very great difficulties.

2001, David Cruz-Uribe (translator), Javier Duoandikoetxea Zuazo, Fourier Analysis, [1995, Javier Duoandikoetxea Zuazo, Análisis de Fourier], American Mathematical Society, page 20,
There are excellent discussions of Fourier series and integrals in Katznelson [10] and Dym and McKean [4]. The book by R. E. Edwards [5] is an exhaustive study of Fourier series from a more modern perspective.

Unlike Taylor series, a Fourier series can describe functions that are not everywhere continuous and/or differentiable.

2006, K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, 3rd edition, Cambridge University Press, page 415:

mathematical analysis
mathematical analysis

(mathematics, mathematical analysis) Any series resulting from the decomposition of a periodic function into terms involving cosines and sines (or, equivalently, complex exponentials).

フーリエ級数

フーリエきゅうすう,fūrie kyūsū


arccotangent

(trigonometry) An inverse of the cotangent function. Symbol: arccot


sine ratio

right-angled triangle
right-angled triangle

(trigonometry) The ratio of the side of a right-angled triangle opposite to a specified angle to the hypotenuse; when expressed as a real number between 0 and 1, it defines the sine of the angle.


arccosecant

(trigonometry) function that is the compositional inverse of the cosecant function. Symbol: arccsc


arcsecant

(trigonometry) function that is the compositional inverse of the secant function. Symbol: arcsec


magic circle

(mathematics) An arrangement of natural numbers on circles such that the sum of the numbers on each circle and the sum of numbers on diameter are identical.


horoball

Coined by William Thurston (1946-2012), American mathematician. See horosphere and ball.

(hyperbolic geometry) A horosphere together with the space above it.


horosphere

A modern scientific coinage, probably from a Latinized spelling of Ancient Greek ὅρος (hóros, “boundary”) (not to be confused with ὧρος (hôros, “year”), which gave rise to many time-related terms) + sphere

An n-1-dimensional hyperplane in hyperbolic n-dimensional space: it is (in the Poincaré disk) Euclidean-tangent at infinity to the boundary sphere or (in the upper-half-space model) Euclidean-parallel to the boundary hyperplane.


Poincaré disk

The hyperbolic plane, viewed as a disk so that Euclidean circles and lines Euclidean-perpendicular to the disk's surface are hyperbolic lines.


nine-point circle

nine-point circle (plural nine-point circles)

So called because the circle passes through nine points of interest: the feet of the three altitudes, the centers of the three sides, and the midpoints of the segments which join the triangle's vertices with its orthocenter.

That circle which passes through the feet of the altitudes of a given triangle.


ar-

Shortened from area (because the function describes the area under a hyperbola), by analogy with arc-, the corresponding prefix for the circular trigonometric functions.

inverse hyperbolic function
inverse hyperbolic function#English

(trigonometry) Used to form the names of inverse hyperbolic functions, and the symbols for these functions.


square wave

(calculus, engineering, signal processing) A function or waveform that alternates regularly and instantaneously between two levels.


Apollonian circle

Apollonian circle (plural Apollonian circles)

(geometry) Any circle in two families of circles such that every circle in the first family intersects every circle in the second family orthogonally and vice versa.


circumcircle

LL-Q1860 (eng)-Flame, not lame-circumcircle.wav

(geometry) A circle that passes through every vertex of a given triangle (or other polygon where possible).

circumferència circumscrita

a circle that passes through every vertex of a given polygon


square the circle

en-verb

square the circle (third-person singular simple present squares the circle, present participle squaring the circle, simple past and past participle squared the circle)

squares the circle (present singular third-person)
squaring the circle (participle present)
squared the circle (participle past)
squared the circle (past)

The problem of squaring the circle is the problem of determining the ratio of the circumference of a circle to the diameter, which is called π.

1946, Bertrand Russell, History of Western Philosophy, I.24:

(geometry) To construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge, a task which is mathematically impossible.

رَبَّعَ الدائِرة

to construct a square with the same area as a circle

square peg in a round hole

semantic overlap regarding a difficult problem


law of cosines

(trigonometry) A generalization of the Pythagorean theorem that relates the lengths of the sides of a triangle to the cosine of one of its angles.


gyrocircle

(hyperbolic geometry) A hyperbolic circle in gyrovector space.


Fourier sine series

en-noun

Fourier sine series (plural Fourier sine series)

a Fourier series consisting solely of sine functions


Fourier cosine series

Fourier cosine series (plural Fourier cosine series)

a Fourier series consisting solely of cosine functions


law of sines

(trigonometry) A statement that relates the lengths of the sides of a triangle to the sines of its angles.


SOHCAHTOA

LL-Q1860 (eng)-Flame, not lame-SOHCAHTOA.wav

(mnemonic, trigonometry) sine: opposite over hypotenuse, cosine: adjacent over hypotenuse, tangent: opposite over adjacent


Pythagorean trigonometric identity

Pythagorean trigonometric identity (plural Pythagorean trigonometric identities)

Pythagorean trigonometric identities (plural)

(trigonometry) The equality sin (x)²+ cos (x)²=1, holding for all real and complex x, equivalent to the application of the Pythagorean theorem to the unit circle.

jedynka trygonometryczna

statement that sum of squares of sine and cosine equals one


closed curve

(topology) A map from the circle, S¹, to a topological space.^((McLarty, p. 4))


inscribed circle

inscribed circle (plural inscribed circles)


circumscribed circle

circumscribed circle (plural circumscribed circles)


generalized circle

generalized circle (plural generalized circles)

We shall speak of lines and circles as generalized circles. A line is a generalized circle that passes through the point at infinity. Under the inversion #x5C;zeta#x3D;1#x2F;z the generalized circle (53.3) goes into
(53-4) β|ζ|²-pζ-◌̅p◌̅ζ+α=0,

1965, H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, page 247

which is a generalized circle in the ζ-plane.

For any Apollonian family of circles defined by the point circles A and B, the Coaxal Circles Theorem states that every generalized circle through A and B meets each of the Apollonian circles at right angles.

1999, David A. Brannan, Matthew F. Esplen, Jeremy J. Gray, Geometry, page 252:

This is why it makes sense to call every circle and every straight line a generalized circle. Thus the geodesics of ds consist of all arcs of generalized circles which lie wholly within Ω and are orthogonal to w (see Figure 2).

2001, Elie Zahar, Poincaré's Philosophy: From Conventionalism to Phenomenology, page 84:

(geometry, inversive geometry) A circle or a line, the two being regarded as types of a single form.


circle inversion

circle inversion (plural circle inversions)

The first of these is obtained by the method of circle inversion first discovered by Peaucellier.

1875, A. B. Kempe, On Some New Limkages: Messenger of Mathematics, volume 4, page 121:

Nikolai Joukowski (1847-1921) was a Russian mathematician who did research in aerodynamics (James and James, 1992). He used circle inversion in the complex number plane to study airfoil shapes.

1997, Mathematical Association of America, MAA Notes, numbers 41-42, page 169:

As is common with nonlinear projections, it is possible to come up with many variants of circle inversions.[…]An obvious (but perhaps not very useful) extension of circle inversion is sphere inversion, where the spaces inside and outside a sphere are swapped.

2011, David Salomon, The Computer Graphics Manual, page 88:

(geometry, inversive geometry, for a given circle with centre O and radius r) A transformation that maps any point P ≠ O to the point P’ on the ray from O through P for which OP × OP’ = r², and which maps O to the point at infinity.


law of tangents

(trigonometry) A statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.


circular angle

(geometry) An angle that defines a position on a circle.


Malfatti circle

Malfatti circle (plural Malfatti circles)

Named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle.

(geometry) Any of the three circles inside a given triangle such that each circle is tangent to the others and to two sides of the triangle.


orthotomic circle

orthotomic circle (plural orthotomic circles)

(geometry) A circle that cuts three other given circles at right angles.


Ford circle

Named after American mathematician Lester Randolph Ford, Sr., who wrote about them in 1938.

Every Ford circle is tangent to the horizontal axis y#x3D;0, and any two Ford circles are either disjoint or meet at a tangent.

There is a unique Ford circle associated with every rational number. Additionally, the axis y#x3D;0 can be considered a Ford circle with infinite radius, corresponding to the case p#x3D;1,#x5C;q#x3D;0.

Thus the Ford circle [1], drawn tangent to the real axis at #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, and having radius #x5C;textstylea#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D;, must contain in its interior some points belonging to #x5C;textstyle#x7B;p#x5F;n#x7D;#x2F;#x7B;q#x5F;n#x7D;, such as #x5C;textstyle#x5C;theta#x2B;ia#x5F;#x7B;n#x2B;1#x7D;#x7B;-1#x7D;q#x5F;n#x7B;-2#x7D; whose imaginary part lies between #x5C;textstyleq#x5F;n#x7B;-2#x7D; and #x5C;textstyleq#x5F;#x7B;n#x2B;1#x7D;#x7B;-2#x7D;.

1949, American Journal of Mathematics, volume 71, Johns Hopkins University Press, page 413:

A Farey fraction #x5C;textstyle#x5C;frackc defines a Ford circle #x5C;textstyle#x5C;mathcal#x7B;C#x7D;(k,c) in #x5C;textstyle#x5C;mathbbC. Its center is given by #x5C;textstylekc#x2B;i#x5C;frac 1#x7B;2c²#x7D; and its radius is #x5C;textstyle#x5C;frac1#x7B;2c²#x7D;. Two Ford circles #x5C;textstyle#x5C;mathcalC(a,b) and #x5C;textstyle#x5C;mathcalC(c,d) are tangent whenever #x5C;textstylead-bc#x3D;#x5C;pm 1. This is the case for Ford circles related to consecutive Farey fractions in a sequence #x5C;textstyleF#x5F;N.

2008, Jan Manschot, Partition Functions for Supersymmetric Black Holes, Amsterdam University Press, page 78:

Ford circles are a collection of horocycles in #x5C;mathbbH used by Ford to study continued fractions in papers such as [2, 3].[…]Two Ford circles intersect in at most a single point, and the interiors of the two circles are disjoint. In fact, one can check that the Ford circles C#x5F;#x7B;a#x2F;b#x7D; and C#x5F;#x7B;c#x2F;d#x7D; are tangent if and only if #x5C;vertad-bc#x5C;vert#x3D;1.

2016, Ian Short, Mairi Walker, “Even-Integer Continued Fractions and the Farey Tree”, in Jozef Širáň, Robert Jajcay, editors, Symmetries in Graphs, Maps, and Polytopes: 5th SIGMAP Workshop, Springer, page 298:

(geometry) Any one of a class of circles with centre at (p/q, 1/(2q²)) and radius 1/(2q²), where p/q is an irreducible fraction (i.e., p and q are coprime integers).

Ford-Kreis

any of a class of circles, each unique to a rational number and pairwise either tangent to or disjoint from each other circle


quarter circle

(geometry) A quarter of a circle. The shape formed from an arc that is a quarter of the circumference of a circle.


tetragonism

quadrature of the circle
quadrature of the circle

(dated, mathematics) quadrature of the circle


inverse tangent

inverse tangent (plural inverse tangents)


Archimedes' constant

(mathematics, chiefly historical) Synonym of pi


unit disk

A set of points on a plane whose distance from a given point is less than or equal to one.


n-sphere

An n-dimensional hypersphere: the 2-sphere is a circle, the 3-sphere is the usual sphere, the 4-sphere is a glome, etc.


goat grazing problem

goat grazing problem (plural goat grazing problems)

The problem: given a field of radius 1, and a goat tied to a point on the edge of said field; how long a rope should the goat be tied to, so that the goat has access to exactly half of the field? Equivalently: given a unit circle C and a point P on the rim of C, what is the value of r such that exactly half of the area within C, is within distance r of point P?

The same problem generalized to any number of dimensions: given a unit n-sphere (n-dimensional hypersphere) S, and a point P on the edge of S, what is r such that exactly one-half of the space (content) within S, is within distance r of P?


goat problem

goat grazing problem
goat grazing problem#English


versed cosine

(trigonometry) The trigonometric function 1 + cos(x).


Brocard circle

Named after Henri Brocard (1845–1922), French mathematician.

(geometry) A circle that passes through the circumcenter and symmedian of a given triangle, and is centered at the midpoint of the line segment joining them.


Simson line

Named after Robert Simson, though it had been published prior to his work.

(geometry) Given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through these points is the Simson line of P.