Angle 1196 225466232 2008-07-13T21:03:58Z AdrianLozano 760758 /* The dot product and generalisation */ Mathified "<·,·>" {{otheruses4|angles in geometry}} [[Image:Angle Symbol.svg|120px|thumb|right|"∠", the angle symbol.]] In [[geometry]] and [[trigonometry]], an '''angle''' (in full, '''plane angle''') is the figure formed by two [[Ray (geometry)|rays]] sharing a common [[endpoint]], called the [[vertex (geometry)|vertex]] of the angle {{harv|Sidorov|2001}}. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below). Where there is no possibility of confusion, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity). The word ''angle'' comes from the [[Latin]] word ''angulus'', meaning "a corner". The word ''angulus'' is a diminutive, of which the primitive form, ''angus'', does not occur in Latin. [[Cognate]] words are the Latin ''angere'', meaning "to compress into a bend" or "to strangle", the [[Greek language|Greek]] {{polytonic|{{Unicode|ἀ}}γκύλος}} ''(ankylοs)'', meaning "crooked, curved," and the [[English language|English]] word "[[ankle]]." All three are connected with the [[Proto-Indo-European language|Proto-Indo-European]] root ''*ank-'', meaning "to bend" or "bow" {{harv|Slocum|2007}}. ==History== [[Euclid]] defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to [[Proclus]] an angle must be either a quality or a quantity, or a relationship. The first concept was used by [[Eudemus]], who regarded an angle as a deviation from a [[straight line]]; the second by [[Carpus of Antioch]], who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative. ==Measuring angles==<!-- linked from [[Degree (angle)]] --> [[Image:Angle measure.svg|right|thumb|The angle <var>θ</var> is the quotient of <var>s</var> and <var>r</var>.]] In order to measure an angle <var>θ</var>, a [[circular arc]] centered at the vertex of the angle is drawn, e.g. with a pair of [[Compasses (drafting)|compasses]]. The length of the arc <var>s</var> is then divided by the radius of the circle <var>r</var>, and possibly multiplied by a scaling constant <var>k</var> (which depends on the units of measurement that are chosen): :<math> \theta = \frac{s}{r}(k). </math> The value of <var>θ</var> thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio ''s''/''r'' is unaltered. In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a [[spiral]] using [[polar coordinates]], an extra full turn gives rise to a quite different point on the curve. ===Units=== Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant <var>k</var> in the formula above. Of these units, treated in more detail below, the ''degree'' and the ''radian'' are by far the most common. With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one revolution) is equal to ''n'' units, for some whole number ''n''. For example, in the case of degrees, {{nowrap|1= ''n'' = 360.}} A full circle of ''n'' units is obtained by setting {{nowrap|1= ''k'' = ''n''/(2π)}} in the formula above. (Proof. The formula above can be rewritten as {{nowrap|1= ''k'' = θ''r''/''s''.}} One full circle, for which {{nowrap|1= θ = ''n''}} units, corresponds to an arc equal in length to the circle's [[circumference]], which is 2π''r'', so {{nowrap|1= ''s'' = 2π''r''}}. Substituting ''n'' for θ and 2π''r'' for ''s'' in the formula, results in {{nowrap|1= ''k'' = ''nr''/(2π''r'') = ''n''/(2π).}}) *The '''[[degree (angle)|degree]]''', denoted by a small superscript circle (°) is 1/360 of a full circle, so one full circle is 360°. One advantage of this old [[sexagesimal]] subunit is that many angles common in simple geometry are measured as a whole number of degrees. (The goal of having ''all'' "interesting" angles measured as whole numbers is of course unattainable.) Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the following sexagesimal subunits of the "degree-minute-second" system are also in use, especially for [[Geographic coordinate system|geographical coordinates]] and in [[astronomy]] and [[ballistics]]: ** The '''[[minute of arc]]''' (or '''MOA''', '''arcminute''', or just '''minute''') is 1/60 of a degree. It is denoted by a single prime (&nbsp;&prime;&nbsp;). For example, 3°&nbsp;30&prime; is equal to 3&nbsp;+&nbsp;30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3°&nbsp;5.72&prime; = 3&nbsp;+&nbsp;5.72/60 degrees. A [[nautical mile]] was historically defined as a minute of arc along a [[great circle]] of the Earth. ** The '''[[second of arc]]''' (or '''arcsecond''', or just '''second''') is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime (&nbsp;&Prime;&nbsp;). For example, 3°&nbsp;7&prime;&nbsp;30&Prime; is equal to 3 + 7/60 + 30/3600 degrees, or 3.125&nbsp;degrees. [[Image:Angle radian.svg|right|thumb|<var>θ</var> = <var>s</var>/<var>r</var> rad = 1 rad.]] *The '''[[radian]]''' is the angle subtended by an arc of a circle that has the same length as the circle's radius (<var>k</var> = 1 in the formula given earlier). One full circle is 2''π'' radians, and one radian is 180/''π'' degrees, or about 57.2958 degrees. The radian is abbreviated ''rad'', though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the [[trigonometric function]]s display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the [[SI]] system. *The '''[[angular mil|mil]]''' is ''approximately'' equal to a [[milliradian]]. There are several definitions. *The '''full circle''' (or '''revolution''', '''rotation''', '''full [[Turn (geometry)|turn]]''' or '''cycle''') is one complete revolution. The revolution and rotation are abbreviated ''rev'' and ''rot'', respectively, but just ''r'' in ''[[Revolutions per minute|rpm]]'' (revolutions per minute). 1 full circle = 360° = 2''π'' rad = 400 gon = 4 right angles. *The '''[[right angle]]''' is 1/4 of a full circle. It is the unit used in [[Euclid's Elements]]. 1 right angle = 90° = ''π''/2&nbsp;rad = 100&nbsp;gon. *The '''angle of the [[equilateral triangle]]''' is 1/6 of a full circle. It was the unit used by the [[Babylonia]]ns, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are [[sexagesimal]] subunits of the Babylonian unit. 1 Babylonian unit = 60° = ''π''/3&nbsp;rad ≈ 1.047197551&nbsp;rad. *The '''[[grad (angle)|grad]]''', also called '''grade''', '''gradian''', or '''gon''' is 1/400 of a full circle, so one full circle is 400 grads and a [[right angle]] is 100 grads. It is a decimal subunit of the right angle. A [[kilometer]] was historically defined as a [[centi]]-gon of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in [[triangulation]]. *The '''point''', used in [[navigation]], is 1/32 of a full circle. It is a binary subunit of the full circle. Naming all 32 points on a [[compass rose]] is called "[[boxing the compass]]". 1&nbsp;point = 1/8 of a right angle = 11.25° = 12.5&nbsp;gon. *The astronomical '''[[hour angle]]''' is 1/24 of a full circle. The sexagesimal subunits were called '''minute of time''' and '''second of time''' (even though they are units of angle). 1&nbsp;hour = 15° = ''π''/12&nbsp;rad = 1/6&nbsp;right&nbsp;angle ≈ 16.667&nbsp;gon. *The '''binary degree''', also known as the '''binary radian''' (or '''brad'''), is 1/256 of a full circle. The binary degree is used in computing so that an angle can be efficiently represented in a single [[byte]]. *The '''[[grade (slope)|grade of a slope]]''', or '''gradient''', is not truly an angle measure (unless it is explicitly given in degrees, as is occasionally the case). Instead it is equal to the [[tangent (trigonometric function)|tangent]] of the angle, or sometimes the [[sine]]. Gradients are often expressed as a percentage. For the usual small values encountered (less than 5%), the grade of a slope is approximately the measure of an angle in radians. ===Positive and negative angles=== A convention universally adopted in mathematical writing is that angles given a sign are '''positive angles''' if measured [[Clockwise and counterclockwise|counterclockwise]], and '''negative angles''' if measured [[Clockwise and counterclockwise|clockwise]], from a given line. If no line is specified, it can be assumed to be the [[x-axis]] in the [[Cartesian plane]]. In many geometrical situations a negative angle of &minus;''θ'' is effectively equivalent to a positive angle of "one full rotation less ''θ''". For example, a clockwise rotation of 45° (that is, an angle of &minus;45°) is often effectively equivalent to a counterclockwise rotation of 360°&nbsp;&minus;&nbsp;45° (that is, an angle of 315°). In three dimensional geometry, "clockwise" and "counterclockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a [[Vector (spatial)|vector]] passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In [[navigation]], [[bearing (navigation)|bearings]] are measured from north, increasing clockwise, so a bearing of 45 degrees is north-east. Negative bearings are not used in navigation, so north-west is 315 degrees. ===Approximations=== * 1° is approximately the width of a little finger at arm's length * 10° is approximately the width of a closed fist at arm's length. * 20° is approximately the width of a handspan at arm's length. ==Identifying angles== In mathematical expressions, it is common to use [[Greek letter]]s (<var>α</var>, <var>β</var>, <var>γ</var>, <var>θ</var>, <var>φ</var>, ...) to serve as [[variable]]s standing for the size of some angle. (To avoid confusion with its other meaning, the symbol [[Pi|π]] is not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. See the figures in this article for examples. In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A"). Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see [[#Positive and negative angles|Positive and negative angles]]). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180° degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B. == Types of angles == {| |- style="vertical-align: top;" |[[Image:Right angle.svg|thumb|134px|Right angle.]] |[[Image:Reflex angle.svg|thumb|96px|Reflex angle.]] |[[Image:Complement angle.svg|thumb|134px|The [[complementary angles]] <var>a</var> and <var>b</var> (<var>b<var> is the complement of <var>a</var>, and <var>a<var> is the complement of <var>b</var>).]] |[[Image:Angle obtuse acute straight.svg|thumb|241px|Acute (<var>a</var>), obtuse (<var>b</var>), and straight (<var>c</var>) angles. Here, <var>a</var> and <var>b</var> are [[supplementary angles]].]] |} <!-- old images | [[image:angle acute.png|thumb|150px|Acute angle]] | [[image:angle obtuse.png|thumb|200px|Obtuse angle]] | [[image:angle straight.png|thumb|200px|Straight angle]] --> *An angle of 90° (''[[pi|π]]''/2 radians, or one-quarter of the full circle) is called a '''[[right angle]]'''. *:Two lines that form a right angle are said to be '''[[perpendicular]]''' or '''[[orthogonality|orthogonal]]'''. *Angles smaller than a right angle (less than 90°) are called '''acute angles''' ("acute" meaning "sharp"). *Angles larger than a right angle and smaller than two right angles (between 90° and 180°) are called '''obtuse angles''' ("obtuse" meaning "blunt"). *Angles equal to two right angles (180°) are called '''straight angles'''. *Angles larger than two right angles but less than a full circle (between 180° and 360°) are called '''reflex angles'''. *Angles that have the same measure are said to be '''[[congruence|congruent]]'''. *Two angles opposite each other, formed by two intersecting straight lines that form an "X" like shape, are called '''[[Vertical (angles)|vertical angles]]''' or '''opposite angles'''. These angles are congruent. *Angles that share a common vertex and edge but do not share any interior points are called '''[[adjacent angles]]'''. *Two angles that sum to one right angle (90°) are called '''[[complementary angle]]s'''. *:The difference between an angle and a right angle is termed the '''complement''' of the angle. *Two angles that sum to a straight angle (180°) are called '''[[supplementary angle]]s'''. *:The difference between an angle and a straight angle is termed the '''supplement''' of the angle. *Two angles that sum to one full circle (360°) are called '''explementary angles''' or '''conjugate angles'''. *An angle that is part of a [[simple polygon]] is called an '''[[interior angle]]''' if it lies in the inside of that the simple polygon. Note that in a simple polygon that is concave, at least one interior angle exceeds 180°. *:In [[Euclidean geometry]], the measures of the interior angles of a [[triangle (geometry)|triangle]] add up to ''π'' radians, or 180°; the measures of the interior angles of a simple [[quadrilateral]] add up to 2''π'' radians, or 360°. In general, the measures of the interior angles of a [[polygon|simple polygon]] with ''n'' sides add up to [(''n''&nbsp;&minus;&nbsp;2)&nbsp;&times;&nbsp;''π''] radians, or [(''n''&nbsp;&minus;&nbsp;2)&nbsp;&times;&nbsp;180]°. *The angle supplementary to the interior angle is called the '''[[exterior angle]]'''. It measures the amount of "turn" one has to make at this vertex to trace out the polygon. If the corresponding interior angle exceeds 180°, the exterior angle should be considered [[negative]]. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an [[orientation (mathematics)|orientation]] of the [[plane (mathematics)|plane]] (or [[surface]]) to decide the sign of the exterior angle measure. *:In Euclidean geometry, the sum of the exterior angles of a simple polygon will be 360°, one full turn. *Some authors use the name '''exterior angle''' of a simple polygon to simply mean the explementary (''not'' supplementary!) of the interior angle [http://mathworld.wolfram.com/ExteriorAngle.html]. This conflicts with the above usage. *The angle between two [[Plane (mathematics)|planes]] (such as two adjacent faces of a [[polyhedron]]) is called a '''[[dihedral angle]]'''. It may be defined as the acute angle between two lines [[Surface normal|normal]] to the planes. *The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane. *If a straight [[transversal line]] intersects two [[Parallel (geometry)|parallel]] lines, corresponding (alternate) angles at the two points of intersection are congruent; [[adjacent angles]] are [[supplementary angles|supplementary]] (that is, their measures add to ''π'' radians, or 180°). ==A formal definition== ===Using trigonometric functions=== A Euclidean angle is completely determined by the corresponding right triangle. In particular, if <math>\theta</math> is a Euclidean angle, it is true that :<math>\cos \theta = \frac{x}{\sqrt{x^2 + y^2}}</math> and :<math>\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}</math> for two numbers <math>x</math> and <math>y</math>. So an angle in the Euclidean plane can be legitimately given by two numbers <math>x</math> and <math>y</math>. To the ratio <math>\frac{y}{x}</math> there correspond two angles in the geometric range <math>0 < \theta < 2\pi </math>, since :<math>\frac{\sin \theta }{\cos \theta } = \frac{\frac{y}{\sqrt{x^2 + y^2}}}{\frac{x}{\sqrt{x^2 + y^2}}} = \frac{y}{x} = \frac{-y}{-x} = \frac{\sin (\theta + \pi)}{\cos (\theta + \pi) }. </math> ===Using rotations=== Suppose we have two unit vectors <math>\vec{u}</math> and <math>\vec{v}</math> in the euclidean plane <math>\mathbb{R}^2</math>. Then there exists one positive [[isometry]] (a rotation), and one only, from <math>\mathbb{R}^2</math> to <math>\mathbb{R}^2</math> that maps <math>u</math> onto <math>v</math>. Let ''r'' be such a rotation. Then the relation <math>\vec{a}\mathcal{R}\vec{b}</math> defined by <math>\vec{b}=r(\vec{a})</math> is an equivalence relation and we call '''angle of the rotation ''r''''' the [[equivalence class]] <math>\mathbb{T}/\mathcal{R}</math>, where <math>\mathbb{T}</math> denotes the unit circle of <math>\mathbb{R}^2</math>. The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector <math>(1,0)</math>, then for any point M on <math>\mathbb{T}</math> at distance <math>\theta</math> from <math>(1,0)</math> (on the circle), let <math>\vec{u}=\overrightarrow{OM}</math>. If we call <math>r_\theta</math> the rotation that transforms <math>(1,0)</math> into <math>\vec{u}</math>, then <math>\left[r_\theta\right]\mapsto\theta</math> is a bijection, which means we can identify any angle with a number between 0 and <math>2\pi</math>. ==Angles between curves== [[Image:Curve angles.svg|thumb|right|The angle between the two curves is defined as the angle between the tangents <var>A</var> and <var>B</var> at <var>P</var>]] The angle between a line and a [[curve]] (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the [[tangent]]s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:&mdash;''amphicyrtic'' (Gr. ''{{Unicode|ἀ}}μφί'', on both sides, ''κυρτόσ'', convex) or ''cissoidal'' (Gr. ''κισσόσ'', ivy), biconvex; ''xystroidal'' or ''sistroidal'' (Gr. ''ξυστρίσ'', a tool for scraping), concavo-convex; ''amphicoelic'' (Gr. ''κοίλη'', a hollow) or ''angulus lunularis'', biconcave. ==The dot product and generalisation== In the [[Euclidean space|Euclidean plane]], the angle θ between two [[Vector (spatial)|vector]]s '''u''' and '''v''' is related to their [[dot product]] and their lengths by the formula :<math>\mathbf{u} \cdot \mathbf{v} = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.</math> This allows one to define angles in any real [[inner product space]], replacing the Euclidean dot product · by the [[Hilbert space]] inner product <math>\langle\cdot,\cdot\rangle</math>. ==Angles in Riemannian geometry== In [[Riemannian geometry]], the [[metric tensor]] is used to define the angle between two [[tangent]]s. Where ''U'' and ''V'' are tangent vectors and ''g''<sub>''ij''</sub> are the components of the metric tensor ''G'', :<math> \cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}. </math> ==Angles in geography and astronomy== In [[geography]] we specify the location of any point on the Earth using a '''[[Geographic coordinate system]]'''. This system specifies the [[latitude]] and [[longitude]] of any location, in terms of angles subtended at the centre of the Earth, using the [[equator]] and (usually) the [[Greenwich meridian]] as references. In [[astronomy]], we similarly specify a given point on the [[celestial sphere]] using any of several '''[[Astronomical coordinate systems]]''', where the references vary according to the particular system. Astronomers can also measure the '''angular separation''' of two [[star]]s by imagining two lines through the centre of the [[Earth]], each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars. Astronomers also measure the '''[[apparent size]]''' of objects. For example, the [[full moon]] has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The [[small-angle formula]] can be used to convert such an angular measurement into a distance/size ratio. == References == * {{springer|id=A/a012500|title=Angle|year=2001|last=Sidorov|first=L.A.}}. * {{citation|last=Slocum|first=Jonathan|year=2007|url=http://www.utexas.edu/cola/centers/lrc/ielex/PokornyMaster-X.html|title=Preliminary Indo-European lexicon - Pokorny PIE data |accessdate=[[16 July]], [[2007]]|publisher=[[Linguistics Research Center at UT Austin|University of Texas research department: linguistics research center]]}}. ==See also== *[[Complementary angles]] *[[Supplementary angles]] *[[Central angle]] *[[Inscribed angle]] *[[Solid angle]] for a concept of angle in three dimensions. *[[Astrological aspect]] *[[Protractor]] *[[Clock angle problem]] ==External links== {{CommonsCat|Angles}} * [http://www.cut-the-knot.org/Curriculum/Geometry/CyQuadri.shtml Angle Bisectors in a Quadrilateral] at [[cut-the-knot]] * [http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml Constructing a triangle from its angle bisectors] at [[cut-the-knot]] * [http://sundials.org/links/local/pages/dd_dms.htm Convert angles in sexagesimal degree format to decimal degrees, and vice-versa] * [http://www.austinastro.org/angles.html Angle Estimation] -- for basic [[astronomy]]. * [http://www.mathopenref.com/tocs/anglestoc.html Angle definition pages] with interactive applets. * [http://www.mathopenref.com/tocs/constructionstoc.html Various angle constructions with compass and straightedge] Animated demonstrations [[Category:Elementary geometry]] [[Category:Trigonometry]] [[Category:Angle| ]] {{Link FA|nl}} <!-- interwiki --> [[af:Hoek (meetkunde)]] [[als:Winkel (Geometrie)]] [[ar:زاوية(هندسة)]] [[ast:Ángulu]] [[gn:Takamby]] [[ay:K'uchu]] [[az:Bucaq]] [[bn:সমকোণ]] [[be-x-old:Кут]] [[bs:Ugao]] [[bg:Ъгъл]] [[ca:Angle]] [[cs:Úhel]] [[da:Vinkel]] [[de:Winkel]] [[el:Γωνία]] [[et:Nurk]] [[es:Ángulo]] [[eo:Angulo]] [[fa:زاویه]] [[fr:Angle]] [[gd:Ceàrn (Matamataig)]] [[gl:Ángulo]] [[zh-classical:角]] [[ko:각도]] [[hr:Kut]] [[io:Angulo]] [[id:Sudut (geometri)]] [[is:Bogagráða]] [[it:Angolo]] [[he:זווית]] [[sw:Pembe (jiometria)]] [[ht:Ang]] [[lv:Leņķis]] [[lt:Kampas]] [[hu:Szög]] [[mk:Агол]] [[mr:कोन]] [[ms:Sudut]] [[nl:Hoek (meetkunde)]] [[ja:角度]] [[pl:Kąt płaski]] [[pt:Ângulo]] [[ro:Ângulo]] [[qu:Chhuka]] [[ru:Угол]] [[simple:Angle]] [[sk:Uhol]] [[sl:Kot]] [[sr:Угао]] [[fi:Kulma]] [[sv:Vinkel]] [[ta:கோணம்]] [[th:มุม]] [[vi:Góc]] [[tr:Açı]] [[uk:Кут]] [[zh:角]]