See also: Angles

In Geometry, the general concept of angle is declined in several related concepts.

In its old direction, the angle is a plane figure, portion of plan delimited by two right S secant. Thus one speaks about the angles of a Polygone. However, the use is now to employ the angular word of sector for such a figure. The angle can also indicate a portion of space delimited by two plans ( Angle diédral ). It is possible to define the measurement of such angles, and this measurement carries usually but wrongly the name of angle it too.

In a more abstract direction, the angle is a Classe of equivalence, i.e. a unit obtained by assimilating between them all the identifiable angle-figures by Isométrie. Any of the identified figures is then called representing angle. All these representatives having even measurement, one can speak about measurement of the abstract angle.

It is possible to define a concept of angle directed in Euclidean Géométrie of the plan, as to extend the concept of angle to the framework of the vector spaces préhilbertiens or varieties riemanniennes.

The word angle derives from Latin angulus , the corner.

The angle like appears of the plan or space

Angular sector and angle

A angular sector is a plane figure obtained by intersection or meeting of two half-planes delimited by right secant or confused .

The angle of an angular sector is the positive real number which measures the proportion of the plan occupied by the angular sector. The Unit S used to quantify it are the Radian, the Quadrant and its subdivisions the degree and the rank. The angles are frequently noted by a small Greek letter, for example α, β, θ, ρ… When the angle is at the top of a Polygone and that there is no ambiguity, one then uses the name of the overcome top of a hat, for example  .

The angle can be also interpreted like the opening of the angular sector, i.e. “speed” to which move away the lines one from the other when one moves away from the point of intersection. It is the measurement of the slope of a line compared to the other.

Value of an angle

In mathematics, the value of the angle formed by two secant lines of an Euclidean plan…

There are actually two angles: the acute angle (which with the smallest value) and the angle obtu (or two flat angles , in the event of equality).

In a broader context, one defines the angle in the intersection of two curves while being reduced to the tangent plan defines by the tangent ones in the curves in the point of intersection.

This value of the angle corresponds in physics to a measurement in radians. The Radian is the international unit of measurement of the angles. However… --> To evaluate this angle, this “proportion of surface”, one takes a disc centered at the point of intersection, and one carries out the relationship between the surface of the portion of disc intercepted by the angular sector and the total surface of the disc. One can show that also amounts submitting the relationship between the length of the intercepted arc and the circumference of the circle; this value lower than 1 is called number of turn . Value 1/4 (quarter of turn) corresponds to the Quadrant.

A unit usually used is the degree, which consists in subdividing the quadrant in 90 equal shares. The full rotation thus corresponds to 360 degrees. The minute of arc is a submultiple of the degree, equal to 1/60 of degree. In the same way, the second of arc is equal to 1/60 of the minute of arc, that is to say 1/3600 of degree. One more rarely uses the rank, which corresponds to a centesimal subdivision of the quadrant.

The international unit of measurement of the angles is however the Radian, definite like the relationship between the length of the intercepted arc and the radius of the circle. The full rotation thus corresponds to $2\; \backslash \; pi$ radians.

The angles can be calculated starting from the lengths on the sides of Polygone S, in particular of Triangle S, by using the Trigonométrie.

In certain cases, the angles are expressed by their tangent. For example, a slope is expressed in Pourcent, it is the number of meters which one assembles (or goes down) when one traverses 100 m compared to the horizontal one; if α is the angle between the line of greater slope and the horizontal one, then the slope in % is equal to 100×tan (α). In Gliding (Aeronautical), the smoothness of a sail is the number of meters which one goes down when 100 m were traversed horizontally (in absence of wind); they are also one hundred times the tangent of the slope.

The measuring unit of the angles used mainly by the soldiers is the Thousandth. It is the angle under which one sees 1 meter with 1 kilometer. 6400 thousandth corresponds to 2π radians or 360 degrees.

“On the ground”, the angles can be measured with an apparatus called Goniomètre; it in general comprises a curved rule graduated in degrees, called Rapporteur.

Name of the angles

The angles corresponding to an integer of quadrants bear a particular name

The right angle is obtained by considering two lines which divide the plan into four equal sectors. Such lines are known as “orthogonal” or “perpendiculars”.

The qualifiers according to are employed for the angles taking of the intermediate values between these remarkable values

• the angle returning is an angle higher than the flat angle;
• the projecting angle is an angle lower than the flat angle;
• a salient angle is blunt when it is higher than the right angle;
• it is acute when it is lower than the right angle.

To qualify the relative values of two angles the following expressions are employed:

• two angles are complementary when their nap made 90°;
• two angles are additional when their nap made 180°.

One still employs other expressions to qualify the position of the angles on a figure, i.e. more precisely, the relative position of angular sectors.

• Two angular sectors are opposed by the top, when they have the same top and that the sides of the one are in the prolongation of those of the other. In this case the angles corresponding are equal.
• Two angular sectors are adjacent when they have the same top, a common side, and that their intersection is equal to this common side. The angles are added when the meeting of these sectors is considered.
• the alternate-external angles and the angles alternate-interns relate to parallel straight lines cut by a secant.

Remark : two complementary or additional angles are not necessarily adjacent: For example, in a right-angled triangle ABE out of B, the angles  and E are complementary.

By extension, one also defines the angles between half-lines, segments of right-hand side and Vecteur S, by prolonging the lines carrying these objects until their intersection. The definition by half-lines or vectors makes it possible to raise the indetermination between the additional angles, i.e. to define without ambiguity which angular sector to use to define the slope of the directions.

Geometrical angle

A geometrical angle is a mathematical object being able to be represented by an angular sector. One can interpret it in several ways: divergence between two directions, directions of the faces of an object (corner) the direction aimed compared to north (angle given by a compass)…

$\backslash \; widehat\; \{VAT\}$ is a geometrical angle.

One has in addition:

$\backslash \; widehat\; \{VAT\}\; =\; \backslash \; widehat\; \{CAB\}$

One frequently confuses “measurement of the angle” and “angle”. Thus for example an angle " plat" angle “equal” to 180 is called wrongly.

Note: This abuse is voluntarily applied largely and in the continuation of this article. In addition a right angle for example, can be represented by several different angular sectors, but as they all are “superposable”, they represent all the same angle. In mathematics one speaks about " classify équivalence". Note: This problem also arises when one tries to distinguish " fraction" and " rationnel".

Angles directed in the plan

If the plan is directed, then the angles can be positive or negative according to the direction in which they “turn”.

$(\backslash \; vec\; \{AB\},\; \backslash \; vec\; \{AC\})$ is a directed angle.

By convention, one directs the plan in the direction known as “trigonometrical”, i.e. in the opposite direction of the needles of a watch (or “anti-clockwise direction”). If one considers two half-lines or vectors, then the order in which one quotes the half-lines or the vectors defines the direction of the angle, therefore its sign; as follows:

$(\backslash \; widehat\; \{\backslash \; vec\; \{U\},\; \backslash \; vec\; \{v\}\})\; =\; -\; (\backslash \; widehat\; \{\backslash \; vec\; \{v\},\; \backslash \; vec\; \{U\}\})$

The angles are defined with a margin of an integer of turns. Thus, the complete plan can be defined by a full rotation in the positive direction, two full rotations in the positive direction, a full rotation in the negative direction… In radians, one says that the angles are definite with 2π close (“to two pi near”). For example, if the angle α is right direct direction, it is noted:

$\backslash \; alpha\; =\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; +\; 2k\; \backslash \; pi,\; K\; \backslash \; in\; \backslash \; mathbb\; \{Z\}$

or

$\backslash \; alpha\; \backslash \; equiv\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}$

This last notation is read: “alpha is adequate with pi on two Modulo two pi”.

It is noticed in particular that for two half-lines (or two vectors) data, the fact of choosing the “small one” or “the great” portion of plan imports little, since α  ≡  α  -   2π (cf illustration above).

Angles of vectors

The angles are defined starting from classes of equivalence in the following way:
In the usual Euclidean plan (normalized), one defines the isométries, transformations of the plan preserving the standard of the vectors. The isométries have a determinant equal to 1 or to -1.

The isométries of determinant 1 (known as “positive”) transform a vector unit (of standard 1) into another vector unit. For a couple of vectors units $(\backslash \; vec\; \{U\},\; \backslash \; vec\; \{v\})$ given, there exists a positive isometry F transforming $\backslash \; vec\; \{U\}$ into $\backslash \; vec\; \{v\}$, one has

$\backslash \; vec\; \{v\}\; =\; F\; (\backslash \; vec\; \{U\})$.

That is to say another positive isometry G and $\backslash \; vec\; \{u\text{'}\}$ and $\backslash \; vec\; \{v\text{'}\}$ two other vectors such as

$\backslash \; vec\; \{u\text{'}\}\; =g\; (\backslash \; vec\; \{U\})$ and $\backslash \; vec\; \{v\text{'}\}\; =\; G\; (\backslash \; vec\; \{v\})$.

We can show that

$\backslash \; vec\; \{v\text{'}\}\; =\; F\; (\backslash \; vec\; \{u\text{'}\})$

and that the whole of the couples of vectors units $(\backslash \; vec\; \{U$}, \ vec {v }) checking

$\backslash \; vec\; \{v$} = F (\ vec {U })

is a class of equivalence on F , each isometry F determines a class of equivalence.

We call angle θ the class of equivalence of this couple, the associated isometry is the rotation of angle θ.

Definition to re-examine, supplement and illustrate

Angles in space

Two secant lines are necessarily coplanar, therefore the angle between the lines is defined in this plan, in the same manner as above. For to direct the plan, one chooses a normal vector in the plan: the plan is then directed in the trigonometrical direction when the normal vector points towards the observer. If one defined a base $(\backslash \; vec\; \{I\},\; \backslash \; vec\; \{J\})$ in this plan, then one chooses for normal vector $\backslash \; vec\; \{I\}\; \backslash \; wedge\; \backslash \; vec\; \{J\}$.

Orientation of a plan by a normal vector

To define the angle between two plans, one considers the angle which their normal vectors form.

To define the angle between a plan and a line, one considers the angle α between the line and his orthogonal projection on the plan, or the angle complementary between the line and the normal to the plan: one cuts off the angle β between the line and the normal in the plan from the right angle (α  =  π/2  -   β in radians).

To define the angle between two unspecified lines of space, one considers the angle which their directing vectors form (of which the cosine is equal to the scalar product of these unit vectors), or the planar angle which one of the two lines with any forms parallel with the other which cuts it. This angle east defines modulo the same choices of orientation evoked above.

One also defines the solid angles: one takes a point (sometimes called “not of observation”) and a surface in space (“surface observed”), the solid angle is the proportion of space delimited by the cone having for top the point considered and being based on the contour of surface. The unit is the Stéradian (Sr in summary), complete space made 4π  Sr.

Use of the angles

• In Geodesy (Geography)

• Azimuth: angle compared to the North-South axis on a level containing this axis and the point concerned, counted compared to the North counted in the direction of the needles of a watch;
• Latitude: angle which a vertical on the basis of a point forms and going in the center of the ground compared to the plan of the equator; the points having the same latitude form a circle called “parallel”
• Longitude: angle allowing to locate itself on Earth: angle which the plan containing forms the North-South axis and the point considered (called “meridian plan”) with a datum-line also containing the North-South axis; the intersection of a meridian plan with the surface of the Earth is a large half Cercle

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