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Attitude and heading reference system

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Flight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of gravity (cg), known as pitch , roll and yaw . These are collectively known as aircraft attitude , often principally relative to the atmospheric frame in normal flight, but also relative to terrain during takeoff or landing, or when operating at low elevation. The concept of attitude is not specific to fixed-wing aircraft, but also extends to rotary aircraft such as helicopters, and dirigibles , where the flight dynamics involved in establishing and controlling attitude are entirely different.

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66-556: An attitude and heading reference system ( AHRS ) consists of sensors on three axes that provide attitude information for aircraft, including roll , pitch , and yaw . These are sometimes referred to as MARG (Magnetic, Angular Rate, and Gravity) sensors and consist of either solid-state or microelectromechanical systems (MEMS) gyroscopes , accelerometers and magnetometers . They are designed to replace traditional mechanical gyroscopic flight instruments . The main difference between an Inertial measurement unit (IMU) and an AHRS

132-726: A r sin ⁡ θ cos ⁡ φ , y = 1 b r sin ⁡ θ sin ⁡ φ , z = 1 c r cos ⁡ θ , r 2 = a x 2 + b y 2 + c z 2 . {\displaystyle {\begin{aligned}x&={\frac {1}{\sqrt {a}}}r\sin \theta \,\cos \varphi ,\\y&={\frac {1}{\sqrt {b}}}r\sin \theta \,\sin \varphi ,\\z&={\frac {1}{\sqrt {c}}}r\cos \theta ,\\r^{2}&=ax^{2}+by^{2}+cz^{2}.\end{aligned}}} An infinitesimal volume element

198-461: A x 2 + b y 2 + c z 2 = d . {\displaystyle ax^{2}+by^{2}+cz^{2}=d.} The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: radius r , inclination θ , azimuth φ ) can be obtained from its Cartesian coordinates ( x , y , z ) by the formulae x = 1

264-400: A unit sphere , where the radius is set to unity and then can generally be ignored, see graphic.) This (unit sphere) simplification is also useful when dealing with objects such as rotational matrices . Spherical coordinates are also useful in analyzing systems that have some degree of symmetry about a point, including: volume integrals inside a sphere; the potential energy field surrounding

330-400: A body through a flow is considered, in flight dynamics, as continuum current. In the outer layer of the space that surrounds the body viscosity will be negligible. However viscosity effects will have to be considered when analysing the flow in the nearness of the boundary layer . Depending on the compressibility of the flow, different kinds of currents can be considered: If the geometry of

396-482: A concentrated mass or charge; or global weather simulation in a planet's atmosphere. Three dimensional modeling of loudspeaker output patterns can be used to predict their performance. A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. Polar plots help to show that many loudspeakers tend toward omnidirectionality at lower frequencies. An important application of spherical coordinates provides for

462-414: A more cost effective solution than conventional high-grade IMUs that only integrate gyroscopes and rely on a high bias stability of the gyroscopes. In addition to attitude determination an AHRS may also form part of an inertial navigation system . A form of non-linear estimation such as an Extended Kalman filter is typically used to compute the solution from these multiple sources. AHRS is reliable and

528-417: A new approximate parabolic drag equation can be traced leaving the minimum drag value at zero lift value. The Coefficient of pressure varies with Mach number by the relation given below: where This relation is reasonably accurate for 0.3 < M < 0.7 and when M = 1 it becomes ∞ which is impossible physical situation and is called Prandtl–Glauert singularity . see Aerodynamic force Stability

594-421: A point P then are defined as follows: The sign of the azimuth is determined by designating the rotation that is the positive sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system definition. (If the inclination is either zero or 180 degrees (= π radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.) The elevation

660-582: A radius from the z- axis that is not from the point of origin. Particular care must be taken to check the meaning of the symbols . According to the conventions of geographical coordinate systems , positions are measured by latitude, longitude, and height (altitude). There are a number of celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees ; (note 90 degrees equals π /2 radians). And these systems of

726-524: A single point of three-dimensional space. On the reverse view, any single point has infinitely many equivalent spherical coordinates. That is, the user can add or subtract any number of full turns to the angular measures without changing the angles themselves, and therefore without changing the point. It is convenient in many contexts to use negative radial distances, the convention being ( − r , θ , φ ) {\displaystyle (-r,\theta ,\varphi )} , which

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792-462: A wide set of applications—on a planar surface, a two-dimensional spherical coordinate system is useful on the surface of a sphere. For example, one sphere that is described in Cartesian coordinates with the equation x + y + z = c can be described in spherical coordinates by the simple equation r = c . (In this system— shown here in the mathematics convention —the sphere is adapted as

858-457: Is −180° ≤ λ ≤ 180° and a given reading is typically designated "East" or "West". For positions on the Earth or other solid celestial body , the reference plane is usually taken to be the plane perpendicular to the axis of rotation . Instead of the radial distance r geographers commonly use altitude above or below some local reference surface ( vertical datum ), which, for example, may be

924-403: Is a convenient frame to express aircraft translational and rotational kinematics. The Earth frame is also useful in that, under certain assumptions, it can be approximated as inertial. Additionally, one force acting on the aircraft, weight, is fixed in the + z E direction. The body frame is often of interest because the origin and the axes remain fixed relative to the aircraft. This means that

990-434: Is a second typical decomposition taking into account the definition of the drag coefficient equation. This decomposition separates the effect of the lift coefficient in the equation, obtaining two terms C D0 and C Di . C D0 is known as the parasitic drag coefficient and it is the base drag coefficient at zero lift. C Di is known as the induced drag coefficient and it is produced by the body lift. A good attempt for

1056-575: Is also commonly used in 3D game development to rotate the camera around the player's position Instead of inclination, the geographic coordinate system uses elevation angle (or latitude ), in the range (aka domain ) −90° ≤ φ ≤ 90° and rotated north from the equator plane. Latitude (i.e., the angle of latitude) may be either geocentric latitude , measured (rotated) from the Earth's center—and designated variously by ψ , q , φ ′, φ c , φ g —or geodetic latitude , measured (rotated) from

1122-468: Is also possible to get the dependency of the drag coefficient respect to the lift coefficient . This relation is known as the drag coefficient equation: The aerodynamic efficiency has a maximum value, E max , respect to C L where the tangent line from the coordinate origin touches the drag coefficient equation plot. The drag coefficient, C D , can be decomposed in two ways. First typical decomposition separates pressure and friction effects: There

1188-474: Is common in commercial and business aircraft. AHRS is typically integrated with electronic flight instrument systems (EFIS) which are the central part of glass cockpits , to form the primary flight display. AHRS can be combined with air data computers to form an Air data, attitude and heading reference system (ADAHRS), which provide additional information such as airspeed, altitude and outside air temperature. Aircraft attitude Control systems adjust

1254-557: Is equivalent to ( r , θ + 180 ∘ , φ ) {\displaystyle (r,\theta {+}180^{\circ },\varphi )} or ( r , 90 ∘ − θ , φ + 180 ∘ ) {\displaystyle (r,90^{\circ }{-}\theta ,\varphi {+}180^{\circ })} for any r , θ , and φ . Moreover, ( r , − θ , φ ) {\displaystyle (r,-\theta ,\varphi )}

1320-399: Is equivalent to ( r , θ , φ + 180 ∘ ) {\displaystyle (r,\theta ,\varphi {+}180^{\circ })} . When necessary to define a unique set of spherical coordinates for each point, the user must restrict the range, aka interval , of each coordinate. A common choice is: But instead of the interval [0°, 360°) ,

1386-414: Is the ability of the aircraft to counteract disturbances to its flight path. Spherical coordinate system In mathematics , a spherical coordinate system is a coordinate system for three-dimensional space where the position of a given point in space is specified by three real numbers : the radial distance r along the radial line connecting the point to the fixed point of origin ;

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1452-435: Is the addition of an on-board processing system in an AHRS, which provides attitude and heading information. This is in contrast to an IMU, which delivers sensor data to an additional device that computes attitude and heading. With sensor fusion , drift from the gyroscopes integration is compensated for by reference vectors, namely gravity, and the Earth's magnetic field . This results in a drift-free orientation, making an AHRS

1518-428: Is the signed angle from the x-y reference plane to the radial line segment OP , where positive angles are designated as upward, towards the zenith reference. Elevation is 90 degrees (= ⁠ π / 2 ⁠ radians) minus inclination . Thus, if the inclination is 60 degrees (= ⁠ π / 3 ⁠ radians), then the elevation is 30 degrees (= ⁠ π / 6 ⁠ radians). In linear algebra ,

1584-437: Is usual to consider perturbations about a nominal steady flight state. So the analysis would be applied, for example, assuming: The speed, height and trim angle of attack are different for each flight condition, in addition, the aircraft will be configured differently, e.g. at low speed flaps may be deployed and the undercarriage may be down. Except for asymmetric designs (or symmetric designs at significant sideslip),

1650-431: Is usually determined by the context, as occurs in applications of the 'unit sphere', see applications . When the system is used to designate physical three-space, it is customary to assign positive to azimuth angles measured in the counterclockwise sense from the reference direction on the reference plane—as seen from the "zenith" side of the plane. This convention is used in particular for geographical coordinates, where

1716-431: The mathematics convention may measure the azimuthal angle counterclockwise (i.e., from the south direction x -axis, or 180°, towards the east direction y -axis, or +90°)—rather than measure clockwise (i.e., from the north direction x-axis, or 0°, towards the east direction y-axis, or +90°), as done in the horizontal coordinate system . (See graphic re "mathematics convention".) The spherical coordinate system of

1782-467: The World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21 km or 13 miles) and many other details. Planetary coordinate systems use formulations analogous to the geographic coordinate system. A series of astronomical coordinate systems are used to measure the elevation angle from several fundamental planes . These reference planes include:

1848-417: The azimuth reference direction. These choices determine a reference plane that is typically defined as containing the point of origin and the x– and y–axes , either of which may be designated as the azimuth reference direction. The reference plane is perpendicular (orthogonal) to the zenith direction, and typically is desiginated "horizontal" to the zenith direction's "vertical". The spherical coordinates of

1914-444: The inclination angle and use the elevation angle instead, which is measured upward between the reference plane and the radial line—i.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The depression angle is the negative of the elevation angle. (See graphic re the "physics convention"—not "mathematics convention".) Both the use of symbols and the naming order of tuple coordinates differ among

1980-458: The mean sea level . When needed, the radial distance can be computed from the altitude by adding the radius of Earth , which is approximately 6,360 ± 11 km (3,952 ± 7 miles). However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be inaccurate by several kilometers. The precise standard meanings of latitude, longitude and altitude are currently defined by

2046-409: The physics convention can be seen as a generalization of the polar coordinate system in three-dimensional space . It can be further extended to higher-dimensional spaces, and is then referred to as a hyperspherical coordinate system . To define a spherical coordinate system, one must designate an origin point in space, O , and two orthogonal directions: the zenith reference direction and

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2112-438: The polar angle θ between the radial line and a given polar axis ; and the azimuthal angle φ as the angle of rotation of the radial line around the polar axis. (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates ( r , θ , φ ), known as a 3- tuple , provide a coordinate system on a sphere , typically called the spherical polar coordinates . The plane passing through

2178-541: The separation of variables in two partial differential equations —the Laplace and the Helmholtz equations —that arise in many physical problems. The angular portions of the solutions to such equations take the form of spherical harmonics . Another application is ergonomic design , where r is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out. The spherical coordinate system

2244-421: The vector from the origin O to the point P is often called the position vector of P . Several different conventions exist for representing spherical coordinates and prescribing the naming order of their symbols. The 3-tuple number set ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} denotes radial distance, the polar angle—"inclination", or as

2310-469: The "zenith" direction is north and the positive azimuth (longitude) angles are measured eastwards from some prime meridian . Note: Easting ( E ), Northing ( N ) , Upwardness ( U ). In the case of ( U , S , E ) the local azimuth angle would be measured counterclockwise from S to E . Any spherical coordinate triplet (or tuple) ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} specifies

2376-516: The alternative, "elevation"—and the azimuthal angle. It is the common practice within the physics convention, as specified by ISO standard 80000-2:2019 , and earlier in ISO 31-11 (1992). As stated above, this article describes the ISO "physics convention"—unless otherwise noted. However, some authors (including mathematicians) use the symbol ρ (rho) for radius, or radial distance, φ for inclination (or elevation) and θ for azimuth—while others keep

2442-423: The azimuth φ is typically restricted to the half-open interval (−180°, +180°] , or (− π , + π ] radians, which is the standard convention for geographic longitude. For the polar angle θ , the range (interval) for inclination is [0°, 180°] , which is equivalent to elevation range (interval) [−90°, +90°] . In geography, the latitude is the elevation. Even with these restrictions, if

2508-430: The body is fixed and in case of symmetric flight (β=0 and Q=0), pressure and friction coefficients are functions depending on: where: Under these conditions, drag and lift coefficient are functions depending exclusively on the angle of attack of the body and Mach and Reynolds numbers . Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well. It

2574-409: The center of the Earth. The other two reference frames are body-fixed, with origins moving along with the aircraft, typically at the center of gravity. For an aircraft that is symmetric from right-to-left, the frames can be defined as: Asymmetric aircraft have analogous body-fixed frames, but different conventions must be used to choose the precise directions of the x and z axes. The Earth frame

2640-513: The correct quadrant of ( x , y ) , as done in the equations above. See the article on atan2 . Alternatively, the conversion can be considered as two sequential rectangular to polar conversions : the first in the Cartesian xy plane from ( x , y ) to ( R , φ ) , where R is the projection of r onto the xy -plane, and the second in the Cartesian zR -plane from ( z , R ) to ( r , θ ) . The correct quadrants for φ and θ are implied by

2706-431: The correctness of the planar rectangular to polar conversions. These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xy plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ = +90° ). If θ measures elevation from the reference plane instead of inclination from

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2772-573: The formulas r = ρ 2 + z 2 , θ = arctan ⁡ ρ z = arccos ⁡ z ρ 2 + z 2 , φ = φ . {\displaystyle {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}},\\\theta &=\arctan {\frac {\rho }{z}}=\arccos {\frac {z}{\sqrt {\rho ^{2}+z^{2}}}},\\\varphi &=\varphi .\end{aligned}}} Conversely,

2838-612: The further complication of taking the motion of the control surfaces into account. Furthermore, the flight is assumed to take place in still air, and the aircraft is treated as a rigid body . Three forces act on an aircraft in flight: weight , thrust , and the aerodynamic force . The expression to calculate the aerodynamic force is: where: projected on wind axes we obtain: where: In absence of thermal effects, there are three remarkable dimensionless numbers: where: According to λ there are three possible rarefaction grades and their corresponding motions are called: The motion of

2904-452: The induced drag coefficient is to assume a parabolic dependency of the lift Aerodynamic efficiency is now calculated as: If the configuration of the plane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane. In case the configuration is asymmetrical respect to the XY plane, however, minimum drag differs from the parasitic drag. On these cases,

2970-649: The lift generated by the wings when it pitches nose up or down by increasing or decreasing the angle of attack (AOA). The roll angle is also known as bank angle on a fixed-wing aircraft, which usually "banks" to change the horizontal direction of flight. An aircraft is streamlined from nose to tail to reduce drag making it advantageous to keep the sideslip angle near zero, though an aircraft may be deliberately "sideslipped" to increase drag and descent rate during landing, to keep aircraft heading same as runway heading during cross-wind landings and during flight with asymmetric power. Roll, pitch and yaw refer to rotations about

3036-401: The longitudinal equations of motion (involving pitch and lift forces) may be treated independently of the lateral motion (involving roll and yaw). The following considers perturbations about a nominal straight and level flight path. To keep the analysis (relatively) simple, the control surfaces are assumed fixed throughout the motion, this is stick-fixed stability. Stick-free analysis requires

3102-495: The longitudinal plane of symmetry, positive nose up. Three right-handed , Cartesian coordinate systems see frequent use in flight dynamics. The first coordinate system has an origin fixed in the reference frame of the Earth: In many flight dynamics applications, the Earth frame is assumed to be inertial with a flat x E , y E -plane, though the Earth frame can also be considered a spherical coordinate system with origin at

3168-434: The naming order differently as: radial distance, "azimuthal angle", "polar angle", and ( ρ , θ , φ ) {\displaystyle (\rho ,\theta ,\varphi )} or ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} —which switches the uses and meanings of symbols θ and φ . Other conventions may also be used, such as r for

3234-564: The observer's horizon , the galactic equator (defined by the rotation of the Milky Way ), the celestial equator (defined by Earth's rotation), the plane of the ecliptic (defined by Earth's orbit around the Sun ), and the plane of the earth terminator (normal to the instantaneous direction to the Sun ). As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between

3300-504: The observer's local vertical , and typically designated φ . The polar angle (inclination), which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography. The azimuth angle (or longitude ) of a given position on Earth, commonly denoted by λ , is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian ); thus its domain (or range)

3366-409: The orientation of a vehicle about its cg. A control system includes control surfaces which, when deflected, generate a moment (or couple from ailerons) about the cg which rotates the aircraft in pitch, roll, and yaw. For example, a pitching moment comes from a force applied at a distance forward or aft of the cg, causing the aircraft to pitch up or down. A fixed-wing aircraft increases or decreases

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3432-426: The origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ). The radial distance from the fixed point of origin is also called the radius , or radial line , or radial coordinate . The polar angle may be called inclination angle , zenith angle , normal angle , or the colatitude . The user may choose to ignore

3498-403: The polar angle (inclination) is 0° or 180°—elevation is −90° or +90°—then the azimuth angle is arbitrary; and if r is zero, both azimuth and polar angles are arbitrary. To define the coordinates as unique, the user can assert the convention that (in these cases) the arbitrary coordinates are set to zero. To plot any dot from its spherical coordinates ( r , θ , φ ) , where θ is inclination,

3564-436: The relative orientation of the Earth and body frames describes the aircraft attitude. Also, the direction of the force of thrust is generally fixed in the body frame, though some aircraft can vary this direction, for example by thrust vectoring . The wind frame is a convenient frame to express the aerodynamic forces and moments acting on an aircraft. In particular, the net aerodynamic force can be divided into components along

3630-405: The respective axes starting from a defined steady flight equilibrium state. The equilibrium roll angle is known as wings level or zero bank angle. The most common aeronautical convention defines roll as acting about the longitudinal axis, positive with the starboard (right) wing down. Yaw is about the vertical body axis, positive with the nose to starboard. Pitch is about an axis perpendicular to

3696-409: The rotation sequences presented below use the z-y'-x" convention. This convention corresponds to a type of Tait-Bryan angles , which are commonly referred to as Euler angles. This convention is described in detail below for the roll, pitch, and yaw Euler angles that describe the body frame orientation relative to the Earth frame. The other sets of Euler angles are described below by analogy. Based on

3762-427: The rotations and axes conventions above: When performing the rotations described above to obtain the body frame from the Earth frame, there is this analogy between angles: When performing the rotations described earlier to obtain the body frame from the Earth frame, there is this analogy between angles: Between the three reference frames there are hence these analogies: In analyzing the stability of an aircraft, it

3828-436: The several sources and disciplines. This article will use the ISO convention frequently encountered in physics , where the naming tuple gives the order as: radial distance, polar angle, azimuthal angle, or ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} . (See graphic re the "physics convention".) In contrast, the conventions in many mathematics books and texts give

3894-3131: The spherical coordinate system and others. The spherical coordinates of a point in the ISO convention (i.e. for physics: radius r , inclination θ , azimuth φ ) can be obtained from its Cartesian coordinates ( x , y , z ) by the formulae r = x 2 + y 2 + z 2 θ = arccos ⁡ z x 2 + y 2 + z 2 = arccos ⁡ z r = { arctan ⁡ x 2 + y 2 z if  z > 0 π + arctan ⁡ x 2 + y 2 z if  z < 0 + π 2 if  z = 0  and  x 2 + y 2 ≠ 0 undefined if  x = y = z = 0 φ = sgn ⁡ ( y ) arccos ⁡ x x 2 + y 2 = { arctan ⁡ ( y x ) if  x > 0 , arctan ⁡ ( y x ) + π if  x < 0  and  y ≥ 0 , arctan ⁡ ( y x ) − π if  x < 0  and  y < 0 , + π 2 if  x = 0  and  y > 0 , − π 2 if  x = 0  and  y < 0 , undefined if  x = 0  and  y = 0. {\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos {\frac {z}{\sqrt {x^{2}+y^{2}+z^{2}}}}=\arccos {\frac {z}{r}}={\begin{cases}\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z>0\\\pi +\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}&{\text{if }}z<0\\+{\frac {\pi }{2}}&{\text{if }}z=0{\text{ and }}{\sqrt {x^{2}+y^{2}}}\neq 0\\{\text{undefined}}&{\text{if }}x=y=z=0\\\end{cases}}\\\varphi &=\operatorname {sgn}(y)\arccos {\frac {x}{\sqrt {x^{2}+y^{2}}}}={\begin{cases}\arctan({\frac {y}{x}})&{\text{if }}x>0,\\\arctan({\frac {y}{x}})+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\\arctan({\frac {y}{x}})-\pi &{\text{if }}x<0{\text{ and }}y<0,\\+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}\end{aligned}}} The inverse tangent denoted in φ = arctan ⁠ y / x ⁠ must be suitably defined, taking into account

3960-467: The spherical coordinates may be converted into cylindrical coordinates by the formulae ρ = r sin ⁡ θ , φ = φ , z = r cos ⁡ θ . {\displaystyle {\begin{aligned}\rho &=r\sin \theta ,\\\varphi &=\varphi ,\\z&=r\cos \theta .\end{aligned}}} These formulae assume that

4026-391: The two systems have the same origin and same reference plane, measure the azimuth angle φ in the same senses from the same axis, and that the spherical angle θ is inclination from the cylindrical z axis. It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates. Let P be an ellipsoid specified by the level set

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4092-459: The use of r for the radius; all which "provides a logical extension of the usual polar coordinates notation". As to order, some authors list the azimuth before the inclination (or the elevation) angle. Some combinations of these choices result in a left-handed coordinate system. The standard "physics convention" 3-tuple set ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} conflicts with

4158-412: The user would: move r units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle ( φ ) about the origin from the designated azimuth reference direction, (i.e., either the x– or y–axis, see Definition , above); and then rotate from the z-axis by the amount of the θ angle. Just as the two-dimensional Cartesian coordinate system is useful—has

4224-431: The usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates , where θ is often used for the azimuth. Angles are typically measured in degrees (°) or in radians (rad), where 360° = 2 π rad. The use of degrees is most common in geography, astronomy, and engineering, where radians are commonly used in mathematics and theoretical physics. The unit for radial distance

4290-454: The wind frame axes, with the drag force in the − x w direction and the lift force in the − z w direction. In addition to defining the reference frames, the relative orientation of the reference frames can be determined. The relative orientation can be expressed in a variety of forms, including: The various Euler angles relating the three reference frames are important to flight dynamics. Many Euler angle conventions exist, but all of

4356-963: The zenith the arccos above becomes an arcsin, and the cos θ and sin θ below become switched. Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates ( radius r , inclination θ , azimuth φ ), where r ∈ [0, ∞) , θ ∈ [0, π ] , φ ∈ [0, 2 π ) , by x = r sin ⁡ θ cos ⁡ φ , y = r sin ⁡ θ sin ⁡ φ , z = r cos ⁡ θ . {\displaystyle {\begin{aligned}x&=r\sin \theta \,\cos \varphi ,\\y&=r\sin \theta \,\sin \varphi ,\\z&=r\cos \theta .\end{aligned}}} Cylindrical coordinates ( axial radius ρ , azimuth φ , elevation z ) may be converted into spherical coordinates ( central radius r , inclination θ , azimuth φ ), by

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