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Topic: Euler angle

Related Topics

Euler angle

SO(3)

Spherical coordinate system

Rotation matrix

Coordinate rotation

Quaternions and spatial rotation

Coordinate rotations and reflections

Charts on SO(3)

Angle

Yaw, pitch and roll

Gimbal lock

Rotation

Flight dynamics

Quaternion

Gimbal

	Euler angles - Wikipedia, the free encyclopedia
		Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler.
		Euler angles are one of several ways of specifying the relative position of two such coordinate systems.
		Euler angles are used extensively in the classical mechanics of rigid bodies, and in the quantum mechanics of angular momentum.
	en.wikipedia.org /wiki/Euler_angles (1313 words)

	Maths - Euler Angles - Martin Baker
		Euler Angles are one possible way to represent the orientation, or other rotational quantity, associated with a solid 3D object.
		Its not just the names of the angles that changes in different Euler angle conventions, there is also the order that the angles are applied (in mathematical terms rotations are not commutative), and also whether the rotations are left or right handed.
		However there is a third angle, we can rotate about the line to the satellite, to correctly align with the horizontal and vertically polarised signal from the satellite, this third angle is dependant on the others so we cant escape from this issue.
	www.euclideanspace.com /maths/geometry/rotations/euler/index.htm (1763 words)

	STK: Orientation Methods
		Users familiar with Euler angle sequences will recognize this as a 323 rotation sequence where the first rotation is by the azimuth angle, the second rotation is 90 degrees minus the elevation angle, and the third rotation angle is zero.
		The Euler Angles Method uses a 3-rotation sequence about a local axis starting from the parent reference frame and rotating to the sensor or antenna frame where the initial sensor or antenna boresight is along the reference Z-axis as shown in the figure above.
		Unlike in the Euler angles method, the angles Roll, Pitch and Yaw are measured about the reference axes X, Y and Z respectively (not the moving axes).
	www.stk.com /resources/help/stk613/helpSystem/stk/sn-orientation.htm (1464 words)

	GPlatesGMLDev.RotationTerminology (Site not responding. Last check: 2007-09-10)
		Euler Poles remain fixed for long periods of time; however, it is quite common for an Euler Pole to jump to a new position ([1], p.221).
		Note that the angle of a Finite Rotation is not called an "Euler Angle": Euler Angles ([2]; [3]) are a distinct mathematical construction, a method of describing an arbitrary rotation using three angles of rotation around three pre-defined, fixed coordinate axes.
		Euler's Rotation Theorem is also the basis of the Euler Angles construction; again, there is obviously a connection between an Euler Angle Sequence and a Finite Rotation, since the first two rotations of an Euler Angle Sequence may be used to position the Finite Rotation axis.
	www.geosci.usyd.edu.au /pmwiki/pmwiki.php?n=GPlatesGMLDev.RotationTerminology?n=GPlatesGMLDev.RotationTerminology (1559 words)

	GameDev.net - Quaternion Powers
		As rotations in the Euler representation are done with respect to the global axis, a rotation in one axis could 'override' a rotation in another, making you lose a degree of freedom.
		Since you can convert the Euler angles to three independent quaternions by setting the arbitrary axis to the coordinate axes, you can then multiply the three quaternions together to obtain the final quaternion.
		Note the Euler angles will be incorrect if you rotate more than 1 axis (because it counts the keypress rather than getting the Euler angles from the camera quaternion).
	www.gamedev.net /reference/articles/article1095.asp (2765 words)

	Angle -- from Wolfram MathWorld
		Half a full rotation is called a straight angle, and a quarter of a full rotation is called a right angle.
		straight angle) is called an obtuse angle, and an angle greater than a straight angle (but less than a full angle) is called a reflex angle.
		The use of degrees to measure angles harks back to the Babylonians, whose sexagesimal number system was based on the number 60.
	mathworld.wolfram.com /Angle.html (343 words)

	Euler angles from matrix - CGAFaq
		As discussed in the article on Euler angles, a triple of angles can represent a 3D rotation as a composition of coordinate axis-aligned rotations in 24 different ways.
		One caution: Extraction differs critically from generation in that, while a triple of angles generates a unique rotation matrix for a given system, extraction of angles suffers from three kinds of ambiguity no matter which system is used.
		This is a rotation by the negative of angle atan2(s,c), as required to undo it.
	cgafaq.info /wiki/Euler_angles_from_matrix (1013 words)

	Gamasutra - Features - "Rotating Objects Using Quaternions" [07.03.98]
		Euler angle representation is very efficient because it uses only three variables to represent three DOF.
		Euler angles also don't have to obey any constraints, so they're not prone to drifting and don't have to be readjusted.
		Euler angles also introduce the problem of "Gimbal lock" or a loss of one degree of rotational freedom.
	www.gamasutra.com /features/19980703/quaternions_01.htm (3465 words)

	Maths - Conversion Axis Angle to Euler - Martin Baker
		Euler angles represent 3 rotations about the x,y and z axis in some given order.
		As explained in euler section there are different types of Euler angles and the result will depend on the Euler definition used.
		Euler can be defined in terms of a quaternion as shown here.
	www.euclideanspace.com /maths/geometry/rotations/conversions/angleToEuler (1191 words)

	Euler angles (Site not responding. Last check: 2007-09-10)
		Euler angles one way of representing rotations in 3-dimensional Euclidean space as a product of three successive 2D coordinate rotations θ, φ, ψ; about the x-, y- and z-axes.
		Most often, the rotations are carried out first around Z axis, then around the X axis and finally around the Z axis again, but other definitions are possible.
		The Euler angles form a chart on SO(3), the mathematical group of rotations in 3D space.
	bopedia.com /en/wikipedia/e/eu/euler_angles.html (176 words)

	Euler's formula - Wikipedia, the free encyclopedia
		Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function.
		Euler's formula was proven (in an obscured form) for the first time by Roger Cotes in 1714, then rediscovered and popularized by Euler in 1748.
		Euler and his beautiful and extraordinary formula by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
	en.wikipedia.org /wiki/Euler's_formula (846 words)

	[No title]
		The second kind of singularity occurs when any of the Euler angles corresponding to a matrix is at one of the endpoints of its range, for example, when the first angle has the value pi.
		Euler angles near the limits of their ranges should be regarded with suspicion.
		The Euler angle routines could be used for conversion between `RA, Dec, and Twist' (as defined by the Voyager project) and a `C-matrix'.
	www-ssc.igpp.ucla.edu /~leel/spice/html/rotations.html (3609 words)

	JSBSim Flight Dynamics Model: JSBSim::FGQuaternion Class Reference
		Transformations and euler angles are therefore computed once they are requested for the first time.
		Initialize the quaternion with the single euler angle where its index is given in the first argument.
		the sine of the Euler angle theta (pitch attitude) corresponding to this quaternion rotation.
	jsbsim.sourceforge.net /JSBSim/classJSBSim_1_1FGQuaternion.html (763 words)

	Multi-Dimensional Rotations, Including Boosts
		The plane containing the two vectors is the plane of rotation, and the angle between the two vectors tells us something about the angle of rotation; specifically, it tells us the half-angle, as will be discussed in section 2.1.
		That is, given two vectors in the plane of rotation, we define the rotor angle to be the angle between the two vectors.
		Because the Euler angles depend on a particular choice of basis, they represent rotations in a way that is not rotation-invariant...
	www.av8n.com /physics/rotations.htm (9082 words)

	Quaternions to Euler Angles (Aerospace Blockset) (Site not responding. Last check: 2007-09-10)
		The conversion is generated by comparing elements in the direction cosine matrix (DCM), as functions of the Euler rotation angles, with elements in the DCM, as functions of a unit quaternion vector.
		The output is a 3-by-1 vector of Euler angles.
		for an example of the use of the Quaternions to Euler Angles block in an implementation of the equations of motion of a rigid body.
	www.weizmann.ac.il /matlab/toolbox/aeroblks/quaternionstoeulerangles.html (123 words)

	Euler Angles (Site not responding. Last check: 2007-09-10)
		The subject is particularly tricky because the effect of the three Euler rotations depends upon which order they are applied and about which axes the rotations are performed.
		Animations 2 show how the orientation of the object changes if the Euler angles are varied, one at a time.
		Euler angle Mathematica notebook which is to be used with
	www.mhl.soton.ac.uk /research/help/Euler/index.html (407 words)

	Statistical Calculations of 3D-Orientation Parameters of Flat Symmetrical Polyhedrons -- from Mathematica Information ...
		The orientation of 3-dimensional objects can be described with the three Euler angles phi (horizontal rotation), Theta (vertical rotation) and psi (rotation in the plane of the object that originally was horizontal).
		To choose which facet has before the Euler rotations been horizontal one has six alternatives and to choose which side of the facet has originally been parallel to the s-axis one has still four alternatives.
		In this paper an algorithm for calculating the mean and standard deviation of the horizontal and vertical Euler angle of a set of flat polyhderons, whose main facet is symmetric with respect to a line, is presented in detail.
	library.wolfram.com /infocenter/Articles/1868 (418 words)

	CS 15-497/15-861 Computer Animation
		The human model (skeleton) is represented using a hierarchy, composed of a root node and several children nodes, corresponding to the various joints.
		We read the joint angles (for the various joints, in euler angles) and translation (of the root node) of the hierarchy from these files and store them in a data structure.
		We represent the orientations as Euler angles, you will also be required to represent the joint angles using quaternions and perform the same interpolations in quaternion space.
	www.cs.cmu.edu /afs/cs.cmu.edu/user/kiranb/www/animation/asst1.html (701 words)

	6DoF (Euler Angles) :: Blocks (Aerospace Blockset)
		The origin of the body-fixed coordinate frame is the center of gravity of the body, and the body is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass.
		The third output is a three-element vector containing the Euler rotation angles [roll, pitch, yaw], in radians.
		The fourth output is a 3-by-3 matrix for the coordinate transformation from Earth-fixed axes to body-fixed axes.
	www.mathworks.com /access/helpdesk/help/toolbox/aeroblks/6dofeulerangles.html (483 words)

	Euler Angles -- from Wolfram MathWorld
		According to Euler's rotation theorem, any rotation may be described using three angles.
		The three angles giving the three rotation matrices are called Euler angles.
		There are several conventions for Euler angles, depending on the axes about which the rotations are carried out.
	mathworld.wolfram.com /EulerAngles.html (431 words)

	[No title]
		In addition, the Euler angle parameterization has singularities at certain angles which limits the generality of their usage.
		Euler's theorem states "the general displacement of a rigid body with one point fixed is a rotation about some axis".
		For small angles, the cosine terms in equation 3 can be replaced with a value of one.
	www.resonancepub.com /quaterni.htm (1112 words)

	Insight
		For assigning Euler angles to subsequentprojection images we search for symmetry-related peaks in the SSCF of thenew projections and in the Cross Sinogram Correlation Functions ("CSCF") ofthe oldprojection images (which already have been assigned Euler angleorientations) simultaneously.
		For our first 3D reconstruction we assigned Euler angles to the 30 bestclass averages taken out of a total of 200 characteristic views which hadresulted from the first rounds of alignments and classification.
		The assignment of Euler angles by using only the CSCFs of agiven class average with respect to the, say 30, reprojected images of theanchor set is thus more sensitive and precise than the Euler-angleassignments of the first round of processing.
	ncmi.bcm.tmc.edu /~irina/papers/Nature1-.html (3672 words)

	EulerAngle class Reference
		Partial derivative of a quaternion with respect to the y-axis rotation of a Euler angle.
		Partial derivative of a quaternion with respect to the x-axis rotation of a Euler angle.
		Partial derivative of a quaternion with respect to the z-axis rotation of a Euler angle.
	ligwww.epfl.ch /~lorna/doxygen/FittingGraph/classEulerAngle.html (498 words)

	Euler Angles (Site not responding. Last check: 2007-09-10)
		Another account detailing some of the "pleasures" of dealing with Euler angles is provided by David Drascic, including some code snippets from Graphics Gems.
		This is described in the topic Determining Euler Angles.
		Using the Euler angles, this three-dimensional problem can be dissected into a sequence of two-dimensional rotations, whereby in each rotation one axis remains invariant.
	casgm3.anorg.chemie.uni-tuebingen.de /klaus/nmr/conventions/euler/euler.html (1004 words)

	Quaternion - CGAFaq
		Popular non-quaternion options are 3×3 special orthogonal matrices (9 numbers with constraints), Euler angles (3 numbers), axis-angle (4 numbers), and angular velocity vectors (3 numbers).
		The distinction is important because, for example, it is common to use an axis-angle with an angle greater than 360° to tell an animation system to spin an object more than a full turn, something a matrix cannot say.
		Comparisons to Euler angles may mention gimbal lock (frequently misspelled) as a disadvantage quaternions avoid.
	cgafaq.info /wiki/Quaternion (1227 words)

	CSPICE Routines: EUL2XF_C
		This routine computes a state transformation from an Euler angle factorization of a rotation and the derivatives of those Euler angles.
		This routine is related to the routine eul2xf_c which produces a state transformation from an input set of axes, Euler angles and derivatives.
		Suppose you have a set of Euler angles and their derivatives for a 3 1 3 rotation, and that you would like to determine the equivalent angles and derivatives for a 1 2 3 rotation.
	www.gps.caltech.edu /~marsdata/cspice/eul2xf_c.html (728 words)

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