query: the earth axis of rotation is always inclined at an angle of

pos: Axial tilt Axial tilt In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, or, equivalently, the angle between its equatorial plane and orbital plane. It differs from orbital inclination. At an obliquity of 0 degrees, the two axes point in the same direction; i.e., the rotational axis is perpendicular to the orbital plane. Earth's obliquity oscillates between 22.1 and 24.5 degrees on a 41,000-year cycle; Earth's mean obliquity is currently and decreasing. Over the course of an orbital period, the obliquity usually does not change considerably, and the orientation of the

neg: Theta Aurigae Venaticorum type variable star and has a surface magnetic field of about 1 kG. Its projected rotational velocity is , with the star completing a rotation in only 3.6 days. The axis of rotation is inclined by an angle of to the line of sight from the Earth. The secondary, Theta Aurigae B, is a +7.2 magnitude companion, 4.5 magnitudes fainter than the primary, located at an angular separation of 3.91 arcseconds along a position angle of 304.9° as of 2002. This is an F-type main sequence star with a stellar classification in the range F2-5 V. The mean combined

neg: Alpha Ophiuchi is calculated to be and the equatorial temperature . The energy radiated by an oblate star is higher along its axis of rotation because of the larger projected area and the Stefan–Boltzmann law. The axis of rotation of α Ophiuchi is inclined about to the line of sight from the Earth, so that it is being observed from nearly equator-on. The apparent bolometric luminosity seen at this angle is , but the true luminosity of the star is thought to be . Since a star's effective temperature is simply the temperature which would produce its total energy output from a

neg: Rotation is no real eigenvalue, meaning that no real vector in the plane is kept unchanged by A. Knowing that the trace is an invariant, the rotation angle formula_5 for a proper orthogonal 3x3 rotation matrix formula_6 is found by formula_7 Using the principal arc-cosine, this formula gives a rotation angle satisfying formula_8. The corresponding rotation axis must be defined to point in a direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis formula_9 can always be written as a rotation having formula_8

neg: Lambda Virginis consisting of this system, Kappa Virginis, Iota Virginis and Phi Virginis. Consequently, Lambda Virginis itself is known as (, .) Lambda Virginis is a double-lined spectroscopic binary with an orbital period of 206.7 days and an eccentricity of 0.0610. The semi-major axis has an angular size of 0.02 arcseconds, which, at the distance of this system, is equivalent to a physical span of AU. The orbit is inclined by an angle of 110° to the line of sight from the Earth. Tidal theory predicts that eventually the orbit of the stars will circularize and their rotation rates will become synchronized

neg: Line moiré superposition of two layers with identically inclined lines forms moiré lines inclined at the same angle. Figure 5 is obtained from Figure 1 with a vertical shearing. In Figure 5 the layer lines and the moiré lines are inclined by 10 degrees. Since the inclination is not a rotation, during the inclination the distance ("p") between the layer lines along the vertical axis is conserved, but the true distance ("T") between the lines (along an axis perpendicular to these lines) is changed. The difference between the vertical periods "p", "p", and the distances "T", "T" is shown in the diagram

neg: Latitude constant latitude and constant longitude, which are constructed with reference to the rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the meridians; and the angle between any one meridian plane and that through Greenwich (the Prime Meridian) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. Planes parallel to the

neg: Foucault pendulum vector diagrams time increases because the central axis of the pendulum is aligned with the axis of rotation of the Earth at the North Pole and then the angle of misalignment increases as the latitude decreases to the point of perpendicularity at the Equator. The angular velocity in relation to the rotation of the Earth's axis that is imparted to the pendulum bob decreases with the cosine of the degree of misalignment of the central axis of the pendulum in comparison to the axis of rotation of the Earth. There are zero degrees of misalignment at the North Pole and the cosine

neg: Omega Carinae break up, is . The star's axis of rotation is inclined by an estimated angle of 70.8° to the line of sight from the Earth. In the next 7500 years, the south Celestial pole will pass close to this stars and I Carinae (5800 CE). In Chinese, (), meaning "Southern Boat", refers to an asterism consisting of ω Carinae, V337 Carinae, PP Carinae, θ Carinae and β Carinae . Consequently, ω Carinae itself is known as (, .) Omega Carinae Omega Carinae, Latinized from ω Carinae, is a star in the constellation Carina. With a declination greater than 70 degrees

neg: Axis–angle representation with its scalar and vector , the axis–angle coordinates can be extracted using the following: A more numerically stable expression of the rotation angle uses the atan2 function: where is the Euclidean norm of the 3-vector . Axis–angle representation In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector rooted at the origin

neg: Rotation matrix main diagonal of a matrix is called the trace; it does not change if we reorient the coordinate system, and always equals the sum of the eigenvalues. This has the convenient implication for and rotation matrices that the trace reveals the angle of rotation, , in the two-dimensional space (or subspace). For a matrix the trace is , and for a matrix it is . In the three-dimensional case, the subspace consists of all vectors perpendicular to the rotation axis (the invariant direction, with eigenvalue 1). Thus we can extract from any rotation matrix a rotation axis and an angle,

neg: Rotation matrix axis of rotation. In 3-space (), there will be an axis of rotation (a 1D manifold, or a line through the origin) or the rotation will be null. In 4-space (), there may be no axes of rotation, or there may be a two-dimensional axis, a plane through the origin, called the "axis plane". As always, when all eigenvalues are unity, the rotation is a null rotation. The trace of a rotation matrix will be equal to the sum of its eigenvalues. For the two eigenvalues are and the trace will be where is the rotation angle about the origin.

neg: Axis–angle representation Axis–angle representation In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector rooted at the origin because the magnitude of is constrained. For example, the elevation and azimuth angles of suffice to locate it in any particular Cartesian coordinate frame. The angle scalar multiplied by the unit vector is the axis-angle vector The

neg: Foucault pendulum the Scottish mathematician Edward Sang contrived and explained the precession of a spinning top. In 1851, Charles Wheatstone Similarly, consider a nonspinning, perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle formula_2 with the disk. When the disk undergoes a full clockwise revolution, the bicycle wheel will not return to its original position, but will have undergone a net rotation of formula_5. Foucault-like precession is observed in a virtual system wherein a massless particle is constrained to remain on a rotating plane that is inclined with respect to the axis of rotation.

neg: Govind Swarup 530 m long and 30 m wide parabolic-cylindrical radio telescope of a unique and innovative design at Ooty in South India, which was placed on a suitably inclined hill so as to make its long axis of rotation parallel to that of the earth, enabling it to track celestial radio sources in hour angle for 9.5 hrs. Using the method of lunar occultation, it provided for the first time high-resolution angular data (1 to 10 arc sec) for more than 1,000 weak radio sources, which provided an independent evidence for the Big Bang model. Ooty Occultation observations of the galactic

neg: Euler angles to its intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top can wobble up and down; the inclination angle is the nutation angle. The same example can be seen with the movements of the earth. Though all three movements can be represented by a rotation operator with constant coefficients in some frame, they cannot be represented by these operators all at the same time. Given a reference frame, at most one of them will be coefficient-free. Only precession can be expressed in general

neg: Axis–angle representation orthogonal axes in a Cartesian representation into Mercer's theorem is a convenient construction of the Cartesian representation of the Rotation Matrix in three dimensions. Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a Euclidean vector, given a rotation axis and an angle of rotation. In other words, Rodrigues' formula provides an algorithm to compute the exponential map from to without computing the full matrix exponential. If is a vector in and is a unit vector rooted at the origin describing an axis of rotation about which is rotated by an angle , Rodrigues' rotation formula

neg: External ballistics spin-stabilized projectile will experience a spin-induced sideways component, due to a gyroscopic phenomenon known as "yaw of repose." For a right hand (clockwise) direction of rotation this component will always be to the right. For a left hand (counterclockwise) direction of rotation this component will always be to the left. This is because the projectile's longitudinal axis (its axis of rotation) and the direction of the velocity vector of the center of gravity (CG) deviate by a small angle, which is said to be the equilibrium yaw or the yaw of repose. The magnitude of the yaw of repose angle

neg: 3D rotation group of rotation together with an angle of rotation about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be clockwise or counterclockwise with respect to this orientation). For example, counterclockwise rotation about the positive "z"-axis by angle "φ" is given by Given a unit vector n in R and an angle "φ", let "R"("φ", n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then Using these properties one can show that any rotation can be represented by a unique angle "φ" in the range

neg: Solstice frame of reference, planet Earth is seen to rotate about an axis and revolve around the Sun in an elliptical path with the Sun at one focus. Earth's axis is tilted with respect to the plane of Earth's orbit and this axis maintains a position that changes little with respect to the background of stars. An observer on Earth therefore sees a solar path that is the result of both rotation and revolution. The component of the Sun's motion seen by an earthbound observer caused by the revolution of the tilted axis – which, keeping the same angle in space,

neg: Axial precession stars, centered on the ecliptic north pole (the blue "E") and with an angular radius of about 23.4°, an angle known as the obliquity of the ecliptic. The direction of precession is opposite to the daily rotation of the Earth on its axis. The orange axis was the Earth's rotation axis 5,000 years ago, when it pointed to the star Thuban. The yellow axis, pointing to Polaris, marks the axis now. The equinoxes occur where the celestial equator intersects the ecliptic (red line), that is, where the Earth's axis is perpendicular to the line connecting the centers of the Sun

neg: Rotation matrix to be orthogonal, not a rotation matrix. This is unavoidable; an with negative determinant has no uniquely defined closest rotation matrix. To efficiently construct a rotation matrix from an angle and a unit axis , we can take advantage of symmetry and skew-symmetry within the entries. If , , and are the components of the unit vector representing the axis, and then Determining an axis and angle, like determining a quaternion, is only possible up to the sign; that is, and correspond to the same rotation matrix, just like and . Additionally, axis–angle extraction presents additional difficulties. The angle can

neg: Angular displacement matrix exponential series for For any skew-symmetric matrix , exp() is always a rotation matrix. An important practical example is the case. In rotation group SO(3), it is shown that one can identify every with an Euler vector , where is a unit magnitude vector. By the properties of the identification , is in the null space of . Thus, is left invariant by and is hence a rotation axis. Using Rodrigues' rotation formula on matrix form with , together with standard double angle formulae one obtains, where , . This is the matrix for a rotation around axis by

neg: Plane of rotation can be described by giving this axis, with the angle through which the rotation turns about it; this is the axis angle representation of a rotation. The plane of rotation is the plane orthogonal to this axis, so the axis is a surface normal of the plane. The rotation then rotates this plane through the same angle as it rotates around the axis, that is everything in the plane rotates by the same angle about the origin. One example is shown in the diagram, where the rotation takes place about the -axis. The plane of rotation is the -plane, so

neg: Allene two terminal carbon atoms are sp-hybridized. The bond angle formed by the three carbon atoms is 180°, indicating linear geometry for the carbon atoms of allene. It can also be viewed as an "extended tetrahedral" with a similar shape to methane. The symmetry and isomerism of allenes has long fascinated organic chemists. For allenes with four identical substituents, there exist two twofold axes of rotation through the center carbon, inclined at 45° to the CH planes at either end of the molecule. The molecule can thus be thought of as a two-bladed propeller. A third twofold axis of rotation passes

neg: Axis–angle representation axis. Viewing the axis-angle representation as an ordered pair, this would be The above example can be represented as a rotation vector with a magnitude of pointing in the direction, The axis–angle representation is convenient when dealing with rigid body dynamics. It is useful to both characterize rotations, and also for converting between different representations of rigid body motion, such as homogeneous transformations and twists. When a rigid body rotates around a fixed axis, its axis–angle data are a constant rotation axis and the rotation angle continuously dependent on time. Plugging the three eigenvalues 1 and and their associated three

neg: Axis–angle representation vector. In this case, both the rotation axis and the angle are represented by a vector codirectional with the rotation axis whose length is the rotation angle , It is used for the exponential and logarithm maps involving this representation. Note that many rotation vectors correspond to the same rotation. In particular, a rotation vector of length , for any integer , encodes exactly the same rotation as a rotation vector of length . Thus, there are at least a countable infinity of rotation vectors corresponding to any rotation. Furthermore, all rotations by are the same as no rotation at

neg: Rotation matrix to an eigenvalue of 1. To find the angle of a rotation, once the axis of the rotation is known, select a vector perpendicular to the axis. Then the angle of the rotation is the angle between and . A more direct method, however, is to simply calculate the trace, i.e., the sum of the diagonal elements of the rotation matrix. Care should be taken to select the right sign for the angle to match the chosen axis: from which follows that the angle's absolute value is For some applications, it is helpful to be able to make a rotation

neg: Effect of Sun angle on climate Effect of Sun angle on climate The amount of heat energy received at any location on the globe is a direct effect of Sun angle on climate, as the angle at which sunlight strikes the Earth varies by location, time of day, and season due to the Earth's orbit around the Sun and the Earth's rotation around its tilted axis. Seasonal change in the angle of sunlight, caused by the tilt of the Earth's axis, is the basic mechanism that results in warmer weather in summer than in winter. Change in day length is another factor. (See also season.) When

neg: Phi Phoenicis revealed that the spot configurations vary on monthly or yearly time scales. The spectral lines of the irregulary distributed elements show variations that allowed a precise rotation period of 9.53 days to be determined, and also show evidence of long term abundance changes. The analysis of the spots suggests that the rotation axis is inclined to the line of sight by an angle of about 53°, and shows evidence of very weak differential rotation. The starspots probably cause milimagnitude variations in the brightness of Phi Phoenicis, even though there are no precise observations to confirm this. The origin of the

neg: Gyrocompass NS-axis by formula_3. We establish coordinates attached to this observer as so that the unit formula_5 versor formula_6 is mapped to the point formula_7. For the 2-O neither the Earth nor the barycenter of the gyroscope is moving. The rotation of 2-O relative to 1-O is performed with angular velocity formula_8. We suppose that the formula_9 axis denotes points with zero longitude (the prime, or Greenwich, meridian). We now rotate about the formula_10 axis, so that the formula_11-axis has the longitude of the barycenter. In this case we have With the next rotation (about the axis formula_13 of an angle

neg: Foucault pendulum to describe the rotation angle of the swing plane of Foucault's pendulum. From the perspective of an Earth-bound coordinate system with its "x"-axis pointing east and its "y"-axis pointing north, the precession of the pendulum is described by the Coriolis force. Consider a planar pendulum with natural frequency "ω" in the small angle approximation. There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and the Coriolis force. The Coriolis force at latitude "φ" is horizontal in the small angle approximation and is given by where Ω is the rotational frequency of

neg: Rotation Rotation A rotation is a circular movement of an object around a center (or point) of rotation. A three-dimensional object can always be rotated around an infinite number of imaginary lines called "rotation axes" ( ). If the axis passes through the body's center of mass, the body is said to rotate upon itself, or spin. A rotation about an external point, e.g. the Earth about the Sun, is called a revolution or orbital revolution, typically when it is produced by gravity. The axis is called a pole. Mathematically, a rotation is a rigid body movement which, unlike a translation,

neg: Axis–angle representation vector itself does not perform rotations, but is used to construct transformations on vectors that correspond to rotations. The rotation occurs in the sense prescribed by the right-hand rule. The rotation axis is sometimes called the Euler axis. It is one of many rotation formalisms in three dimensions. The axis–angle representation is predicated on Euler's rotation theorem, which dictates that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis. The axis–angle representation is equivalent to the more concise rotation vector, also called the Euler

neg: Rotation matrix is not commutative) only if they are applied in the specified order (see Ambiguities for more details). Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis (Euler rotation theorem). There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation). Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.

neg: Rotation matrix For the three eigenvalues are 1 and where is the rotation angle about the axis line. The trace will be . For , the four eigenvalues are of the form and and the trace will be . If one of the angles, say , is equal to zero, then the rotation will be a simple rotation, with two unit eigenvalues and the other angle () will be the angle of rotation about the axis plane spanned by the two eigenvectors with eigenvalues of 1. Otherwise, there will be no axis plane of rotation. If (an "isoclinic" rotation), the eigenvalues will

neg: Foucault pendulum vector diagrams be interpreted at different latitudes. In order to observe the rotation of the Earth in relation to the plane of the pendulum swing there must be a basic difference in the two types of motion that are being compared. This basic difference is then manifested as (1) being able to 'observe' the change in position of the Earth in relation to the pendulum swing and (2) the time to observe a complete 'relative rotation' decreases with the sine of the latitude (decreases with an increase in the angle of alignment with the Earth's axis of rotation). At the North Pole:

neg: Euler's rotation theorem If is an eigenvector of with eigenvalue 1, then is also an eigenvector of , also with eigenvalue 1. Unless , and are different. Suppose we specify an axis of rotation by a unit vector , and suppose we have an infinitely small rotation of angle about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix is represented as: A finite rotation through angle about this axis may be seen as a succession of small rotations about the same axis. Approximating as where is a large number, a rotation of

neg: Orthogonal matrix a rotation matrix, and the product of two reflection matrices is also a rotation matrix. Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for matrices and larger the non-rotational matrices can be more complicated than reflections. For example, represent an "inversion" through the origin and a "rotoinversion", respectively, about the -axis. Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a rotation matrix in terms of an axis and

neg: X-ray crystallography a camera focused on the crystal. The most common type of goniometer is the "kappa goniometer", which offers three angles of rotation: the ω angle, which rotates about an axis perpendicular to the beam; the κ angle, about an axis at ~50° to the ω axis; and, finally, the φ angle about the loop/capillary axis. When the κ angle is zero, the ω and φ axes are aligned. The κ rotation allows for convenient mounting of the crystal, since the arm in which the crystal is mounted may be swung out towards the crystallographer. The oscillations carried out during data

neg: Dove prism Refraction at the entrance and exit surfaces results in substantial image astigmatism when used in convergent light. Thus the Dove prism is used almost exclusively for images appearing at infinity. If the flat hypoteneuse surface of a Dove prism is cut into a roof shape, the result is an Amici roof prism. A reflection in one axis of a plane is always equivalent to a reflection and rotation in two other axes (with the angle between the source and image being twice the angle between the source and reflection axis of the prism). Thus, Dove prisms can be used to

neg: Rotation formalisms in three dimensions theorem we know that any rotation can be expressed as a single rotation about some axis. The axis is the unit vector (unique except for sign) which remains unchanged by the rotation. The magnitude of the angle is also unique, with its sign being determined by the sign of the rotation axis. The axis can be represented as a three-dimensional unit vector and the angle by a scalar . Since the axis is normalized, it has only two degrees of freedom. The angle adds the third degree of freedom to this rotation representation. One may wish to express rotation as

neg: Rotation if the axis is replaced with formula_11.) Every proper rotation formula_6 in 3D space has an axis of rotation, which is defined such that any vector formula_13 that is aligned with the rotation axis will not be affected by rotation. Accordingly, formula_14, and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle formula_5 is nonzero (i.e., the rotation is not the identity tensor), there is one and only one such direction. Because A has only real components, there is at least one real eigenvalue, and

neg: Orbital effects on climate average, every 50 to 100 million years, where as geomagnetic storms occur periodically through the sun's eleven-year activity cycle. With all of these factors affecting climate in their own way, it becomes easy to see that Earth's climate is in fact, largely dependent on various solar effects/circumstances. Perhaps one of the most apparent factors contributing to Earth climate change is the angle at which the earth is tilted. This is the angle at which Earth's axis of rotation is from the vertical, also known as Earth's obliquity. Earth's current tilt angle is approximately 23.5 degrees. The axial tilt angle affects

neg: Quaternions and spatial rotation be described by a rotation by some angle about some axis; for our purposes, we will use an axis "vector" to establish handedness for our angle. Consider the special case in which the axis of rotation lies in the "xy" plane. We can then specify the axis of one of these rotations by a point on a circle through which the vector crosses, and we can select the radius of the circle to denote the angle of rotation. Similarly, a rotation whose axis of rotation lies in the "xy" plane can be described as a point on a sphere of

neg: Sidereal time 23 h 56 min 4.1 s (86,164.1 s). Earth's rotation is not a simple rotation around an axis that would always remain parallel to itself. Earth's rotational axis itself rotates about a second axis, orthogonal to Earth's orbit, taking about 25,800 years to perform a complete rotation. This phenomenon is called the precession of the equinoxes. Because of this precession, the stars appear to move around Earth in a manner more complicated than a simple constant rotation. For this reason, to simplify the description of Earth's orientation in astronomy and geodesy, it was conventional to chart the positions of the

neg: Rotational symmetry about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry (i.e. rotational symmetry with respect to a central axis) like a doughnut (torus). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties). In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding

neg: Season Egypt. The seasons result from the Earth's axis of rotation being tilted with respect to its orbital plane by an angle of approximately 23.4 degrees. (This tilt is also known as "obliquity of the ecliptic".) Regardless of the time of year, the northern and southern hemispheres always experience opposite seasons. This is because during summer or winter, one part of the planet is more directly exposed to the rays of the Sun (see "Fig. 1") than the other, and this exposure alternates as the Earth revolves in its orbit. For approximately half of the year (from around March 20 to

neg: Orientation (geometry) matrices were developed. Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads to it from the reference frame. When used to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector. A similar method, called axis–angle representation, describes a rotation or orientation using a unit vector aligned with the rotation axis, and a separate value to indicate the angle (see

neg: Foucault pendulum vector diagrams the force to complete a full circle since it takes one day for the Earth to rotate. At the equator: The central axis of the pendulum is perpendicular with the axis of rotation of the Earth. The central axis of the pendulum is always determined by the force of gravity directed towards the center of the Earth. A pendulum bob at rest at the Equator is still rotating with the Earth and there is no spin on the bob. If the bob is displaced from the central axis of the pendulum in preparation for swinging and held in place, then

neg: Axis–angle representation derivation, in contrast to the geometric one in the article Rodrigues' rotation formula. Due to the existence of the above-mentioned exponential map, the unit vector representing the rotation axis, and the angle are sometimes called the "exponential coordinates" of the rotation matrix . Let continue to denote the 3 × 3 matrix that effects the cross product with the rotation axis : for all vectors in what follows. To retrieve the axis–angle representation of a rotation matrix, calculate the angle of rotation from the trace of the rotation matrix and then use that to find the normalized axis, Note also

neg: Three-dimensional rotation operator formula_30 there is only the eigenvalue formula_25 with the one-dimensional eigenspace of the vectors on the rotation axis: The rotation matrix by angle formula_11 around a general axis of rotation formula_34 is given by Rodrigues' rotation formula. where formula_36 is the identity matrix and formula_37 is the dual 2-form of formula_38 or cross product matrix, Note that formula_40 satisfies formula_41 for all formula_42. The operator "Rotation with the angle formula_11 around a specified axis" discussed above is an orthogonal mapping and its matrix relative to any base vector system is therefore an orthogonal matrix . Furthermore its determinant has the

neg: Conversion between quaternions and Euler angles an axis by the following expression where α is a simple rotation angle (the value in radians of the angle of rotation) and cos(β), cos(β) and cos(β) are the "direction cosines" locating the axis of rotation (Euler's Rotation Theorem). Similarly for Euler angles, we use the Tait Bryan angles (in terms of flight dynamics): where the X-axis points forward, Y-axis to the right and Z-axis downward with angles defined for clockwise/lefthand rotation. In the conversion example above the rotation occurs in the order heading, attitude, bank (about intrinsic axes). The orthogonal matrix (post-multiplying a column vector) corresponding to a clockwise/left-handed

neg: Rodrigues' rotation formula vector describing an axis of rotation about which rotates by an angle according to the right hand rule, the Rodrigues formula for the rotated vector is An alternative statement is to write the axis vector as a cross product of any two nonzero vectors and which define the plane of rotation, and the sense of the angle is measured away from and towards . Letting denote the angle between these vectors, the two angles and are not necessarily equal, but they are measured in the same sense. Then the unit axis vector can be written This form may be more

neg: Rotation matrix exponential map, which is defined using the standard matrix exponential series for For any skew-symmetric matrix , is always a rotation matrix. An important practical example is the case. In rotation group SO(3), it is shown that one can identify every with an Euler vector , where is a unit magnitude vector. By the properties of the identification , is in the null space of . Thus, is left invariant by and is hence a rotation axis. Using Rodrigues' rotation formula on matrix form with , together with standard double angle formulae one obtains, where , . This is the

neg: Extraterrestrial skies despite the fact that Triton is more than 4 times as intrinsically bright as our moon and orbits much closer to Neptune. Triton, Neptune's largest moon, has an atmosphere, but it is so thin that its sky is still black, possibly with some pale haze at the horizon. Because Triton orbits with synchronous rotation, Neptune always appears in the same position in its sky. Triton's rotation axis is inclined 130° to Neptune's orbital plane and thus points within 40° of the Sun twice per Neptunian year, much like Uranus's. As Neptune orbits the Sun, Triton's polar regions take turns facing

neg: Symmetry in quantum mechanics where r′ are the rotated coordinates about an axis defined by a unit vector formula_27 through an angular increment Δ"θ", given by: where formula_29 is a rotation matrix dependent on the axis and angle. In group theoretic language, the rotation matrices are group elements, and the angles and axis formula_30 are the parameters, of the three-dimensional special orthogonal group, SO(3). The rotation matrices about the standard Cartesian basis vector formula_31 through angle , and the corresponding generators of rotations , are: More generally for rotations about an axis defined by formula_32, the rotation matrix elements are: where "δ" is the

neg: Flight dynamics (fixed-wing aircraft) angle "φ" represents a rotation around the aircraft longitudinal axis after rotating by yaw and pitch. To transform from the Earth frame to the wind frame, the three Euler angles are the bank angle "μ", the flight path angle "γ", and the heading angle "σ". When performing the rotations described above to obtain the wind frame from the Earth frame, ("μ","γ","σ") are analogous to ("φ","θ","ψ"), respectively. The heading angle "σ" is the angle between north and the horizontal component of the velocity vector, which describes which direction the aircraft is moving relative to cardinal directions. The flight path angle "γ"

neg: Euler's rotation theorem by a unit vector . Its product by the rotation angle is known as an axis-angle. The extension of the theorem to kinematics yields the concept of instant axis of rotation, a line of fixed points. In linear algebra terms, the theorem states that, in 3D space, any two Cartesian coordinate systems with a common origin are related by a rotation about some fixed axis. This also means that the product of two rotation matrices is again a rotation matrix and that for a non-identity rotation matrix one eigenvalue is 1 and the other two are both complex, or both

neg: Rotation formalisms in three dimensions a rotation vector, or Euler vector, an un-normalized three-dimensional vector the direction of which specifies the axis, and the length of which is , The rotation vector is in some contexts useful, as it represents a three-dimensional rotation with only three scalar values (its components), representing the three degrees of freedom. This is also true for representations based on sequences of three Euler angles (see below). If the rotation angle is zero, the axis is not uniquely defined. Combining two successive rotations, each represented by an Euler axis and angle, is not straightforward, and in fact does not satisfy the

neg: SS 433 the original primary object and was therefore longer lived. Estimates for its mass range from 3 to 30 solar masses. The primary and secondary orbit each other at a very close distance in stellar terms, with an orbital period of 13.1 days. The jets from the primary are emitted perpendicular to its accretion disk. The jets and disk precess around an axis inclined about 79° to a line between Earth and SS 433. The angle between the jets and the axis is around 20°, and the precessional period is around 162.5 days. Precession means that the jets sometimes point more

neg: Angular displacement by a unit vector ["x", "y", "z"] , and suppose we have an infinitely small rotation of angle Δθ about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix Δ"R" is represented as: A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ as θ/"N" where "N" is a large number, a rotation of θ about the axis may be represented as: It can be seen that Euler's theorem essentially states that all rotations may be

neg: Rotation which the rotation axis is arbitrary). A spectral analysis is not required to find the rotation axis. If formula_17 denotes the unit eigenvector aligned with the rotation axis, and if formula_5 denotes the rotation angle, then it can be shown that formula_19. Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector "if it has a nonzero magnitude." On the other hand, if this vector has a zero magnitude, it means that formula_20. In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the

neg: Charts on SO(3) we specify the point at the north pole to represent the identity rotation. In the case of the identity rotation, no axis of rotation is defined, and the angle of rotation (zero) is irrelevant. A rotation with its axis contained in the "xy"-plane and a very small rotation angle can be specified by a slice through the sphere parallel to the "xy"-plane and very near the north pole. The circle defined by this slice will be very small, corresponding to the small angle of the rotation. As the rotation angles become larger, the slice moves southward, and the circles become

neg: Rodrigues' rotation formula Rodrigues' rotation formula In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in , the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula provides an algorithm to compute the exponential map from , the Lie algebra of , to without actually computing the full matrix exponential. If is a vector in and is a unit

neg: Quaternions and spatial rotation the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle φ this axis. The associated quaternion is given by, Then the composition of the rotation R with R is the rotation R=RR with rotation axis and angle defined by the product of the quaternions that is Expand this product to obtain Divide both sides of this equation by the identity, which is the law of cosines on a sphere, and compute This is Rodrigues formula for the axis of a composite rotation defined in terms of the axes of the two

neg: 3D rotation group the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle "φ" this axis. The associated quaternion is given by, Then the composition of the rotation R with R is the rotation R=RR with rotation axis and angle defined by the product of the quaternions that is Expand this product to obtain Divide both sides of this equation by the identity, which is the law of cosines on a sphere, and compute This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two

neg: Improper rotation they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection. An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis. This is called an "n"-fold improper rotation if the angle of rotation is 360°/"n". There are several different systems for naming individual improper rotations: The direct subgroup, index 2, is C, ["n"], ("nn"), order "n", as the rotoreflection generator applied twice. "S" for odd "n" contain inversion, with is the group generated by inversion. "S" contain indirect isometries but not

neg: Photovoltaics described as a ladder mounted on a turnable disk. Each step of that ladder is the middle axis of a rectangular solar panel. In case the zenith distance of the Sun reaches zero, the "ladder" may be rotated to the north or the south to avoid a solar panel producing a shadow on a lower solar panel. Instead of an exactly vertical tower one can choose a tower with an axis directed to the polar star, meaning that it is parallel to the rotation axis of the Earth. In this case the angle between the axis and the Sun is

neg: Messier 75 densely concentrated globular clusters known. It shows a slow rotation around an axis that is inclined along a position angle of . The absolute magnitude of M75 is about −8.5 or some 180,000 times more luminous than the Sun. The cluster has a half-light radius of with a core radius of about and appears not to have undergone core collapse yet. The mass density at the core is ·pc. There are 38 RR Lyrae variable stars and the cluster appears to be Oosterhoff-intermediate in terms of metallicity. 62 candidate blue stragglers have been identified in the cluster field, with 60%

neg: Rotation formalisms in three dimensions arranged by rows) The elements of the rotation matrix are not all independent—as Euler's rotation theorem dictates, the rotation matrix has only three degrees of freedom. The rotation matrix has the following properties: The angle which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation. The eigenvector corresponding to the eigenvalue of 1 is the accompanying Euler axis, since the axis is the only (nonzero) vector which remains unchanged by left-multiplying (rotating) it with the rotation matrix. The above properties are equivalent to: which is another way of stating that form a 3D

neg: Commentariolus the Moon shares with the Earth in his system, Copernicus explains the Moon's motion as composed of five independent motions. Its motion around the Earth lies in a plane which is inclined at an angle of 5° to the plane of the Earth's orbit, and which precesses from east to west around an axis perpendicular to that plane, with a period of between 18 and 19 years with respect to the fixed stars. The remaining three motions, which take place within this orbital plane, are depicted in the diagram to the right. The first of these is that of the

neg: Quaternions and spatial rotation necessarily zero. The quaternion inverse of a rotation is the opposite rotation, since formula_7. The square of a quaternion rotation is a rotation by twice the angle around the same axis. More generally is a rotation by times the angle around the same axis as . This can be extended to arbitrary real , allowing for smooth interpolation between spatial orientations; see Slerp. Two rotation quaternions can be combined into one equivalent quaternion by the relation: in which corresponds to the rotation followed by the rotation . (Note that quaternion multiplication is not commutative.) Thus, an arbitrary number of rotations

neg: Equatorial coordinate system angle"), a left-handed system, measures the angular distance of an object westward along the celestial equator from the observer's meridian to the hour circle passing through the object. Unlike right ascension, hour angle is always increasing with the rotation of Earth. Hour angle may be considered a means of measuring the time since upper culmination, the moment when an object contacts the meridian overhead. A culminating star on the observer's meridian is said to have a zero hour angle (0). One sidereal hour (approximately 0.9973 solar hours) later, Earth's rotation will carry the star to the west of the meridian,

neg: Rodrigues' rotation formula exponential, and that the formulas match for infinitesimal . For an alternative derivation based on this exponential relationship, see exponential map from to. For the inverse mapping, see log map from to. Rodrigues' rotation formula In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in , the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula

neg: Euler's rotation theorem of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the axis corresponding to the third coordinate) and a reflection on a plane perpendicular to that axis. If we only restrict ourselves to matrices with determinant 1, we can thus see that they must be proper rotations. This result implies that any orthogonal matrix corresponding to a proper rotation is equivalent to a rotation over an angle around an axis . The trace (sum of diagonal elements) of the real rotation matrix given above is . Since a trace is invariant under

neg: Foucault pendulum vector diagrams The central axis of the pendulum aligns with the axis of rotation of the Earth. The central axis of the pendulum is always determined by the force of gravity directed towards the center of the Earth. A pendulum bob at rest at the North Pole still has spin on the bob. The Foucault pendulum connection is constructed such that the pendulum is free to swing in any direction. If the bob is displaced from the central axis of the pendulum in preparation for swinging and held in place, then the bob will revolve about the central axis of the pendulum

neg: Charts on SO(3) neighborhood of rotations which are nearly the same, and this neighborhood becomes flat as the neighborhood shrinks. Also, each rotation is actually represented by two antipodal points on the sphere, which are at opposite ends of a line through the center of the sphere. This reflects the fact that each rotation can be represented as a rotation about some axis, or, equivalently, as a negative rotation about an axis pointing in the opposite direction (a so-called double cover). The "latitude" of a circle representing a particular rotation angle will be half of the angle represented by that rotation, since as

neg: Quaternions and spatial rotation above. The expression formula_43 rotates any vector quaternion formula_44 around an axis given by the vector formula_45 by the angle formula_46, where formula_45 and formula_46 depends on the quaternion formula_17. formula_45 and formula_46 can be found from the following equations: where formula_54 is the two-argument arctangent. Care should be taken when the quaternion approaches a scalar, since due to degeneracy the axis of an identity rotation is not well-defined. A benefit of the quaternion formulation of the composition of two rotations R and R is that it yields directly the rotation axis and angle of the composite rotation R=RR. Let

neg: Rigid body dynamics former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. Based on this fact he introduced a vectorial way to describe any rotation, with a vector on the rotation axis and module equal to the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads to it from the reference frame. When used to represent an orientation, the rotation vector is commonly called orientation vector, or attitude vector. A similar method, called axis-angle representation, describes a rotation or orientation

neg: Aircraft flight mechanics load factor greater than one, determined by the aircraft's angle of bank. There are three primary ways for an aircraft to change its orientation relative to the passing air. "Pitch" (movement of the nose up or down, rotation around the transversal axis), "roll" (rotation around the longitudinal axis, that is, the axis which runs along the length of the aircraft) and "yaw" (movement of the nose to left or right, rotation about the vertical axis). Turning the aircraft (change of heading) requires the aircraft firstly to roll to achieve an angle of bank (in order to balance the centrifugal force);

neg: Woodturning lathe bed, as if a log were mounted in the lathe. Grain is thus always perpendicular to the direction of rotation under the tool. In bowl turning, the grain runs at right angles to the axis, as if a plank were mounted across the chuck. When a bowl blank rotates, the angle that the grain makes with the cutting tool continually changes between the easy cuts to two places per rotation where the tool is cutting across the grain and even upwards across it. This varying grain angle limits some of the tools that may be used and requires additional

neg: Precession past ice ages. Orbital nodes also precess over time. Precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called "nutation". In physics, there are two types of precession:

neg: Precession Precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called "nutation". In physics, there are two types of precession: torque-free and torque-induced. In astronomy, "precession" refers to any

neg: Magic angle ("i.e.", the edge central angle also known as the tetrahedral angle). In nuclear magnetic resonance (NMR) spectroscopy, three prominent nuclear magnetic interactions, dipolar coupling, chemical shift anisotropy (CSA), and first-order quadrupolar coupling, depend on the orientation of the interaction tensor with the external magnetic field. By spinning the sample around a given axis, their average angular dependence becomes: where "θ" is the angle between the principal axis of the interaction and the magnetic field, "θ" is the angle of the axis of rotation relative to the magnetic field and "β" is the (arbitrary) angle between the axis of rotation and

neg: Effect of Sun angle on climate and this results in change in energy level of receiving bodies in Earth's domain. Different materials have different properties for transmitting back received energy in the form of "heat energy" at different rates. Effect of Sun angle on climate The amount of heat energy received at any location on the globe is a direct effect of Sun angle on climate, as the angle at which sunlight strikes the Earth varies by location, time of day, and season due to the Earth's orbit around the Sun and the Earth's rotation around its tilted axis. Seasonal change in the angle of sunlight,

neg: South Atlantic Anomaly field at less than 32,000 nanotesla at sea level, which corresponds to the dipolar magnetic field at ionospheric altitudes. However, the field itself varies in intensity as a gradient. The Van Allen radiation belts are symmetrical about the Earth's magnetic axis, which is tilted with respect to the Earth's rotational axis by an angle of approximately 11°. The intersection between the magnetic and rotation axes of the Earth is located not at the Earth's center, but some away. Because of this asymmetry, the inner Van Allen belt is closest to the Earth's surface over the south Atlantic Ocean where it

neg: Orbit of the Moon orbit, but is inclined to it by a constant value of 6.688° (this is the obliquity). As was discovered by Jacques Cassini in 1722, the rotational axis of the Moon precesses with the same rate as its orbital plane, but is 180° out of phase "(see Cassini's Laws)". Therefore, the angle between the ecliptic and the lunar equator is always 1.543°, even though the rotational axis of the Moon is not fixed with respect to the stars. The nodes are points at which the Moon's orbit crosses the ecliptic. The Moon crosses the same node every 27.2122 days, an interval

neg: Foucault pendulum vector diagrams Earth in relation to the pendulum swing increases in time (decreases in effect) with decreasing latitude is related to the Coriolis Effect. As summarized in the Coriolis effect article, the effect is greatest in polar regions where the surface of the Earth is at right angles to the axis of rotation (the central axis of the pendulum aligns with the Earth's axis of rotation). The Coriolis effect decreases nearer the equator because the surface of the Earth is parallel to the axis of rotation (the central axis of the pendulum is perpendicular with the Earth's axis of rotation). Refer to

neg: Commentariolus at an angle of about 23° to the perpendicular to the plane of its orbit. The third motion is a precession of the Earth's axis of rotation about an axis perpendicular to the plane of its orbit. Copernicus specified the rate of this precession with respect to the radial line from the Earth to the centre of its orbit as being slightly less than a year, with an implied direction as being from west to east. "With respect to the fixed stars", this precession is very slow, and in the opposite direction—from east to west—and explains the phenomenon of the

neg: Commentariolus 2 and 3 months. The Moon's sphere, however, revolves around the Earth in a period of one month, and moves with it around the Sun like an epicycle. This section explains how the apparent motion of the Sun could arise from three separate motions of the Earth. The first motion is a uniform revolution, with a period of one year, from west to east along a circular orbit whose centre is offset from the Sun by 1/25 of the orbit's radius. The second motion is the daily rotation about an axis which passes through the Earth's centre and is inclined

neg: Epsilon Ursae Majoris it is listed as one of the 57 navigational stars. According to "Hipparcos", Alioth is from the Sun. Its spectral type is A1p; the "p" stands for "peculiar", as its spectrum is characteristic of an Alpha2 Canum Venaticorum variable. Alioth, as a representative of this type, may harbor two interacting processes. First, the star's strong magnetic field separating different elements in the star's hydrogen 'fuel'. In addition, a rotation axis at an angle to the magnetic axis may be spinning different bands of magnetically sorted elements into the line of sight between Alioth and the Earth. The intervening elements react

neg: Plane of rotation the plane of rotation is uniquely defined. Together with the angle of rotation it fully describes the rotation. Or in a continuously rotating object the rotational properties such as the rate of rotation can be described in terms of the plane of rotation. It is perpendicular to, and so is defined by and defines, an axis of rotation, so any description of a rotation in terms of a plane of rotation can be described in terms of an axis of rotation, and vice versa. But unlike the axis of rotation the plane generalises into other, in particular higher, dimensions. A

neg: Chasles' theorem (kinematics) line "AK" be selected parallel to the axis of the given rotation, with "K" the foot of a perpendicular from "B". The appropriate screw displacement is about an axis parallel to "AK" such that "K" is moved to "B". The method corresponds to Euclidean plane isometry where a composition of rotation and translation can be replaced by rotation about an appropriate center. In Whittaker's terms, "A rotation about any axis is equivalent to a rotation through the same angle about any axis parallel to it, together with a simple translation in a direction perpendicular to the axis." Chasles' theorem (kinematics)

neg: Beta Cassiopeiae the Hertzsprung–Russell diagram as that of classical Cepheids. Delta Scuti stars are located at the intersection of the strip with the main sequence. This star is rotating at about 92% of its critical speed, completing 1.12 rotations every day. This is giving the star an oblate spheroid shape with an equatorial bulge that is 24% larger than the polar radius. This shape is causing the polar region to have a higher temperature than the equator: the temperature difference is about 1,000 K. The axis of rotation is inclined about 20° to the line of sight from the Earth. β Cassiopeiae

neg: 3D rotation group rotating around "y" and then "x". The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem. Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R which is called the "axis of rotation" (this is Euler's rotation theorem). Each such rotation acts as an ordinary 2-dimensional rotation in the plane orthogonal to this axis. Since every 2-dimensional rotation can be represented by an angle "φ", an arbitrary 3-dimensional rotation can be specified by an axis

neg: 99 Herculis period of 56.3 years and an eccentricity of 0.766. The semi-major axis of their orbit spans an angle of 1.06 arcseconds, which corresponds to a physical dimension of 16.5 AU. The plane of their orbit is inclined by an angle of about 39° to the line-of-sight from the Earth. Reports of a third component of this system now appear doubtful. The primary component, 99 Herculis A, is an F-type main sequence star with a stellar classification of F7 V. It has 94% of the Sun's mass and nearly double the luminosity, but has an estimated 10% greater radius. The effective

neg: Conjugation of isometries in Euclidean space conjugate by a translation of a rotation about an axis is the corresponding rotation about the translated axis. Such a conjugation produces he screw displacement known to express an arbitrary Euclidean motion according to Chasles' theorem. The conjugacy class within the Euclidean group "E"(3) of a rotation about an axis is a rotation by the same angle about any axis. The conjugate closure of a singleton containing a rotation in 3D is "E"(3). In 2D it is different in the case of a "k"-fold rotation: the conjugate closure contains "k" rotations (including the identity) combined with all translations. "E"(2) has

neg: Mercury (planet) on Earth. Combined with a 3:2 spin–orbit resonance of the planet's rotation around its axis, it also results in complex variations of the surface temperature. The resonance makes a single solar day on Mercury last exactly two Mercury years, or about 176 Earth days. Mercury's orbit is inclined by 7 degrees to the plane of Earth's orbit (the ecliptic), as shown in the diagram on the right. As a result, transits of Mercury across the face of the Sun can only occur when the planet is crossing the plane of the ecliptic at the time it lies between Earth and

neg: Quaternions and spatial rotation coordinate system), they are called orientation quaternions or attitude quaternions. In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the "Euler axis") that runs through the fixed point. The Euler axis is typically represented by a unit vector . Therefore, any rotation in three dimensions can be represented as a combination of a vector and a scalar . Quaternions give a simple way to encode this axis–angle representation in

neg: Axial precession Earth's axis and the shift in the equinoxes. These images show the position of the Earth's axis on the "celestial sphere", a fictitious sphere which places the stars according to their position as seen from Earth, regardless of their actual distance. The first image shows the celestial sphere from the outside, with the constellations in mirror image. The second image shows the perspective of a near-Earth position as seen through a very wide angle lens (from which the apparent distortion arises). The rotation axis of the Earth describes, over a period of 25,700 years, a small circle (blue) among the