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This article is about the orientation or attitude of an object or a shape in a space. For the orientation of a space, see Orientability.
In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body toàn thân – is part of the mô tả tìm kiếm of how it is placed in the space it occupies. More specifically, it refers vĩ đại the imaginary rotation that is needed vĩ đại move the object from a reference placement vĩ đại its current placement. A rotation may not be enough vĩ đại reach the current placement, in which case it may be necessary vĩ đại add an imaginary translation vĩ đại change the object's position (or linear position). The position and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought vĩ đại occur in any order, as the orientation of an object does not change when it translates, and its position does not change when it rotates.
Euler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. This gives one common way of representing the orientation using an axis–angle representation. Other widely used methods include rotation quaternions, rotors, Euler angles, or rotation matrices. More specialist uses include Miller indices in crystallography, strike and dip in geology and grade on maps and signs. Unit vector may also be used vĩ đại represent an object's normal vector orientation or the relative direction between two points.
Typically, the orientation is given relative vĩ đại a frame of reference, usually specified by a Cartesian coordinate system. Two objects sharing the same direction are said vĩ đại be codirectional (as in parallel lines). Two directions are said vĩ đại be opposite if they are the additive inverse of one another, as in an arbitrary unit vector and its multiplication by -1. Two directions are obtuse if they size an obtuse angle (greater than thở a right angle) or, equivalently, if their scalar product or scalar projection is negative.
In general the position and orientation in space of a rigid body toàn thân are defined as the position and orientation, relative vĩ đại the main reference frame, of another reference frame, which is fixed relative vĩ đại the body toàn thân, and hence translates and rotates with it (the body's local reference frame, or local coordinate system). At least three independent values are needed vĩ đại describe the orientation of this local frame. Three other values describe the position of a point on the object. All the points of the body toàn thân change their position during a rotation except for those lying on the rotation axis. If the rigid body toàn thân has rotational symmetry not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. For example, the orientation in space of a line, line segment, or vector can be specified with only two values, for example two direction cosines. Another example is the position of a point on the earth, often described using the orientation of a line joining it with the earth's center, measured using the two angles of longitude and latitude. Likewise, the orientation of a plane can be described with two values as well, for instance by specifying the orientation of a line normal vĩ đại that plane, or by using the strike and dip angles.
Further details about the mathematical methods vĩ đại represent the orientation of rigid bodies and planes in three dimensions are given in the following sections.
In two dimensions the orientation of any object (line, vector, or plane figure) is given by a single value: the angle through which it has rotated. There is only one degree of freedom and only one fixed point about which the rotation takes place.
When there are d dimensions, specification of an orientation of an object that does not have any rotational symmetry requires d(d − 1) / 2 independent values.
Rigid body toàn thân in three dimensions
Several methods vĩ đại describe orientations of a rigid body toàn thân in three dimensions have been developed. They are summarized in the following sections.
The first attempt vĩ đại represent an orientation is attributed vĩ đại Leonhard Euler. He imagined three reference frames that could rotate one around the other, and realized that by starting with a fixed reference frame and performing three rotations, he could get any other reference frame in the space (using two rotations vĩ đại fix the vertical axis and another vĩ đại fix the other two axes). The values of these three rotations are called Euler angles.
These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles. Mathematically they constitute a mix of six possibilities inside the twelve possible sets of Euler angles, the ordering being the one best used for describing the orientation of a vehicle such as an airplane. In aerospace engineering they are usually referred vĩ đại as Euler angles.
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Orientation vector 
Euler also realized that the composition of two rotations is equivalent vĩ đại a single rotation about a different fixed axis (Euler's rotation theorem). Therefore, the composition of the former three angles has vĩ đại be equal vĩ đại only one rotation, whose axis was complicated vĩ đại calculate until matrices were developed.
Based on this fact he introduced a vectorial way vĩ đại describe any rotation, with a vector on the rotation axis and module equal vĩ đại the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads vĩ đại it from the reference frame. When used vĩ đại 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 vĩ đại indicate the angle (see figure).
With the introduction of matrices, the Euler theorems were rewritten. The rotations were described by orthogonal matrices referred vĩ đại as rotation matrices or direction cosine matrices. When used vĩ đại represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix.
The above-mentioned Euler vector is the eigenvector of a rotation matrix (a rotation matrix has a unique real eigenvalue). The product of two rotation matrices is the composition of rotations. Therefore, as before, the orientation can be given as the rotation from the initial frame vĩ đại achieve the frame that we want vĩ đại describe.
The configuration space of a non-symmetrical object in n-dimensional space is SO(n) × Rn. Orientation may be visualized by attaching a basis of tangent vectors vĩ đại an object. The direction in which each vector points determines its orientation.
Another way vĩ đại describe rotations is using rotation quaternions, also called versors. They are equivalent vĩ đại rotation matrices and rotation vectors. With respect vĩ đại rotation vectors, they can be more easily converted vĩ đại and from matrices. When used vĩ đại represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions.
Plane in three dimensions
The attitude of a lattice plane is the orientation of the line normal vĩ đại the plane, and is described by the plane's Miller indices. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices (hkl), so sánh the family of planes has an attitude common vĩ đại all its constituent planes.
Strike and dip
Many features observed in geology are planes or lines, and their orientation is commonly referred vĩ đại as their attitude. These attitudes are specified with two angles.
For a line, these angles are called the trend and the plunge. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane.
For a plane, the two angles are called its strike (angle) and its dip (angle). A strike line is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike angle is the bearing of this line (that is, relative vĩ đại geographic north or from magnetic north). The dip is the angle between a horizontal plane and the observed planar feature as observed in a third vertical plane perpendicular vĩ đại the strike line.
The attitude of a rigid body toàn thân is its orientation as described, for example, by the orientation of a frame fixed in the body toàn thân relative vĩ đại a fixed reference frame. The attitude is described by attitude coordinates, and consists of at least three coordinates. One scheme for orienting a rigid body toàn thân is based upon body-axes rotation; successive rotations three times about the axes of the body's fixed reference frame, thereby establishing the body's Euler angles. Another is based upon roll, pitch and yaw, although these terms also refer vĩ đại incremental deviations from the nominal attitude
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- Angular displacement
- Attitude control
- Body relative direction
- Directional statistics
- Oriented area
- Plane of rotation
- Rotation formalisms in three dimensions
- Signed direction
- Triad method
- Media related vĩ đại Orientation (mathematics) at Wikimedia Commons