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.^{[1]}
More specifically, it refers to tát the imaginary rotation that is needed to tát move the object from a reference placement to tát its current placement. A rotation may not be enough to tát reach the current placement, in which case it may be necessary to tát add an imaginary translation to tát 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 to tát 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 to tát represent an object's normal vector orientation or the relative direction between two points.

Typically, the orientation is given relative to tát a frame of reference, usually specified by a Cartesian coordinate system.
Two objects sharing the same direction are said to tát be * codirectional* (as in parallel lines).
Two directions are said to tát be

*if they are the additive inverse of one another, as in an arbitrary unit vector and its multiplication by -1. Two directions are*

**opposite***if they khuông an obtuse angle (greater than vãn a right angle) or, equivalently, if their scalar product or scalar projection is negative.*

**obtuse**

## Mathematical representations[edit]

### Three dimensions[edit]

In general the position and orientation in space of a rigid body toàn thân are defined as the position and orientation, relative to tát the main reference frame, of another reference frame, which is fixed relative to tát 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 to tát 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 to tát that plane, or by using the strike and dip angles.

Further details about the mathematical methods to tát represent the orientation of rigid bodies and planes in three dimensions are given in the following sections.

### Two dimensions[edit]

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.

### Multiple dimensions[edit]

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[edit]

Several methods to tát describe orientations of a rigid body toàn thân in three dimensions have been developed. They are summarized in the following sections.

### Euler angles[edit]

The first attempt to tát represent an orientation is attributed to tát 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 to tát fix the vertical axis and another to tát fix the other two axes). The values of these three rotations are called Euler angles.

#### Tait–Bryan angles[edit]

These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles. Mathematically they constitute a phối 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 to tát as Euler angles.

Xem thêm: tính chất của cạnh tranh là gì

### Orientation vector [edit]

Euler also realized that the composition of two rotations is equivalent to tát a single rotation about a different fixed axis (Euler's rotation theorem). Therefore, the composition of the former three angles has to tát be equal to tát only one rotation, whose axis was complicated to tát calculate until matrices were developed.

Based on this fact he introduced a vectorial way to tát describe any rotation, with a vector on the rotation axis and module equal to tát the value of the angle. Therefore, any orientation can be represented by a rotation vector (also called Euler vector) that leads to tát it from the reference frame. When used to tát 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 tát indicate the angle (see figure).

### Orientation matrix[edit]

With the introduction of matrices, the Euler theorems were rewritten. The rotations were described by orthogonal matrices referred to tát as rotation matrices or direction cosine matrices. When used to tát 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 to tát achieve the frame that we want to tát describe.

The configuration space of a non-symmetrical object in *n*-dimensional space is SO(*n*) × **R**^{n}. Orientation may be visualized by attaching a basis of tangent vectors to tát an object. The direction in which each vector points determines its orientation.

### Orientation quaternion[edit]

Another way to tát describe rotations is using rotation quaternions, also called versors. They are equivalent to tát rotation matrices and rotation vectors. With respect to tát rotation vectors, they can be more easily converted to tát and from matrices. When used to tát represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions.

## Plane in three dimensions[edit]

### Miller indices[edit]

The attitude of a lattice plane is the orientation of the line normal to tát the plane,^{[2]} 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*),^{[3]}^{[4]} sánh the family of planes has an attitude common to tát all its constituent planes.

### Strike and dip[edit]

Many features observed in geology are planes or lines, and their orientation is commonly referred to tát 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.^{[5]}

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 to tát 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 to tát the strike line.

## Usage examples[edit]

### Rigid body[edit]

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 to tát a fixed reference frame. The attitude is described by *attitude coordinates*, and consists of at least three coordinates.^{[6]} 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.^{[7]}^{[8]} Another is based upon roll, pitch and yaw,^{[9]} although these terms also refer to tát incremental deviations from the nominal attitude

Xem thêm: 9/9 là ngày gì

## See also[edit]

- Angular displacement
- Attitude control
- Body relative direction
- Directional statistics
- Oriented area
- Plane of rotation
- Rotation formalisms in three dimensions
- Signed direction
- Triad method

## References[edit]

## External links[edit]

- Media related to tát Orientation (mathematics) at Wikimedia Commons

## Bình luận