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. If the rigid body has rotational symmetry not all orientations are distinguishable, except by observing how the orientation evolves in time from a known starting orientation. Three other values describe the position of a point on the object.Īll the points of the body change their position during a rotation except for those lying on the rotation axis. At least three independent values are needed to describe the orientation of this local frame. In general the position and orientation in space of a rigid body are defined as the position and orientation, relative to the main reference frame, of another reference frame, which is fixed relative to the body, and hence translates and rotates with it (the body's local reference frame, or local coordinate system). Mathematical representations Three dimensions Two directions are obtuse if they form an obtuse angle (greater than a right angle) or, equivalently, if their scalar product or scalar projection is negative. Two directions are said to be opposite if they are the additive inverse of one another, as in an arbitrary unit vector and its multiplication by -1. Two objects sharing the same direction are said to be codirectional (as in parallel lines). Typically, the orientation is given relative to a frame of reference, usually specified by a Cartesian coordinate system. Unit vector may also be used to represent an object's normal vector orientation or the relative direction between two points. More specialist uses include Miller indices in crystallography, strike and dip in geology and grade on maps and signs. Other widely used methods include rotation quaternions, rotors, Euler angles, or rotation matrices. This gives one common way of representing the orientation using an axis–angle representation. ![]() The above-mentioned imaginary rotation and translation may be thought to 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.Įuler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. The position and orientation together fully describe how the object is placed in space. A rotation may not be enough to reach the current placement, in which case it may be necessary to add an imaginary translation to change the object's position (or linear position). More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. ![]() In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. Notion of pointing in a direction Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it.
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