In geometry the orientation, angular position, or attitude 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 is in. Namely, it is the rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement and it may be necessary to add an imaginary translation, called the objects location. The location and orientation together fully describe how the object is placed in space, eulers rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. This gives one way of representing the orientation using an axis–angle representation. Other widely used methods include rotation quaternions, Euler angles, or rotation matrices, more specialist uses include Miller indices in crystallography, strike and dip in geology and grade on maps and signs. Typically, the orientation is given relative to a frame of reference, at least three independent values are needed to describe the orientation of this local frame. Three other values are needed to describe its location, thus, a rigid body free to move in space is said to have six degrees of freedom. All the points of the body change their position during a rotation except for those lying on the rotation axis, 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. For example, the orientation in space of a line, line segment, another example is the position of a point on the earth, often described using the orientation of a line joining it with the earths 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 normal to that plane, or by using the strike. Further details about the methods to 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 is given by a 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, several methods to describe orientations of a rigid body in three dimensions have been developed. They are summarized in the following sections, the first attempt to represent an orientation was owed to Leonhard Euler. The values of three rotations are called Euler angles. These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles, in aerospace engineering they are usually referred to as Euler angles. Euler also realized that the composition of two rotations is equivalent to a rotation about a different fixed axis
Strike line and dip of a plane describing attitude relative to a horizontal plane and a vertical plane perpendicular to the strike line
A rotation represented by an Euler axis and angle.