As a fundamental description of motion, rotations are important aspects of kinematics and engineering. All rotations can be described either as vector rotations, in which a series of vectors are rotated, or frame rotations, in which the entire frame is rotated. Each type of rotation can be expressed through three main notations.
The first is the directional cosine matrix or DCM. Through matrix multiplication of the DCM and a vector, the resultant vector is produced. By applying the product of three DCMs, the Aerospace Rotation sequence can be expressed in terms of the Euler Angles. These three angles can describe the orientation of the body in almost every position. A series of rotations can also be expressed as a single rotation with a known angle and axis of rotation. With these two parameters, the rotation can be expressed through quaternions. Through hyper complex operations, quaternions offer another method of calculating both frame and vector rotations. Each of these representations are investigated and related through computational means.
Although each representation has their own advantages and disadvantages, quaternions are very significant for applications with a known axis of rotation. An example is proving the intersection of any plane and a double cone produces the conic sections. To eliminate a variable and express the intersection in two dimensions, a rotation must be applied. By calculating the axis and angle of rotation, the required rotation is found by method of quaternions.